EXTENSIONS OF p-LOCAL FINITE GROUPS
C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, AND B. OLIVER
Abstract. A p-local finite group consists of a finite p-group S, together*
* with a pair
of categories which encode "conjugacy" relations among subgroups of S, an*
*d which
are modelled on the fusion in a Sylow p-subgroup of a finite group. It co*
*ntains enough
information to define a classifying space which has many of the same prop*
*erties as
p-completed classifying spaces of finite groups. In this paper, we study *
*and classify
extensions of p-local finite groups, and also compute the fundamental gro*
*up of the
classifying space of a p-local finite group.
A p-local finite group consists of a finite p-group S, together with a pair o*
*f categories
(F, L), of which F is modeled on the conjugacy (or fusion) in a Sylow subgroup *
*of a
finite group. The category L is essentially an extension of F and contains just*
* enough
extra information so that its p-completed nerve has many of the same properties*
* as
p-completed classifying spaces of finite groups. We recall the precise definiti*
*ons of these
objects in Section 1, and refer to [BLO2 ] and [5A1 ] for motivation for their*
* study.
In this paper, we study extensions of saturated fusion systems and of p-local*
* finite
groups. This is in continuation of our more general program of trying to unders*
*tand to
what extent properties of finite groups can be extended to properties of p-loca*
*l finite
groups, and to shed light on the question of how many (exotic) p-local finite g*
*roups
there are. While we do not get a completely general theory of extensions of one*
* p-local
finite group by another, we do show how certain types of extensions can be desc*
*ribed
in manner very similar to the situation for finite groups.
From the point of view of group theory, developing an extension theory for p-*
*local
finite groups is related to the question of to what extent the extension proble*
*m for
groups is a local problem, i.e., a problem purely described in terms of a Sylow*
* p-
subgroup and conjugacy relations inside it. In complete generality this is not*
* the
case. For example, strongly closed subgroups of a Sylow p-subgroup S of G need *
*not
correspond to normal subgroups of G. However, special cases where this does hap*
*pen
include the case of existence of p-group quotients (the focal subgroup theorems*
*, see
[Go , xx7.3-7.4]) and central subgroups (described via the Z*-theorem of Glaube*
*rman
[Gl]).
From the point of view of homotopy theory, one of the problems which comes up
when looking for a general theory of extensions of p-local finite groups is tha*
*t while
an extension of groups 1 ! K ! ! G ! 1 always induces a (homotopy) fibration
sequence of classifying spaces, it does not in general induce a fibration seque*
*nce of
p-completed classifying spaces. Two cases where this does happen are those whe*
*re
___________
1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R40, 20D2*
*0.
Key words and phrases. Classifying space, p-completion, finite groups, fusion.
C. Broto is partially supported by MCYT grant BFM2001-2035.
N. Castellana is partially supported by MCYT grant BFM2001-2035.
J. Grodal is partially supported by NSF grants DMS-0104318 and DMS-0354633.
R. Levi is partially supported by EPSRC grant GR/M7831.
B. Oliver is partially supported by UMR 7539 of the CNRS.
1
2 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
G is a p-group, and where the extension is central. In both of these cases, BH*
*^pis
the homotopy fiber of the map B ^p---! BG^p. Thus, also from the point of view
of homotopy theory, it is natural to study extensions of p-local finite groups *
*with p-
group quotient, and to study central extensions of p-local finite groups. The *
*third
case we study is that of extensions with quotient of order prime to p; and the *
*case
of p and p0-group quotients to some extent unify to give a theory of extensions*
* with
p-solvable quotient. Recall in this connection that by a previous result of our*
*s [5A1 ,
Proposition C], solvable p-local finite groups all come from p-solvable groups.*
* In all
three of these situations, we develop a theory of extensions which parallels th*
*e situation
for finite groups.
We now describe the contents of this paper in more detail, stating simplified*
* versions
of our main results on extensions. Stronger and more precise versions of some o*
*f these
theorems will be stated and proven later.
In Section 3, we construct a very general theory of fusion subsystems (Propos*
*ition
3.8) and linking subsystems (Theorem 3.9) with quotient a p-group or a group of*
* order
prime to p. As a result we get the following theorem (Corollary 3.10), which f*
*or a
p-local finite group (S, F, L), describes a correspondence between covering spa*
*ces of
the geometric realization |L| and certain p-local finite subgroups of (S, F, L).
Theorem A. Suppose that (S, F, L) is a p-local finite group. Then there is a no*
*rmal
subgroup H C ss1(|L|) which is minimal among all those whose quotient is finite*
* and p-
solvable. Any covering space of the geometric realization |L| whose fundamental*
* group
contains H is homotopy equivalent to |L0| for some p-local finite group (S0, F0*
*, L0),
where S0 S and F0 F.
Moreover, the p-local finite group (S0, F0, L0) of Theorem A can be explicitl*
*y de-
scribed in terms of L, as we will explain in Section 3.
In order to use this theorem, it is useful to have ways of finding the finite*
* p-solvable
quotients of ss1(|L|). This can be done by iteration, using the next two theore*
*ms. In
them, the maximal p-group quotient of ss1(|L|), and the maximal quotient of ord*
*er
prime to p, are described solely in terms of the fusion system F.
0
When G is an infinite group, we define Op(G) and Op (G) to be the intersection
of all normal subgroups of G of p-power index, or index prime to p, respectivel*
*y.
These clearly generalize the usual definitions for finite G (but are not the on*
*ly possible
generalizations).
Theorem B (Hyperfocal subgroup theorem for p-local finite groups). For a p-local
finite group (S, F, L), the natural homomorphism
S -----! ss1(|L|)=Op(ss1(|L|)) ~=ss1(|L|^p)
is surjective, with kernel equal to
-1 fi p ff
OpF(S) def=g ff(g) 2 S fig 2 P S, ff 2 O (Aut F(P )) .
0
For any saturated fusion system F over a p-group S, we let Op*(F) F be the
smallest fusion subsystem of F (in the sense of Definition 1.1) which contains *
*all
0 p0
automorphism groups Op (Aut F(P )) for P S. Equivalently, O* (F) is the small*
*est
subcategory of F with the same objects, and which contains all restrictions of *
*all
automorphisms in F of p-power order. This subcategory is needed in the statemen*
*t of
the next theorem.
EXTENSIONS OF P-LOCAL FINITE GROUPS 3
Theorem C. For a p-local finite group (S, F, L), the natural map
0
Out F(S) -----! ss1(|L|)=Op (ss1(|L|))
is surjective, with kernel equal to
fi *
* ff
Out 0F(S) def=ff 2 OutF (S) fiff|P 2 Mor Op0*(F)(P, S), some F-centric P *
*S .
Theorem B is proved as Theorem 2.5, and Theorem C is proved as Theorem 5.5.
In fact, we give a purely algebraic description of these subsystems of "p-pow*
*er index"
or of "index prime to p" (Definition 3.1), and then show in Sections 4.1 and 5.*
*1 that
they in fact all arise as finite covering spaces of |L| (see Theorems 4.4 and 5*
*.5). Similar
results were also discovered independently by Puig [Pu4 ]. Then, in Sections 4*
*.2 and
5.2, we establish converses to these concerning the extensions of a p-local fin*
*ite group,
which include the following theorem.
Theorem D. Let (S, F, L) be a p-local finite group. Suppose we are given a fibr*
*ation
sequence |L|^p___! E ___! BG, where G is a finite p-group or has order prime *
*to p.
Then there exists a p-local finite group (S0, F0, L0) such that |L0|^p' E^p.
This is shown as Theorems 4.7 and 5.8. Moreover, when G is a p-group, we give*
* in
Theorem 4.7 an explicit algebraic construction of the p-local finite group (S0,*
* F0, L0).
Finally, in Section 6, we develop the theory of central extensions of p-local*
* finite
groups. Our main results there (Theorems 6.8 and 6.13) give a more elaborate ve*
*rsion
of the following theorem. Here, the center of a p-local finite group (S, F, L) *
*is defined
to be the subgroup of elements x 2 Z(S) such that ff(x) = x for all ff 2 Mor (F*
*c).
Theorem E. Suppose that A is a central subgroup of a p-local finite group (S, F*
*, L).
Then there exists a canonical quotient p-local finite group (S=A, F=A, L=A), an*
*d the
canonical projection of |L| onto |L=A| is a principal fibration with fiber BA.
Conversely, for any principal fibration E ___! |L| with fiber BA, where A is*
* a finite
abelian p-group, there exists a p-local finite group (Se, eF, eL) such that |Le*
*| ' E. Further-
more, this correspondence sets up a 1 - 1 correspondence between central extens*
*ions of
L by A and elements in H2(|L|; A).
One motivation for this study was the question of whether extensions of p-loc*
*al finite
groups coming from finite groups can produce exotic p-local finite groups. In t*
*he case
of central extensions, we are able to show that (S, F, L) comes from a finite g*
*roup
if and only if (S=A, F=A, L=A) comes from a group (Corollary 6.14). For the ot*
*her
types of extensions studied in this paper, this is still an open question. We *
*have so
far failed to produce exotic examples in this way, and yet we have also been un*
*able to
show that exotic examples cannot occur. This question seems to be related to s*
*ome
rather subtle and interesting group theoretic issues relating local to global s*
*tructure;
see Corollary 4.8 for one partial result in this direction.
This paper builds on the earlier paper [5A1 ] by the same authors, and many o*
*f the
results in that paper were originally motivated by this work on extensions.
The authors would like to thank the University of Aberdeen, Universitat Aut`o*
*noma
de Barcelona, Universit'e Paris 13, and Aarhus Universitet for their hospitalit*
*y. In
particular, the origin of this project goes back to a three week period in the *
*spring of
2001, when four of the authors met in Aberdeen.
4 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
1. A quick review of p-local finite groups
We first recall the definitions of a fusion system, and a saturated fusion sy*
*stem, in
the form given in [BLO2 ]. For any group G, and any pair of subgroups H, K G*
*, we
set
NG(H, K) = {x 2 G | xHx-1 K},
let cx denote conjugation by x (cx(g) = xgx-1), and set
fi
Hom G(H, K) = cx 2 Hom (H, K) fix 2 NG(H, K) ~=NG(H, K)=CG(H).
By analogy, we also write
fi
AutG(H) = Hom G(H, H) = cx 2 Aut(H) fix 2 NG(H) ~=NG(H)=CG(H).
Definition 1.1 ([Pu2 ] and [BLO2 , Definition 1.1]). A fusion system over a fi*
*nite p-
group S is a category F, where Ob (F) is the set of all subgroups of S, and whi*
*ch
satisfies the following two properties for all P, Q S:
o Hom S(P, Q) Hom F(P, Q) Inj(P, Q); and
o each ' 2 Hom F(P, Q) is the composite of an isomorphism in F followed by an
inclusion.
The following additional definitions and conditions are needed in order for t*
*hese
systems to be very useful. If F is a fusion system over a finite p-subgroup S, *
*then two
subgroups P, Q S are said to be F-conjugate if they are isomorphic as objects*
* of the
category F.
Definition 1.2 ([Pu2 ], see [BLO2 , Def. 1.2]). Let F be a fusion system over *
*a p-group
S.
o A subgroup P S is fully centralized in F if |CS(P )| |CS(P 0)| for all P *
*0 S
which is F-conjugate to P .
o A subgroup P S is fully normalized in F if |NS(P )| |NS(P 0)| for all P 0*
* S
which is F-conjugate to P .
o F is a saturated fusion system if the following two conditions hold:
(I)For all P S which is fully normalized in F, P is fully centralized in F *
*and
Aut S(P ) 2 Sylp(Aut F(P )).
(II)If P S and ' 2 Hom F(P, S) are such that 'P is fully centralized, and i*
*f we
set
N' = {g 2 NS(P ) | 'cg'-1 2 AutS('P )},
_ _
then there is ' 2 Hom F(N', S) such that '|P = '.
If G is a finite group and S 2 Sylp(G), then by [BLO2 , Proposition 1.3], th*
*e category
FS(G) defined by letting Ob (FS(G)) be the set of all subgroups of S and setting
Mor FS(G)(P, Q) = Hom G(P, Q) is a saturated fusion system.
An alternative pair of axioms for a fusion system being saturated have been g*
*iven
by Radu Stancu [St]. He showed that axioms (I) and (II) above are equivalent to*
* the
two axioms:
(I0)Inn(S) 2 Sylp(Aut F(S)).
EXTENSIONS OF P-LOCAL FINITE GROUPS 5
(II0)If P S and ' 2 Hom F(P, S) are such that 'P is fully normalized, and if *
*we set
N' = {g 2 NS(P ) | 'cg'-1 2 AutS('P )},
_ _
then there is ' 2 Hom F(N', S) such that '|P = '.
The following consequence of conditions (I) and (II) above will be needed sev*
*eral
times throughout the paper.
Lemma 1.3. Let F be a saturated fusion system over a p-group S. Let P, P 0 S be
a pair of F-conjugate subgroups such that P 0is fully normalized in F. Then the*
*re is a
homomorphism ff 2 Hom F(NS(P ), NS(P 0)) such that ff(P ) = P 0.
Proof.This is shown in [BLO2 , Proposition A.2(b)].
In this paper, it will sometimes be necessary to work with fusion systems whi*
*ch are
not saturated. This is why we have emphasized the difference between fusion sys*
*tems,
and saturated fusion systems, in the above definitions.
We next specify certain collections of subgroups relative to a given fusion s*
*ystem.
Definition 1.4. Let F be a fusion system over a finite p-subgroup S.
o A subgroup P S is F-centric if CS(P 0) = Z(P 0) for all P 0 S which is F-
conjugate to P .
o A subgroup P S is F-radical if OutF (P ) is p-reduced; i.e., if Op(Out F(P *
*)) = 1.
o For any P S which is fully centralized in F, the centralizer fusion system *
*CF (P )
is the fusion system over CS(P ) defined by setting
fi 0 0
Hom CF(P)(Q, Q0) = ff|Q fiff 2 Hom F(QP, Q P ), ff|P = IdP, ff(Q) Q .
A subgroup P S is F-quasicentric if for all P 0 S which is F-conjugate to *
*P and
fully centralized in F, CF (P 0) is the fusion system of the p-group CS(P 0).
o Fc Fq F denote the full subcategories of F whose objects are the F-centric
subgroups, and F-quasicentric subgroups, respectively, of S.
If F = FS(G) for some finite group G, then P S is F-centric if and only if
P is p-centric in G (i.e., Z(P ) 2 Sylp(CG(P ))), and P is F-radical if and o*
*nly if
NG(P )=(P .CG(P )) is p-reduced. Also, P is F-quasicentric if and only if CG(P *
*) contains
a normal subgroup of order prime to p and of p-power index.
In fact, when working with p-local finite groups, it suffices to have a fusio*
*n system
Fc defined on the centric subgroups of S, and which satisfies axioms (I) and (I*
*I) above
for those centric subgroups. In other words, fusion systems defined only on the*
* centric
subgroups are equivalent to fusion systems defined on all subgroups, as describ*
*ed in
the following theorem.
Theorem 1.5. Fix a p-group S and a fusion system F over S.
(a)Assume F is saturated. Then each morphism in F is a composite of restrictions
of morphisms between subgroups of S which are F-centric, F-radical, and fully
normalized in F. More precisely, for each P, P 0 S and each ' 2 IsoF(P, P 0*
*), there
are subgroups P = P0, P1, . .,.Pk = P 0, subgroups Qi (i = 1, .*
* .,.k)
which are F-centric, F-radical, and fully normalized in F, and automorphisms
'i2 AutF (Qi), such that 'i(Pi-1) = Pi for all i and ' = 'k O. .O.'1|P.
6 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
(b)Assume conditions (I) and (II) in Definition 1.2 are satisfied for all F-cen*
*tric sub-
groups P S. Assume also that each morphism in F is a composite of restrict*
*ions
of morphisms between F-centric subgroups of S. Then F is saturated.
Proof.Part (a) is Alperin's fusion theorem for saturated fusion systems, in the*
* form
shown in [BLO2 , Theorem A.10]. Part (b) is a special case of [5A1 , Theorem 2*
*.2]: the
case where H is the set of all F-centric subgroups of S.
Theorem 1.5(a) will be used repeatedly throughout this paper. The following l*
*emma
is a first easy application of the theorem, and provides a very useful criterio*
*n for a
subgroup to be quasicentric or not.
Lemma 1.6. Let F be a saturated fusion system over a p-group S. Then the follow*
*ing
hold for any P S.
(a)Assume that P Q P .CS(P ) and Id 6= ff 2 AutF (Q) are such that ff|P = I*
*dP
and ff has order prime to p. Then P is not F-quasicentric.
(b)Assume that P is fully centralized in F, and is not F-quasicentric. Then the*
*re are
P Q P .CS(P ) and Id 6= ff 2 AutF (Q) such that Q is F-centric, ff|P = I*
*dP,
and ff has order prime to p.
Proof.(a) Fix any P 0which is F-conjugate to P and fully centralized in F. By a*
*xiom
(II), there is ' 2 Hom F(Q, S) such that '(P ) = P 0; set Q0= '(Q). Thus 'ff'-1*
*|CQ0(P0)
is an automorphism in CF (P 0) whose order is not a power of p, so CF (P 0) is *
*not the
fusion system of CS(P 0), and P is not F-quasicentric.
(b) Assume that P is fully centralized in F and not F-quasicentric. Then CF (*
*P )
strictly contains the fusion system of CS(P ) (since CF (P 0) is isomorphic as *
*a category
to CF (P ) for all P 0which is F-conjugate to P and fully centralized in F). Si*
*nce CF (P )
is saturated by [BLO2 , Proposition A.6], Theorem 1.5(a) implies there is a su*
*bgroup
Q CS(P ) which is CF (P )-centric and fully normalized in CF (P ), and such t*
*hat
AutCF(P)(Q) AutCS(P)(Q). Since Q is fully normalized, Aut CS(P)(Q) is a Sylow*
* p-
subgroup of AutCF(P)(Q), and hence this last group is not a p-group. Also, by [*
*BLO2 ,
Proposition 2.5(a)], P Q is F-centric since Q is CF (P )-centric.
Since orbit categories _ both of fusion systems and of groups _ will play a r*
*ole in
certain proofs in the last three sections, we define them here.
Definition 1.7. (a)If F is a fusion system over a p-group S, then Oc(F) (the ce*
*ntric
orbit category of F) is the category whose objects are the F-centric subgrou*
*ps of
S, and whose morphism sets are given by
Mor Oc(F)(P, Q) = RepF (P, Q) def=Q\ Hom F(P, Q).
Let ZF :Oc(F) ___! Z(p)-mod be the functor which sends P to Z(P ) and ['] (*
*the
'-1
class of ' 2 Hom F(P, Q)) to Z(Q) ----! Z(P ).
(b)If G is a finite group and S 2 Sylp(G), then OcS(G) (the centric orbit categ*
*ory of
G) is the category whose objects are the subgroups of S which are p-centric *
*in G,
and where
MorOcS(G)(P, Q) = Q\NG(P, Q) ~=Map G(G=P, G=Q).
Let ZG :OcS(G) ___! Z(p)-mod be the functor which sends P to Z(P ) and [g] *
*(the
class of g 2 NG(P, Q)) to conjugation by g-1.
EXTENSIONS OF P-LOCAL FINITE GROUPS 7
We now turn to linking systems associated to abstract fusion systems.
Definition 1.8 ([BLO2 , Def. 1.7]). Let F be a fusion system over the p-group *
*S.
A centric linking system associated to F is a category L whose objects are the *
*F-
centric subgroups of S, together with a functor ss :L ---! Fc, and "distinguish*
*ed"
ffiP
monomorphisms P --! AutL (P ) for each F-centric subgroup P S, which satisfy
the following conditions.
(A) ss is the identity on objects. For each pair of objects P, Q 2 L, Z(P ) act*
*s freely
on Mor L(P, Q) by composition (upon identifying Z(P ) with ffiP(Z(P )) Aut*
*L(P )),
and ss induces a bijection
~=
MorL (P, Q)=Z(P ) ------! Hom F(P, Q).
(B) For each F-centric subgroup P S and each x 2 P , ss(ffiP(x)) = cx 2 AutF *
*(P ).
(C) For each f 2 Mor L(P, Q) and each x 2 P , the following square commutes in *
*L:
f
P ______! Q
| |
ffiP(x)| |ffiQ(ss(f)(x))
# #
f
P ______! Q.
A p-local finite group is defined to be a triple (S, F, L), where S is a fini*
*te p-group,
F is a saturated fusion system over S, and L is a centric linking system associ*
*ated to
F. The classifying space of the triple (S, F, L) is the p-completed nerve |L|^p.
For any finite group G with Sylow p-subgroup S, a category LcS(G) was defined*
* in
[BLO1 ], whose objects are the p-centric subgroups of G, and whose morphism se*
*ts are
defined by
Mor LcS(G)(P, Q) = NG(P, Q)=Op(CG(P )).
Since CG(P ) = Z(P ) x Op(CG(P )) when P is p-centric in G, LcS(G) is easily se*
*en
to satisfy conditions (A), (B), and (C) above, and hence is a centric linking s*
*ystem
associated to FS(G). Thus (S, FS(G), LcS(G)) is a p-local finite group, with cl*
*assifying
space |LcS(G)|^p' BG^p(see [BLO1 , Proposition 1.1]).
It will be of crucial importance in this paper that any centric linking syste*
*m asso-
ciated to F can be extended to a quasicentric linking system; a linking system *
*with
similar properties, whose objects are the F-quasicentric subgroups of S. We fir*
*st make
more precise what this means.
Definition 1.9. Let F be any saturated fusion system over a p-group S. A quasic*
*entric
linking system associated to F consists of a category Lq whose objects are the *
*F-
quasicentric subgroups of S, together with a functor ss :Lq ---! Fq, and distin*
*guished
monomorphisms
ffiP
P .CS(P ) ----! AutLq(P ),
which satisfy the following conditions.
(A)q ss is the identity on objects and surjective on morphisms. For each pair o*
*f objects
P, Q 2 Lq such that P is fully centralized, CS(P ) acts freely on Mor Lq(P, *
*Q) by
composition (upon identifying CS(P ) with ffiP(CS(P )) AutLq(P )), and ss *
*induces
a bijection
~=
Mor Lq(P, Q)=CS(P ) ------! Hom F(P, Q).
8 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
(B)q For each F-quasicentric subgroup P S and each g 2 P , ss sends ffiP(g) 2
Aut Lq(P ) to cg 2 AutF (P ).
(C)q For each f 2 Mor Lq(P, Q) and each x 2 P , f O ffiP(x) = ffiQ(ss(f)(x)) O*
*f in
Mor Lq(P, Q).
(D)q For each F-quasicentric subgroup P S, there is a morphism 'P 2 Mor Lq(P,*
* S)
such that ss('P) = inclSP2 Hom (P, S), and such that for each g 2 P .CS(P ),*
* ffiS(g) O
'P = 'P OffiP(g) in Mor Lq(P, S).
If P and P 0are F-conjugate and F-quasicentric, then for any Q S, MorLq(P, *
*Q) ~=
Mor Lq(P 0, Q) and Hom F(P, Q) ~= Hom F(P 0, Q), while the centralizers CS(P *
*) and
CS(P 0) need not have the same order. This is why condition (A)q makes sense on*
*ly if
we assume that P is fully centralized; i.e., that CS(P ) is as large as possibl*
*e. When
P is F-centric, then this condition is irrelevant, since every subgroup P 0whi*
*ch is
F-conjugate to P is fully centralized (CS(P 0) = Z(P 0) ~=Z(P )).
Note that (D)q is a special case of (C)q when P is F-centric; this is why the*
* axiom
is not needed for centric linking systems. We also note the following relation *
*between
these axioms:
Lemma 1.10. In the situation of Definition 1.9, axiom (C)q implies axiom (B)q.
Proof.Fix an F-quasicentric subgroup P S, and an element g 2 P . We apply (C)q
with f = ffiP(g). For each x 2 P , if we set y = ss(ffiP(g))(x), then ffiP(g) *
*OffiP(x) =
ffiP(y) OffiP(g). Since ffiP is an injective homomorphism, this implies that gx*
* = yg, and
thus that y = cg(x). So ss(ffiP(g)) = cg.
When (S, F, L) is a p-local finite group, and Lq is a quasicentric linking sy*
*stem
associated to F, then we say that Lq extends L if the full subcategory of Lq wi*
*th
objects the F-centric subgroups of S is isomorphic to L via a functor which com*
*mutes
with the projection functors to F and with the distinguished monomorphisms. In
[5A1 , Propositions 3.4 & 3.12], we constructed an explicit quasicentric linkin*
*g system
Lq associated to F and extending L, and showed that it is unique up to an isomo*
*rphism
of categories which preserves all of these structures. So from now on, we will *
*simply
refer to Lq as the quasicentric linking system associated to (S, F, L).
Condition (D)q above helps to motivate the following definition of inclusion *
*mor-
phisms in a quasicentric linking system.
Definition 1.11. Fix a p-local finite group (S, F, L), with associated quasicen*
*tric link-
ing system Lq.
(a)A morphism 'P 2 Mor Lq(P, S) is an inclusion morphim if it satisfies the hyp*
*otheses
of axiom (D)q: if ss('P) = inclSP, and if ffiS(g) O'P = 'P OffiP(g) in Mor L*
*q(P, S) for
all g 2 P .CS(P ). If P = S, then we also require that 'S = IdS.
(b)A compatible set of inclusions for Lq is a choice of morphisms {'QP} for all*
* pairs of
F-quasicentric subgroups P Q, such that 'QP2 Mor Lq(P, Q), such that 'RP= *
*'RQO'QP
for all P Q R, and such that 'SPis an inclusion morphism for each P .
The following properties of quasicentric linking systems were also proven in *
*[5A1 ].
Proposition 1.12. The following hold for any p-local finite group (S, F, L), wi*
*th as-
sociated quasicentric linking system Lq.
EXTENSIONS OF P-LOCAL FINITE GROUPS 9
(a)The inclusion L Lq induces a homotopy equivalence |L| ' |Lq| between geome*
*tric
realizations. More generally, for any full subcategory L0 Lq which contai*
*ns as
objects all subgroups of S which are F-centric and F-radical, the inclusion *
*L0 Lq
induces a homotopy equivalence |L0| ' |Lq|.
(b)Let ' 2 Mor Lq(P, R) and _ 2 Mor Lq(Q, R) be any pair of morphisms in Lq with
the same target group such that Im (ss(')) Im (ss(_)). Then there is a un*
*ique
morphism O 2 Mor Lq(P, Q) such that ' = _ OO.
Proof.The homotopy equivalences |Lq| ' |L| ' |L0| are shown in [5A1 , Theorem 3*
*.5].
Point (b) is shown in [5A1 , Lemma 3.6].
Point (b) above will be frequently used throughout the paper. In particular, *
*it makes
it possible to embed the linking system of S (or an appropriate full subcategor*
*y) in
Lq, depending on the choice of an inclusion morphisms 'P as defined above, for *
*each
object P . Such inclusion morphisms always exist by axiom (D)q. In the follow*
*ing
proposition, LS(S)|Ob(Lq)denotes the full subcategory of LS(S) whose objects ar*
*e the
F-quasicentric subgroups of S.
In general, for a functor F :C ___! C0, and objects c, d 2 Ob (C), we let Fc*
*,ddenote
the map from Mor C(c, d) to Mor C0(F (c), F (d)) induced by F .
Proposition 1.13. Fix a p-local finite group (S, F, L). let Lq be its associat*
*ed qua-
sicentric linking system, and let ss :Lq --! Fq be the projection. Then any cho*
*ice of
inclusion morphisms 'P = 'SP2 Mor Lq(P, S), for all F-quasicentric subgroups P *
* S,
extends to a unique inclusion of categories
ffi :LS(S)|Ob(Lq)------! Lq
such that ffiP,S(1) = 'P for all P ; and such that
(a)ffiP,P(g) = ffiP(g) for all g 2 P . CS(P ), and
(b)ss(ffiP,Q(g)) = cg 2 Hom (P, Q) for all g 2 NS(P, Q).
In addition, the following hold.
(c)If we set 'QP= ffiP,Q(1) for all P Q, then {'QP} is a compatible set of in*
*clusions
for Lq.
(d)For any P C Q S, where P and Q are both F-quasicentric and P is fully
centralized in F, and any morphism _ 2 Aut Lq(P ) which normalizes ffiP,P(Q),
there is a unique b_2 AutLq(Q) such that b_O'QP= 'QPO_. Furthermore, for any
g 2 Q, _ffiP,P(g)_-1 = ffiP,P(ss(_b)(g)).
(e)Every morphism ' 2 Mor Lq(P, Q) in Lq is a composite ' = 'QP0O'0 for a unique
morphism '02 IsoLq(P, P 0), where P 0= Im(ss(')).
Proof.For each P and Q, and each g 2 NS(P, Q), there is by Proposition 1.12(b) a
unique morphism ffiP,Q(g) such that
ffiS(g) O'P = 'Q OffiP,Q(g).
This defines ffi on morphism sets, and also allows us to define 'QP= ffiP,Q(1).*
* Then by
the axioms in Definition 1.9, {'QP} is a compatible set of inclusions for Lq, a*
*nd ffi is
a functor which satisfies (a), (b), and (c). Point (e) is a special case of Pr*
*oposition
1.12(b) (where P 0= Im(ss('))).
10 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
If ffiP,Q(g) = ffiP,Q(g0) for g, g0 2 NG(P, Q), then ffiS(g) O'P = ffiS(g0) O*
*'P, and hence
g = g0by [5A1 , Lemma 3.9]. Thus each ffiP,Q is injective.
It remains to prove (d). Set ' = ss(_) 2 Aut F(P ) for short. Since _ norma*
*lizes
ffiP,P(Q), for all g 2 Q there is h 2 Q such that _ffiP,P(g)_-1 = ffiP,P(h), an*
*d this implies
the relation 'cg'-1 = ch in AutF (P ). Thus Q is contained_in N'. So by axiom (*
*II)
(and since P is fully centralized), ' extends to ' 2 IsoF(Q, Q0) for some P C Q*
*0 S.
_ q
Let b_02 IsoLq(Q, Q0) be any lifting of ' to L .
By axiom (A)q (and since _ is an isomorphism), there is x 2 CS(P ) such that
0 Q -1 0
'QPO ffiP(x) O_ = b_0O'P. By (C)q, b_0ffiQ(Q)_b0 = ffiQ0(Q ), and hence after r*
*estriction,
ffiP(x) O_ conjugates ffiP,P(Q) to ffiP,P(Q0). Since _ normalizes ffiP,P(Q), th*
*is shows that
ffiP(x) conjugates ffiP,P(Q) to ffiP,P(Q0), and hence (since ffiP,Pis injective*
*) that xQx-1 =
Q0. We thus have the following commutative diagram:
_ ffiP(x) ffiP(x)-1
P __________!P __________! P __________! P
| Q0| Q |
'QP| 'P | 'P |
# b # #-1
_0 0 ffiQ,Q0(x)
Q _______________________! Q _________! Q .
So if we set b_= ffiQ,Q0(x)-1 Ob_0, then b_2 AutLq(Q) and 'QPO_ = b_O'QP.
The uniqueness of _b follows from [5A1 , Lemma 3.9]. Finally, for any g 2 Q,
b_ffiQ(g)_b-1= ffiQ(ss(_b)(g)) by (C)q, and hence _ffiP,P(g)_-1 = ffiP,P(ss(_b)*
*(g)) since mor-
phisms have unique restrictions (Proposition 1.12(b) again).
Once we have fixed a compatible set of inclusions {'QP} in a linking system L*
*q, then
for any ' 2 Mor Lq(P, Q), and any P 0 P and Q0 Q such that ss(')(P 0) Q0, t*
*here
_ 0 0 Q _ P _
is a unique morphism ' 2 Mor Lq(P , Q ) such that 'Q0O ' = ' O'P0. We think of *
*' as
the restriction of '.
Note, however, that all of this depends on the choice of morphisms 'P 2 Mor L*
*q(P, S)
which satisfy the hypotheses of axiom (D)q, and that not just any lifting of th*
*e inclusion
incl2 Hom F (P, S) can be chosen. To see why, assume for simplicity that P is *
*also
fully centralized. From the axioms in Definition 1.9 and Proposition 1.12(b), *
*we see
that if 'P, '0P2 Mor Lq(P, S) are two liftings of incl2 Hom F(P, S), then '0P= *
*'P OffiP(g)
for some unique g 2 CS(P ). But if 'P satisfies the conditions of (D)q, then '*
*0Palso
satisfies those conditions only if g 2 Z(CS(P )).
One situation where the choice of inclusion morphisms is useful is when descr*
*ibing
the fundamental group of |L| or of its p-completion. For any group , we let B*
*( )
denote the category with one object, and with morphism monoid the group . Reca*
*ll
that |L| ' |Lq| (Proposition 1.12(a)), so we can work with either of these cate*
*gories;
we will mostly state the results for |Lq|. Let the vertex S be the basepoint of*
* |Lq|. For
each morphism ' 2 Mor Lq(P, Q), let J(') 2 ss1(|Lq|) denote the homotopy class *
*of the
loop 'Q.'.'P-1 in |Lq| (where paths are composed from right to left). This defi*
*nes a
functor
J :Lq ------! B(ss1(|Lq|)),
where all objects are sent to the unique object of B(ss1(|Lq|)), and where all *
*inclusion
morphisms are sent to the identity. Let j :S ___! ss1(|Lq|) denote the composi*
*te of J
with the distinguished monomorphism ffiS :S ___! AutL (S).
The next proposition describes how J is universal among functors of this type*
*, and
also includes some other technical results for later use about the structure of*
* ss1(|L|).
EXTENSIONS OF P-LOCAL FINITE GROUPS 11
Proposition 1.14. Let (S, F, L) be a p-local finite group, and let Lq be the as*
*sociated
quasicentric linking system. Assume a compatible set of inclusions {'QP} has be*
*en chosen
for Lq. Then the following hold.
(a)For any group , and any functor ~: Lq ----! B( ) which sends inclusions to
the identity, there is a unique homomorphism ~~:ss1(|Lq|) ----! such that *
*~ =
B(~~) OJ.
(b)For g 2 P S with P F-quasicentric, J(ffiP(g)) = J(ffiS(g)). In particu*
*lar,
J(ffiP(g)) = 1 in ss1(|Lq|) if and only if ffiP(g) is nulhomotopic as a loop*
* based
at the vertex P of |Lq|.
(c)If ff 2 Mor Lq(P, Q), and ss(ff)(x) = y, then j(y) = J(ff)j(x)J(ff)-1 in ss1*
*(|Lq|).
(d)If x and y are F-conjugate elements of S, then j(x) and j(y) are conjugate in
ss1(|Lq|).
Proof.Clearly, any functor ~: Lq ____! B( ) induces a homomorphism ~~= ss1(|~|)
between the fundamental groups of their geometric realizations. If ~ sends inc*
*lusion
morphisms to the identity, then ~ = B(~~) OJ by definition of J.
The other points follow easily, using condition (C)q for quasicentric linking*
* systems.
Point (d) is shown by first reducing to a map between centric subgroups of S wh*
*ich
sends x to y.
We finish this introductory section with two unrelated results which will be *
*needed
later in the paper. The first is a standard, group theoretic lemma.
Lemma 1.15. Let Q C P be p-groups. If ff is a p0-automorphism of P which acts a*
*s the
identity on Q and on P=Q, then ff = IdP. Equivalently, the group of all automor*
*phisms
of P which restrict to the identity on Q and on P=Q is a p-group.
Proof.See [Go , Corollary 5.3.3].
The following proposition will only be used in Section 4, but we include it h*
*ere
because it seems to be of wider interest. Note, for any fusion system F over S*
*, any
subgroup P S fully normalized in F, and any P 0which is S-conjugate to P , th*
*at P 0
is also fully normalized in F since NS(P 0) is S-conjugate to NS(P ).
Proposition 1.16. Let F be a saturated fusion system over a p-group S. Then for
any subgroup P S, the set of S-conjugacy classes of subgroups F-conjugate to *
*P and
fully normalized in F has order prime to p.
Proof.By [BLO2 , Proposition 5.5], there is an (S, S)-biset which, when rega*
*rded as
a set with (S x S)-action, satisfies the following three conditions.
(a)The isotropy subgroup of each point in is of the form
P' def={(x, '(x)) | x 2 P }
for some P S and some ' 2 Hom F(P, S).
(b)For each P S and each ' 2 Hom F(P, S), the two structures of (S x P )-set *
*on
obtained by restriction and by Idx ' are isomorphic.
(c)| |=|S| 1 (mod p).
12 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
Note that by (a), the actions of S x 1 and 1 x S on are both free.
Now fix a subgroup P S. Set S2 = 1 x S for short, and let 0 be the sub*
*set
such that 0=S2 = ( =S2)P. In other words, 0 is the set of all x 2 such that
for each g 2 P , there is some h 2 S satisfying (g, h).x = x. Since the action*
* of S2
on is free, this element h 2 S is uniquely determined for each x 2 0 and g 2*
* P .
Let `(x): P ___! S denote the function such that for each g 2 P , (g, `(x)(g))*
*.x =
x. The isotropy subgroup at x of the (P x S)-action is thus the subgroup P`(x)=
{(g, `(x)(g)) | g 2 P }; and by (a), `(x) 2 Hom F(P, S). This defines a map
` : 0 ___! Hom F (P, S).
By definition, for each ' 2 Hom F(P, S), `-1(') is the set of elements of f*
*ixed by
P'. By condition (b) above, the action of P x P on induced by the homomorphism
1 x ' 2 Hom (P x P, S x S) is isomorphic to the action defined by restriction, *
*and thus
|`-1(')| = |`-1(incl)|. This shows that the point inverses of ` all have the sa*
*me fixed
order k.
Now let Rep F(P, S) = Hom F (P, S)=Inn(S): the set of S-conjugacy classes of *
*mor-
phisms from P to S. Let P be the set of S-conjugacy classes of subgroups F-conj*
*ugate
to P , and let Pfn P be the subset of classes of subgroups fully normalized in*
* F. If
x 2 0 and `(x) = ', then for all s 2 S and g 2 P ,
(1, s).x = (g, s'(g)).x = (g, csO '(g)).(1, s).x,
and this shows that `((1, s).x) = csO `(x). Thus ` induces a map
~`:( =S2)P = 0=S2 --`=S2----!RepF(P, S) --Im(-)----!P,
where Rep F(P, S) = Hom F (P, S)=Inn(S). Furthermore, | =S2| 1 (mod p) by (c*
*),
and thus |( =S2)P| 1 (mod p).
For each P 0which is F-conjugate to P , there are |S|=|NS(P 0)| distinct subg*
*roups in
the S-conjugacy class [P 0]. Hence there are
| AutF(P )|.|S|=|NS(P 0)|
elements of Hom F(P, S) whose image lies in [P 0]. Since each of these is the i*
*mage of k
elements in 0, this shows that
|~`-1([P 0])| = k.| AutF(P )|=|NS(P 0)|.
fi
Thus |NS(P 0)|fik.| AutF(P )| for all P 0F-conjugate to P , and so ~`-1([P 0]) *
*has order a
multiple of p if P 0is not fully normalized in F (if [P 0] 2 Pr Pfn). Hence
|~`-1(Pfn)| |( =S2)P| 1 (mod p) .
So if we set m = |NS(P 0)| for [P 0] 2 Pfn(i.e., the maximal value of |NS(P 0)|*
* for P 0
F-conjugate to P ), then
ik.| Aut (P )|j
|~`-1(Pfn)| = |Pfn|. _______F____ ,
m
and thus |Pfn| is prime to p.
Using a similar argument, one can also show that the set RepfcF(P, S) of elem*
*ents of
RepF (P, S) whose image is fully centralized also has order prime to p.
EXTENSIONS OF P-LOCAL FINITE GROUPS 13
2. The fundamental group of |L|^p
The purpose of this section is to give a simple description of the fundamenta*
*l group
of |L|^p, for any p-local finite group (S, F, L), purely in terms of the fusion*
* system F.
The result is analogous to the (hyper-) focal subgroup theorem for finite group*
*s, as we
explain below.
In Section 1, we defined a functor J :Lq ___! B(ss1(|L|)), for any p-local f*
*inite group
(S, F, L), and a homomorphism j = J OffiS from S to ss1(|L|). Let o :S ! ss1(|L*
*|^p) be
the composite of j with the natural homomorphism from ss1(|L|) to ss1(|L|^p).
In [BLO2 , Proposition 1.12], we proved that o :S ___! ss1(|L|^p) is a surj*
*ection. In
this section, we will show that Ker(o) is the hyperfocal subgroup of F, defined*
* by Puig
[Pu4 ] (see also [Pu3 ]).
Definition 2.1. For any saturated fusion system F over a p-group S, the hyperfo*
*cal
subgroup of F is the normal subgroup of S defined by
-1 fi p ff
OpF(S) = g ff(g) fig 2 P S, ff 2 O (Aut F(P )) .
We will prove, for any p-local finite group (S, F, L), that ss1(|L|^p) ~=S=Op*
*F(S). This
is motivated by Puig's hyperfocal theorem, and we will also need that theorem i*
*n order
to prove it. Before stating Puig's theorem, we first recall the standard focal *
*subgroup
theorem. If G is a finite group and S 2 Sylp(G), then this theorem says that S *
*\ [G, G]
(the focal subgroup) is the subgroup generated by all elements of the form x-1y*
* for
x, y 2 S which are G-conjugate (cf. [Go , Theorem 7.3.4] or [Suz2, 5.2.8]).
The quotient group S=(S \ [G, G]) is isomorphic to the p-power torsion subgro*
*up
of G=[G, G], and can thus be identified as a quotient group of the maximal p-gr*
*oup
quotient G=Op(G). Since G=Op(G) is a p-group, G = S.Op(G), and hence G=Op(G) ~=
S=(S \ Op(G)). Hence S \ Op(G) S \ [G, G]. This subgroup S \ Op(G) is what Pu*
*ig
calls the hyperfocal subgroup, and is described by the hyperfocal subgroup theo*
*rem in
terms of S and fusion.
For S 2 Sylp(G) as above, let OpG(S) be the normal subgroup of S defined by
-1 fi p ff
OpG(S) = OpFS(G)(S)= g ff(g) fig 2 P S, ff 2 O (Aut G(P ))
fi ff
= [g, x] fig 2 P S, x 2 NG(P ) of order prime to p.
Lemma 2.2 ([Pu3 ]). Fix a prime p, a finite group G, and a Sylow subgroup S 2
Sylp(G). Then OpG(S) = S \ Op(G).
Proof.This is stated in [Pu3 , x1.1], but the proof is only sketched there, and*
* so we
elaborate on it here. Following standard notation, for any P S and any A Au*
*t(P ),
we write [P, A] = . Thus OpG(S) is generated by the s*
*ubgroups
[P, Op(Aut G(P ))] for all P S. It is clear that OpG(S) S \ Op(G); the prob*
*lem is to
prove the opposite inclusion.
Set G* = Op(G) and S* = S \ G* for short. Then [G*, G*] has index prime to p
in G*, so it contains S*. By the focal subgroup theorem (cf. [Go , Theorem 7.*
*3.4]),
applied to S* 2 Sylp(G*), S* is generated by all elements of the form x-1y for *
*x, y 2 S*
which are G*-conjugate. Combined with Alperin's fusion theorem (in Alperin's or*
*iginal
version [Al] or in the version of Theorem 1.5(a)), this implies that S* is gene*
*rated by
all subgroups [P, NG*(P )] for P S* such that NS*(P ) 2 Sylp(NG*(P )). (This*
* last
condition is equivalent to P being fully normalized in FS*(G*).) Also, NG*(P *
*) is
14 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
generated by Op(NG*(P )) and the Sylow subgroup NS*(P ), so [P, NG*(P )] is gen*
*erated
by [P, Op(NG*(P ))] OpG(S) and [P, NS*(P )] [S*, S*]. Thus S* = .
Since OpG(S) is normal in S (hence also normal in S*), this shows that S*=OpG(S*
*) is
equal to its commutator subgroup, which for a p-group is possible only if S*=Op*
*G(S) is
trivial, and hence S* = OpG(S).
By Proposition 1.14, the key to getting information about ss1(|L|) is to cons*
*truct
functors from L or Lq to B( ), for a group , which send inclusions to the iden*
*tity.
The next lemma is our main inductive tool for doing this. Whenever ff 2 Mor Lq(*
*P, P 0)
and fi 2 Mor Lq(Q, Q0) are such that P Q, P 0 Q0, we write ff = fi|P to mean*
* that
0 *
* Q
ff is the restriction of fi in the sense defined in Section 1; i.e., 'QP0Off = *
*fi O'P.
Lemma 2.3. Fix a p-local finite group (S, F, L), and let Lq be its associated q*
*uasi-
centric linking system. Assume a compatible set of inclusions {'QP} has been c*
*hosen
for Lq. Let H0 be a set of F-quasicentric subgroups of S which is closed under*
* F-
conjugacy and overgroups. Let P be an F-conjugacy class of F-quasicentric subgr*
*oups
maximal among those not in H0, set H = H0 [ P, and let LH0 LH Lq be the full
subcategories with these objects. Assume, for some group , that
~0: LH0 -----! B( )
is a functor which sends inclusions to the identity. Fix some P 2 P which is f*
*ully
normalized in F, and fix a homomorphism ~P :Aut Lq(P ) ___! . Assume that
(*)for all P Q NS(P ) such that Q is fully normalized in NF (P ), and for a*
*ll
ff 2 AutLq(P ) and fi 2 AutLq(Q) such that ff = fi|P, ~P(ff) = ~0(fi).
Then there is a unique extension of ~0 to a functor ~: LH ! B( ) which sends in*
*clu-
sions to the identity, and such that ~(ff) = ~P(ff) for all ff 2 AutF (P ).
Proof.The uniqueness of the extension is an immediate consequence of Theorem 1.*
*5(a)
(Alperin's fusion theorem).
To prove the existence of the extension ~, we first show that (*) implies the*
* following
(a prori stronger) statement:
(**)for all Q, Q0 S which strictly contain P , and for all fi 2 Mor Lq(Q, Q0)*
* and
ff 2 AutLq(P ) such that ff = fi|P, ~P(ff) = ~0(fi).
To see this, note first that it suffices to consider the case where P is normal*
* in Q and Q0.
By assumption, ss(fi)(P ) =_ss(ff)(P ) = P , hence ss(fi)(NQ(P )) NQ0(P ), an*
*d therefore
fi restricts to a morphism fi2 Mor Lq(NQ(P ), NQ0(P )) by Proposition 1.12(b) (*
*applied
0
with ' = fi O'QNQ(P)and _ = 'QNQ0(P)). Since NQ(P ), NQ0(P ) P , by the indu*
*ction
_ _
hypothesis, ~0(fi) = ~0(fi), so we are reduced to proving that ~P(ff) = ~0(fi).
Thus fi 2 Mor NLq(P)(Q, Q0). We now apply Alperin's fusion theorem (Theorem
1.5(a)) to the morphism ss(fi) in the fusion system NF (P ) (which is saturated*
* by [BLO2 ,
Proposition A.6]). Thus ss(fi) = 'kO. .O.'1, where each 'i2 Hom NF(P)(Qi-1, Qi)*
* is the
restriction to Qi-1of an automorphism b'i2 AutNF(P)(Ri), where Ri Qi-1, Qi is *
*an
NF (P )-centric subgroup of NS(P ) which is fully normalized in NF (P ), and wh*
*ere Q =
Q0 and Q0= Qk. Each Ri contains P , and hence is F-centric by [BLO2 , Lemma 6.*
*2].
For each b'i, also regarded as an automorphism in F, we choose a lifting bfii2 *
*AutL(Ri),
and let fii 2 Mor Lq(Qi-1, Qi) be its restriction. By (A)q, fi = fik O. .O.fi1 *
*OffiQ(g) for
EXTENSIONS OF P-LOCAL FINITE GROUPS 15
some g 2 CS(Q), and hence
~0(fi)= ~0(fik) . .~.0(fi1) . ~0(ffiQ(g)) = ~0(bfik) . .~.0(bfi1) . ~0(f*
*fiNS(P)(g))
= ~P(bfik|P) . .~.P(bfi1|P) . ~P(IdP) = ~P(fi|P) = ~P(ff) ,
where the third equality follows from (*). This finishes the proof of (**).
We can now extend ~ to be defined on all morphisms in LH not in LH0. Fix such
a morphism ' 2 Mor Lq(P1, Q). Set P2 = ss(')(P1) Q; then P1, P2 2 P, and
' = 'QP2O'0 for some unique '0 2 IsoLq(P1, P2). By Lemma 1.3 (and then lifting
_
to the linking category), there are isomorphisms ' i2 IsoLq(NS(Pi), Ni), for so*
*me
Ni NS(P ) containing P , which restrict to isomorphisms 'i 2 IsoLq(Pi, P ). *
*Set
_ = '2O'0O'-112 AutLq(P ). We have thus decomposed '0as the composite '-12O_O'1,
and can now define
_ -1 _
~(') = ~('0) = ~0(' 2) .~P(_).~0(' 1).
Now_let '0= ('02)-1O_0O'01be another such decomposition, where '0iis the rest*
*riction
of '0i2 IsoLq(NS(Pi), N0i). We thus have a commutative diagram
'1 '01
P ______P1 ______! P
| | |
_| |'0 _0|
# # #
'2 '02
P ______P2 ______! P ,
_ _0
where for each i, 'iand '0iare restrictions of isomorphisms 'iand 'idefined on *
*NS(Pi).
To see that the two decompositions give the same value of ~('), it remains to s*
*how
that
_ 0 _ -1 _ 0 _ -1
~P(_0).~0(' 1O(' 1) ) = ~0(' 2O(' 2) ).~P(_).
_ 0 _ -1 0 -1
And this holds since ~0(' iO(' i) ) = ~P('iO('i) ) by (**).
We have now defined ~ on all morphisms in LH , and it sends inclusion morphis*
*ms
to the identity by construction. By construction, ~ sends composites to product*
*s, and
thus the proof is complete.
Lemma 2.3 provides the induction step when proving the following proposition,*
* which
is the main result needed to compute ss1(|L|^p).
Proposition 2.4. Fix a p-local finite group (S, F, L), and let Lq be its associ*
*ated
quasicentric linking system. Assume a compatible set of inclusions {'QP} has be*
*en chosen
for Lq. Then there is a unique functor
~: Lq ------! B(S=OpF(S))
which sends inclusions to the identity, and such that ~(ffiS(g)) = g for all g *
*2 S.
Proof.The functor ~ will be constructed inductively, using Lemma 2.3. Let H0
Ob (Lq) be a subset (possibly empty) which is closed under F-conjugacy and over-
groups. Let P be an F-conjugacy class of F-quasicentric subgroups maximal among
those not in H0, set H = H0 [ P, and let LH0 LH Lq be the full subcategories
with these objects. Assume that
~0: LH0 -----! B(S=OpF(S))
has already been constructed, such that ~0(ffiS(g)) = g for all g 2 S (if S 2 H*
*0), and
such that ~0 sends inclusions to the identity.
16 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
Fix P 2 P which is fully normalized in F, and let ffiP,P:NS(P ) ---! AutLq(*
*P )
be the homomorphism of Proposition 1.13. Then Im (ffiP,P) is a Sylow p-subgrou*
*p of
AutLq(P ), since Aut S(P ) 2 Sylp(Aut F(P )) by axiom (I). We identify NS(P ) a*
*s a
subgroup of AutLq(P ) to simplify notation. Then
ffi p
AutLq(P )=Op(Aut Lq(P )) ~=NS(P ) NS(P ) \ O (Aut Lq(P )) = NS(P )=N0,
where by Lemma 2.2, N0 is the subgroup generated by all commutators [g, x] for *
*g 2
Q NS(P ), and x 2 NAutLq(P)(Q) of order prime to p. In this situation, conjug*
*ation
by x lies in AutF (Q) by Proposition 1.13(d), and thus [g, x] = g.cx(g)-1 2 OpF*
*(S). We
conclude that N0 OpF(S), and hence that the inclusion of NS(P ) into S extend*
*s to a
homomorphism
~P :Aut Lq(P ) ------! S=OpF(S).
We claim that condition (*) in Lemma 2.3 holds for ~0 and ~P. To see this, f*
*ix
P Q S such that P C Q and Q is fully normalized in NF (P ), and fix ff 2 Au*
*tLq(P )
and fi 2 Aut Lq(Q) such that ff = fi|P. We must show that ~P(ff) = ~0(fi). Up*
*on
replacing ff by ffk and fi by fik for some appropriate k 1 (mod p), we can as*
*sume that
both automorphisms have order a power of p. Since Q is fully normalized, AutNS(*
*P)(Q)
is a Sylow subgroup of Aut NF(P)(Q). Hence (since any two Sylow p-subgroups of
a finite group G are conjugate by an element of Op(G)), there is an automorphism
_ p _ _-1 _
fl2 O (Aut NLq(P)(Q))_such that flfifl = ffiQ(g) for some g 2 NS(Q) \ NS(P ). *
* In
particular,_~0(fl) = 1 since it is a composite of automorphisms of order prime *
*to p. Set
fl = fl|P; then fl 2 Op(Aut Lq(P_)) and hence ~P(fl) = 1. Using axiom (C)q and *
*Lemma
_
1.12, we see that flfffl-1 = ffiP(g); and thus (since ~P(fl) = 1 and ~0(fl) = 1*
*) that
_ _
~0(fi) = ~0(ffiQ(g)) = g = ~P(ffiP(g)) = ~P(ff).
Thus, by Lemma 2.3, we can extend ~0 to a functor defined on LH . Upon contin*
*uing
this procedure, we obtain a functor ~ defined on all of Lq.
For any finite group G, ss1(BG^p) ~=G=Op(G). (This is implicit in [BK , xVII.*
*3], and
shown explicitly in [BLO1 , Proposition A.2].) Hence our main result in this *
*section
can be thought of as the hyperfocal theorem for p-local finite groups.
Theorem 2.5 (Hyperfocal subgroup theorem for p-local finite groups). Let (S, F,*
* L)
be a p-local finite group. Then
ss1(|L|^p) ~=S=OpF(S).
More precisely, the natural map o :S ___! ss1(|L|^p) is surjective, and Ker(o)*
* = OpF(S).
Proof.Let ~: L ---! B(S=OpF(S)) be the functor of Proposition 2.4, and let |~| *
*be
the induced map between geometric realizations. Since |B(S=OpF(S))| is the clas*
*sifying
space of a finite p-group and hence p-complete, |~| factors through the p-compl*
*etion
|L|^p. Consider the following commutative diagram:
j
S _________! ss1(|L|)
H H H
H H ^| HH ss (|~|)
H H (-)p| HH 1
o H Hj # ~~=HH Hj
ss1(|L|^p)___ss!S=Op^F(S) .
1(|~|p)
EXTENSIONS OF P-LOCAL FINITE GROUPS 17
Here, o is surjective by [BLO2 , Proposition 1.12]. Also, by construction, the*
* composite
~~Oo = ss1(|~|) Oj is the natural projection. Thus Ker(o) OpF(S), and it rema*
*ins to
show the opposite inclusion.
Fix g 2 P S and ff 2 Aut F(P ) such that ff has order prime to p; we want to
show that g-1ff(g) 2 Ker(o). Since Ker(o) is closed under F-conjugacy (Proposit*
*ion
1.14(d)), for any ' 2 IsoF(P, P 0), g-1ff(g) is in Ker(o) if '(g-1ff(g)) is in *
*Ker(o). In
particular, since '(g-1ff(g)) = '(g)-1.('ff'-1)('(g)), we can assume that P is *
*fully
centralized in F. Then, upon extending ff to an automorphism of P .CS(P ), whi*
*ch
can be assumed also to have order prime to p (replace it by an appropriate powe*
*r if
necessary), we can assume that P is F-centric. In this case, by Proposition 1.*
*14(c),
j(g) and j(ff(g)) are conjugate in ss1(|L|) by an element of order prime to p, *
*and hence
are equal in ss1(|L|^p) since this is a p-group. This shows that g-1ff(g) 2 Ker*
*(o), and
finishes the proof of the theorem.
The following result, which will be useful in Section 5, is of a similar natu*
*re, but
much more elementary.
Proposition 2.6. Fix a p-local finite group (S, F, L), and let Lq be its associ*
*ated
quasicentric linking system. Then the induced maps
ss1(|L|) -----! ss1(|Fc|) and ss1(|Lq|) -----! ss1(|Fq|)
are surjective, and their kernels are generated by elements of p-power order.
Proof.We prove this for |Lq| and |Fq|; a similar argument applies to |L| and |F*
*c|.
Recall from Section 1 that we can regard ss1(|Lq|) as the group generated by th*
*e loops
J(ff), for ff 2 Mor (Lq), with relations given by composition of morphisms and *
*making
inclusion morphisms equal to 1. In a similar way, we regard ss1(|Fq|) as being *
*generated
by loops J(') for ' 2 Mor (Fq). Since every morphism ff 2 Hom Fq(P, Q) has a li*
*fting
to a morphism of Lq, the map ss# :ss1(|Lq|) ___! ss1(|Fq|) induced by the proj*
*ection
functor ss :Lq ___! Fq is an epimorphism.
Write JC for the normal subgroup of ss1(|Lq|) generated by the loops j(g) = J*
*(ffiP(g)),
for all F-quasicentric subgroups P S and all g 2 CS(P ). In particular, JC *
*is
generated by elements of p-power order. Since ss# (j(g)) = 1, JC is contained i*
*n the
kernel of ss# , and we have a factorization bss#:ss1(|Lq|)=JC ___! ss1(|Fq|).
Define an inverse s: ss1(|Fq|) ___! ss1(|Lq|)=JC as follows. Given a morphis*
*m ff 2
Hom Fq(P, Q), choose a lifting ~ffin Lq, and set s(ff) = [J(~ff)]. If ~ff, ~ff0*
*2 Hom Lq(P, Q)
are two liftings of ff, then there is an isomorphism O: P 0! P in Lq where P 0i*
*s fully
centralized, and an element g 2 CS(P 0), such that ~ff0= ~ffO(O OffiP0(g) OO-1)*
*. Since
OOffiP0(g)OO-1 represents a loop that belongs to JC, [J(~ff)] = [J(~ff0)]. Thus*
* the definition
of s does not depend on the choice of ~ff. It remains to show that s preserves *
*the relations
among the generators. But we clearly have that s([J(inclQP)]) = [J('QP)] = 1. A*
*lso, if ff
and fi are composable morphisms of Fq, and ~ffand ~fiare liftings to Lq, then ~*
*ffO~fiis a
lifting of ff Ofi, and hence s(ff)s(fi) = [J(~ff)][J(f~i)] = [J(~ffO~fi)] = s(f*
*f Ofi).
This shows that ss# induces an isomorphim ss1(|Lq|)=JC ~=ss1(|Fq|).
3. Subsystems with p-solvable quotient
In this section, we prove some general results about subsystems of saturated *
*fusion
systems and p-local finite subgroups: subsystems with p-group quotient or quoti*
*ent of
18 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
order prime to p. These will then be used in the next two sections to prove som*
*e more
specific theorems.
It will be convenient in this section to write "p0-group" for a finite group *
*of order
prime to p. Recall that a p-solvable group is a group G with normal series 1 = *
*H0 C
H1 C . . .C Hk = G such that each Hi=Hi-1 is a p-group or a p0-group. As one
consequence of the results of this section, we show that for any p-local finite*
* group
(S, F, L), and any homomorphism ` from ss1(|L|) to a finite p-solvable group, t*
*here is
another p-local finite group (S0, F0, L0) such that |L0| is homotopy equivalent*
* to the
covering space of |L| with fundamental group Ker(`).
0
We start with some definitions. Recall that for any finite group G, Op (G) a*
*nd
Op(G) are the smallest normal subgroups of G of index prime to p and of p-power
0
index, respectively. Equivalently, Op (G) is the subgroup generated by elements*
* of p-
power order in G, and Op(G) is the subgroup generated by elements of order prim*
*e to
p in G.
We want to identify the fusion subsystems0of a given fusion system which are *
*anal-
ogous to subgroups of G which contain Op (G) or Op(G). This motivates the follo*
*wing
definitions.
Definition 3.1. Let F be a saturated fusion system over a p-group S. Let (S0, F*
*0)
(S, F) be a saturated fusion subsystem; i.e., S0 S is a subgroup, and F0 F *
*is a
subcategory which is a saturated fusion system over S0.
(a)(S0, F0) is of p-power index in (S, F) if S0 OpF(S), and AutF0(P ) Op(Aut*
* F(P ))
for all P S0. Equivalently, a saturated fusion subsystem F0 F over S0
OpF(S) has p-power index if it contains all F-automorphisms of order prime t*
*o p
of subgroups of S0.
0
(b)(S0, F0) is of index prime to p in (S, F) if S0= S, and AutF0(P ) Op (Aut *
*F(P ))
for all P S. Equivalently, a saturated fusion subsystem F0 F over S has *
*index
prime to p if it contains all F-automorphisms of p-power order.
This terminology has been chosen for simplicity. Subsystems "of p-power index*
*" or
"of index prime to p" are really analogous to subgroups H G which contain nor*
*mal
subgroups of G of p-power index or index prime to p, respectively. For example,*
* if F is
a saturated fusion system over S, and F0 F is the fusion system of S itself (*
*i.e., the
minimal fusion system over S), then F0 is not in general a subsystem of index p*
*rime
to p under the above definition, despite the inclusion F0 F being analogous t*
*o the
inclusion of a Sylow p-subgroup in a group.
Over the next three sections, we will classify all saturated fusion subsystem*
*s of p-
power index, or of index prime to p, in a given saturated fusion system.0 In bo*
*th
cases, there will be a minimal such subsystem, denoted Op(F) or Op (F); and the
saturated fusion subsystems of the given type will be in bijective corresponden*
*ce with
the subgroups of a given p-group or p0-group.
The following terminology will be useful for describing some of the categorie*
*s we
have to work with.
Definition 3.2. Fix a finite p-group S.
EXTENSIONS OF P-LOCAL FINITE GROUPS 19
(a)A restrictive category over S is a category F such that Ob (F) is the set of*
* subgroups
of S, such that all morphisms in F are group monomorphisms between the sub-
groups, and with the following additional property: for each P 0 P S and *
*Q0
Q S, and each ' 2 Hom F(P, Q) such that '(P 0) Q0, '|P0 2 Hom F(P 0, Q0).
(b)A restrictive category F over S is normalized by an automorphism ff 2 AutF (*
*S)
if for each P, Q S, and each ' 2 Hom F(P, Q), ff'ff-1 2 Hom F(ff(P ), ff(Q*
*)).
(c)For any restrictive category F over S and any subgroup A Aut(S), is*
* the
smallest restrictive category over S which contains F together with all auto*
*mor-
phisms in A and their restrictions.
By definition, any restrictive category is required to contain all inclusion *
*homomor-
phisms (restrictions of IdS). The main difference between a restrictive categor*
*y over S
and a fusion system over S is that the restrictive subcategory need not contain*
* FS(S).
When F is a fusion system over S, then an automorphism ff 2 Aut(S) normalizes
F if and only if it is fusion preserving in the sense used in [BLO1 ]. For the*
* purposes
of this paper, it will be convenient to use both terms, in different situations.
We next define the following subcategories of a given fusion system F.
Definition 3.3. Let F be any fusion system over a p-group S.
(a)Op*(F) F denotes the smallest restrictive subcategory of F whose morphism *
*set
contains Op(Aut F(P )) for all subgroups P S.
0
(b)Op*(F) F denotes the smallest restrictive subcategory of F whose morphism *
*set
0
contains Op (Aut F(P )) for all subgroups P S.
0
By definition, for subgroups P, Q, a morphism ' 2 Hom F(P, Q) lies in Op*(F) *
*if and
0
only if it is a composite of morphisms which are restrictions of elements of Op*
* (Aut F(R))
for subgroups R S. Morphisms in Op*(F) are described in a similar way.
The subcategory Op*(F) is not, in general, a fusion system _ and this is why *
*we
0
had to define restrictive categories. The subcategory Op*(F) is always a fusion*
* system
0
(since Aut S(P ) Op (Aut F(P )) for all P S), but it is not, in general, sa*
*turated.
The subscripts "*" have been put0in as a reminder of these facts, and as a cont*
*rast
with the notation Op(F) and Op (F) which will be used to denote certain minimal
saturated fusion systems.
0
We now check some of the basic properties of the subcategories Op*(F) and Op**
*(F),
stated in terms of these definitions.
Lemma 3.4. The following hold for any fusion system F over a p-group S.
0
(a)Op*(F) and Op*(F) are normalized by AutF (S).
0
(b)If F is saturated, then F = = .
(c)If F0 F is any restrictive subcategory normalized by Aut F(S) and such that
F = , then for each P, Q S and ' 2 Hom F(P, Q), there are
morphisms ff 2 AutF (S), '02 Hom F0(ff(P ), Q), and '002 Hom F0(P, ff-1Q), s*
*uch
that ' = '0Off|P = ff O'00.
Proof.To simplify notation in the following proofs, for ff 2 Aut F(S), we write*
* ff in
composites of morphisms between subgroups of S, rather than specifying the appr*
*o-
priate restriction each time.
20 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
0
(a) Set F0 = Op*(F) or Op*(F). Let ff 2 AutF (S) and let ' 2 Hom F0(P, Q). We *
*must
show that ff'ff-1 2 Hom F0(ff(P ), ff(Q)). By definition of F0, there are subgr*
*oups
P = P0, P1, . .,.Pk = '(P ) Q,
__ __ __
subgroups P 1, . .,.Pk, and automorphisms Oi 2 Op(Aut F(P i)) (if F0 = Op*(F)) *
*or
0 __ 0 p0 __
Oi2 Op (Aut F(P i)) (if F = O* (F)), such that Pi-1, Pi Pi, Oi(Pi-1) = Pi, and
' = inclQPkO(Ok|Pk-1) O. .O.(O2|P1) O(O1|P0).
__ __ __ 0 *
* __
Let Pi0= ff(Pi), P 0i= ff(P i), and O0i= ffOiff-1 2 Op(Aut F(P 0i)) or Op (Aut *
*F(P 0i)).
Then
ff'ff-1 = inclff(Q)P0kO(O0k|P0k-1) O. .O.(O02|P01) O(O01|P00)
is in Hom F0(ff(P ), ff(Q)), as required.
0
(b1) Now assume that F is saturated. We show here that F = ;
0
i.e., that each ' 2 Mor (F) is a composite of morphisms in Op*(F) and in AutF (*
*S). By
Alperin's fusion theorem for saturated fusion systems (Theorem 1.5(a)), it suff*
*ices to
prove this when ' 2 AutF (P ) for some P S which is F-centric and fully norma*
*lized.
The result clearly holds if P = S. So we can assume that P S, and also assume
inductively that the lemma holds for every automorphism of any subgroup P 0 S *
*such
that |P 0| > |P |.
Consider the subgroup
K = ' AutS(P )'-1 = {'cg'-1 2 AutF (P ) | g 2 NS(P )}.
Then K is a p-subgroup of AutF (P ), and hence K AutOp0*(F)(P ). Since P is f*
*ully
normalized in F, Aut S(P ) 2 Sylp(Aut F(P )). Thus K and Aut S(P ) are both Sy*
*low
p-subgroups of AutOp0*(F)(P ), and there is some O 2 AutOp0*(F)(P ) such that O*
*KO-1
AutS(P ). In other words,
NO' def={g 2 NS(P ) | (O')cg(O')-1 2 AutS(P )} = NS(P ).
So by condition (II) in Definition 1.2, O O' can be extended to a homomorphism *
*_ 2
Hom F(NS(P ), S). By the induction hypothesis (and since NS(P ) P ), _, and h*
*ence
0
O O', are composites of morphisms in NF (S) and in Op*(F). Also, O 2 AutOp0*(F)*
*(P )
by assumption, and hence ' is a composite of morphisms in these two subcategori*
*es.
(b2) To see that F = when F is saturated, it again suffices *
*to
restrict to the case where ' 2 Aut F(P ) for some P S which is F-centric and
fully normalized. But in this case, Aut F(P ) is generated by its Sylow p-subg*
*roup
AutS(P ) AutNF(S)(P ), together with Op(Aut F(P )) AutOp*(F)(P ).
(c) Since F = , every morphism in F is a composite of morphisms in
F0 and restrictions of automorphisms in Aut F(S). Assume ' 2 Hom F (P, Q) is t*
*he
composite
'1 ff1 ffn
P = P0 -ff0--!Q0 ---! P1 ---! Q1 ---! . . .---!Pn ---! Qn = Q,
where ffi2 AutF (S), ffi(Pi) = Qi, and 'i2 Hom F0(Qi-1, Pi). Write ffj,i= ffj. *
*.f.fifor
any i j, and set ff = ffn,0= ffn . .f.f0. Then
' = ff O(ffn-1,0-1'nffn-1,0) . .(.ff0,0-1'1ff0,0) = (ffn,n'nffn,n-1) . .(.ffn,*
*1'1ffn,1-1) Off;
where each ffn,i'iffn,i-1and each ffi,0-1'i+1ffi,0is a morphism in F0 since F0 *
*is normal-
ized by AutF (S).
EXTENSIONS OF P-LOCAL FINITE GROUPS 21
The following lemma will also be needed later.
Lemma 3.5. Let F be a saturated fusion system over a p-group S. Fix a normal
subgroup S0 C S which is strongly F-closed; i.e., no element of S0 is F-conjuga*
*te to
any element of Sr S0. Let (S0, F0) be a saturated fusion subsystem of (S, F). T*
*hen for
any P S which is F-centric and F-radical, P \ S0 is F0-centric.
Proof.Assume P S is F-centric and F-radical, and set P0 = P \S0 for short. Ch*
*oose
a subgroup P00 S0 which is F-conjugate to P0 and fully normalized in F. In par*
*ticular,
by (I), P00is fully centralized in F. By Lemma 1.3, there is ' 2 Hom F(NS(P0), *
*NS(P00))
such that '(P0) = P00. Set P 0= '(P ); thus P 0is F-conjugate to P (so P 0is a*
*lso
F-centric and F-radical), and P00= P 0\ S0 since S0 is strongly closed. For any
P000 S0 which is F0-conjugate to P0 (hence F-conjugate to P00), there is a mor*
*phism
_ 2 Hom F (P000.CS(P000), P00.CS(P00)) such that _(P000) = P00(by axiom (II)), *
*and then
_(CS0(P000)) CS0(P00). So if CS0(P00) = Z(P00), then CS0(P000) = Z(P000) for *
*all P000which
is F0-conjugate to P0, and P0 is F0-centric.
We are thus reduced to showing that CS0(P00) = Z(P00); and without loss of ge*
*nerality,
we can assume that P 0= P and P00= P0. Since S0 is strongly closed, every ff 2
AutF (P ) leaves P0 invariant. Let Aut0F(P ) AutF (P ) be the subgroup of el*
*ements
which induce the identity on P0 and on P=P0. This is a normal subgroup of AutF *
*(P )
since all elements of AutF (P ) leave P0 invariant, and is also a p-subgroup by*
* Lemma
1.15. Thus Aut 0F(P ) Op(Aut F(P )), and hence Aut 0F(P ) Inn(P ) since P *
*is F-
radical.
We want to show that CS0(P0) = Z(P0). Fix any x 2 CS0(P0), and assume first
that the coset x.Z(P0) 2 CS0(P0)=Z(P0) is fixed by the conjugation action of P *
*. Thus
x 2 S0, [x, P0] = 1, and [x, P ] Z(P0), so cx 2 Aut0F(P ) Inn(P ), and xg 2*
* CS(P ) for
some g 2 P . Since P is F-centric, this implies that xg 2 P , so x 2 CS0(P0)\P *
*= Z(P0).
In other words, [CS0(P0)=Z(P0)]P = 1, so CS0(P0)=Z(P0) = 1, and thus CS0(P0) =
Z(P0).
The motivation for the next definition comes from considering the situation w*
*hich
arises when one is given a saturated fusion system F with an associated quasice*
*ntric
linking system Lq, and a functor Lq ___! B( ) which sends inclusions to the id*
*entity (or
equivalently a homomorphism from_ss1(|Lq|) to ) for some group . Such a funct*
*or
is equivalent to a function : Mor(Lq) ___! which sends_composites to produ*
*cts
and sends inclusions to the identity; and for any H , -1(H) = Mor (LqH) for
some subcategory LqH Lq with the same objects. Let FqH Fq be the image of LqH
under the canonical projection; then in some sense (to be made precise later), *
*FqHis a
subsystem of Fq of index [ :H].
What we now need is to make sense of such "inverse image subcategories" of the
fusion system F, when we are not assuming that we have an associated linking_sy*
*stem.
Let Sub ( ) denote the set of nonempty subsets of . Given a function as abov*
*e, there
is an obvious associated_function : Mor (Fq) ___! Sub ( ), which sends a morp*
*hism
ff 2 Mor (Fq)_to (ss-1(ff)). Here, ss denotes_the natural projection from Lq*
* to Fq.
Moreover, also induces a homomorphism ` = O ffiS from S to . The maps
and ` are closely related to each other, and satisfy certain properties, none o*
*f which
depend on the existence (or choice) of a quasicentric linking system associated*
* to F.
In fact, we will see that the data encoded in such a pair of functions, if they*
* satisfy the
appropriate conditions, suffices to describe precisely what is meant by "invers*
*e image
22 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
subcategories" of fusion systems; and to show that under certain restrictions, *
*these
categories are (or generate) saturated fusion subsystems.
Definition 3.6. Let F be a saturated fusion system over a p-group S, and let F0
Fq be any full subcategory such that Ob (F0) is closed under F-conjugacy. A fu*
*sion
mapping triple for F0 consists of a triple ( , `, ), where is a group, ` :S *
*--! is
a homomorphism, and
: Mor (F0) -----! Sub ( ),
is a map which satisfies the following conditions for all subgroups P, Q, R S*
* which
lie in F0:
' _
(i)For all P --! Q --! R in F0, and all x 2 (_), (_') = x. (').
(ii)If P is fully centralized in F, then (IdP) = `(CS(P )).
(iii)If ' = cg 2 Hom F(P, Q), where g 2 NS(P, Q), then `(g) 2 (').
(iv)For all ' 2 Hom F(P, Q), all x 2 ('), and all g 2 P , x`(g)x-1 = `('(g)).
For any fusion mapping triple ( , `, ) and any H , we let FoH F be the sma*
*llest
restrictive subcategory which contains all ' 2 Mor (Fq) such that (') \ H 6= ;*
*. Let
FH FoHbe the full subcategory whose objects are the subgroups of `-1(H).
When ` is the trivial homomorphism (which is always the case when is a p0-g*
*roup),
then a fusion mapping triple ( , `, ) on a subcategory F0 Fq is equivalent t*
*o a
functor from F0 to B( ); i.e., (') contains just one element for all ' 2 Mor (*
*F0). By
(i), it suffices to show this for identity morphisms; and by (ii), | (IdP)| = 1*
* if P is fully
centralized. For arbitrary P , it then follows from (i), together with the ass*
*umption
(included in the definition of Sub ( )) that (') 6= ; for all '.
The following additional properties of fusion mapping triples will be needed.
Lemma 3.7. Fix a saturated fusion system F over a p-group S, let F0 be a full
subcategory such that Ob (F0) is closed under F-conjugacy, and let ( , , `) be*
* a fusion
mapping triple for F0. Then the following hold for all P, Q, R 2 Ob (F0).
(v) (IdP) is a subgroup of , and restricts to a homomorphism
P :Aut F(P ) ___! N ( (IdP))= (IdP).
Thus P(ff) = (ff) (as a coset of (IdP)) for all ff 2 AutF (P ).
' _
(vi)For all P --! Q --! R in F0, and all x 2 ('), (_') (_).x, with equali*
*ty
if '(P ) = Q. In particular, if P Q, then (_|P) (_).
(vii)Assume S 2 Ob (F0). Then for any ' 2 Hom F(P, Q), any ff 2 AutF (S), and a*
*ny
x 2 (ff), (ff'ff-1) = x (')x-1, where ff'ff-1 2 Hom F(ff(P ), ff(Q)).
Proof.(v) By (i), for any ff, fi 2 AutF (P ) and any x 2 (ff), (fffi) = x. (f*
*i). When
applied with ff = fi = IdP, this shows that (IdP) is a subgroup of . (Note he*
*re that
(IdP) 6= ; by definition of Sub ( ).) When applied with fi = ff-1, this shows*
* that
x-1 2 (ff-1) if x 2 (ff). Hence (ff) = x. (IdP) implies that (ff-1) = (IdP*
*).x-1
and (ff-1) = x-1. (IdP), and thus that each (ff) is a right coset as well as *
*a left
coset. Thus (ff) N ( (IdP)) for all ff 2 Aut F(P ), and the induced map P *
*is
clearly a homomorphism.
EXTENSIONS OF P-LOCAL FINITE GROUPS 23
(vi) By (i), (_') (_). (') for any pair of composable morphisms ', _ in F0.*
* In
particular, (_') (_).x if x 2 ('). If ' is an isomorphism, then 1 2 (IdP)*
* =
x. ('-1) by (v) and (i), so x-1 2 ('-1). This gives the inclusions
(_) = (_''-1) (_').x-1 (_)xx-1,
and hence these are both equalities. The last statement is the special case whe*
*re P Q
and ' = inclQP; 1 2 (inclQP) by (iii).
(vii) For x 2 (ff), (ff') = x. (') = (ff'ff-1).x by (i) and (vi).
We are now ready to prove the main result about fusion mapping triples.
Proposition 3.8. Let F be a saturated fusion system over a finite p-group S. L*
*et
( , `, ) be any fusion mapping triple on Fq, where is a p-group or a p0-grou*
*p, and
` :S ----! and : Mor (Fq) ----! Sub ( ),
Then the following hold for any subgroup H , where we set SH = `-1(H).
(a)FH is a saturated fusion system over SH .
(b)If is a p-group, then a subgroup P SH is FH -quasicentric if and only if*
* it is
F-quasicentric. Also, Fo1 Op*(F).
(c)If is a p0-group, then SH = S. A subgroup P S is FH -centric (fully cent*
*ralized
in FH , fully normalized in FH ) if and only if it is F-centric (fully centr*
*alized in
0
F, fully normalized in F). Also, Fo1 Op*(F).
Proof.Throughout the proof, (i)-(vii) refer to the conditions in Definition 3.6*
* and
Lemma 3.7. If X is any set of morphisms of F, then we let (X) be the union of *
*the
sets (ff) for ff 2 X. Condition (v) implies that for any P S and any subgro*
*up
A AutF (P ), (A) is a subgroup of .
We first prove the following two additional properties of these subcategories:
(1)For each pair of subgroups P, Q S, and each ' 2 Hom F(P, Q), there are Q0 *
* S,
'02 Hom Fo1(P, Q0), and ff 2 AutF (S) such that ff(Q0) = Q and ' = (ff|Q0) O*
*'0.
(2)For all P S there exists P 0 S which is fully normalized in F, and ' 2
Hom Fo1(NS(P ), NS(P 0)) such that '(P ) = P 0.
By (vii), for any ff 2 AutF (S), any x 2 (ff), and any ' 2 Hom F(P, Q), (ff*
*'ff-1) =
x (')x-1. In particular, (ff'ff-1) = 1 if and only if (') = 1, and thus Fo1is
normalized by AutF (S).
0 c p0 p c p
Let Op*(F) O* (F) and O*(F) O*(F) be the full subcategories whose objec*
*ts
are the F-centric subgroups of S. By (v), for each F-quasicentric subgroup P *
* S,
AutFo1(P ) = Ker( P) where P is a homomorphism to a subquotient of . Hence
0 0 p
AutFo1(P ) contains Op (Aut F(P )) if is a p -group or O (Aut F(P )) if is *
*a p-group.
0 c 0 p c
This shows that Fo1contains either Op*(F) (if is a p -group) or O*(F) (if *
* is a p-
group). Thus Fc by Lemma 3.4(b), and so F = sin*
*ce
F is the smallest restrictive category over S which contains Fc by Theorem 1.5(*
*a)
(Alperin's fusion theorem). Point (1) now follows from Lemma 3.4(c).
To see (2), recall that by Lemma 1.3, if P 00is any subgroup F-conjugate to P
and fully normalized in F, then there is a morphism '0 2 Hom F(NS(P ), NS(P 00*
*))
such that '0(P ) = P 00. By (1), '0 = ff O' for some ff 2 Aut F(S) and some ' 2
24 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
Hom Fo1(NS(P ), NS(ff-1(P 00))). Also, the subgroup P 0= ff-1(P 00) is fully no*
*rmalized in
F since P 00is.
(b) Assume is a p-group, and fix H . Consider the following set of subgro*
*ups
of SH :
fi 0 0 0 0
QH = P SH fi8 P FH -conjugate to P8,P Q P .CSH(P ),
8 ff 2 AutFH (Q) such that ff|P0 = Id,|ff| is a power of*
* p.
This set clearly contains all FH -centric subgroups of SH . By Lemma 1.6, if F*
*H is
saturated, then QH is precisely the set of FH -quasicentric subgroups. We prov*
*e (b)
here with "FH -quasicentric" replaced by "element of QH ". This is all that wi*
*ll be
needed for the proof that FH is saturated; and once that is shown then (b) (in*
* its
original form) will follow.
Assume P SH and P =2QH . Fix P 0which is FH -conjugate to P , Q P 0.CSH(P*
* 0)
which contains P 0, and IdQ6= ff 2 AutFH (Q) of order prime to p, such that ff|*
*P0 = IdP0.
Then P 0, and hence P , is not F-quasicentric by Lemma 1.6(a).
We have shown that if P is F-quasicentric, then P 2 QH , and it remains to ch*
*eck the
converse. Assume P is not F-quasicentric: fix P 0 Q P 0.CS(P 0) and ff 2 Aut*
*F (Q)
as in Lemma 1.6(b). In particular, Q is F-centric, and hence F-quasicentric. *
*Set
Q1 = Q \ S1 (where S1 = `-1(1)). Since 1 2 (ff) by (v) (and since |ff| is prim*
*e to
p), `(g) = `(ff(g)) for all g 2 Q by (iv), and thus ff(Q1) = Q1 and (since g-1f*
*f(g) 2
Ker(`) = S1 for g 2 Q) ff induces the identity on Q=Q1. Since |ff| is not a pow*
*er of p,
it cannot be the identity on both Q1 and Q=Q1 (Lemma 1.15), and hence ff|Q1 6= *
*IdQ1.
Thus P 0 P 0Q1 P 0.CSH(P 0), ff|P0Q1 is a nontrivial automorphism of P 0Q1 o*
*f order
prime to p whose restriction to P 0is the identity.
Finally, by (1), P is FH -conjugate to a subgroup_P 00for which there is some*
* ' 2
IsoF(P 0, P 00) which is the restriction of some ' 2 AutF (S). Since P SH and*
* is FH -
conjugate to_P 00, P 00 _SH by (iv) (applied with_ff 2_IsoFH(P, P 00) and x 2 *
*(ff) \ H).
Set Q00= ' (P 0Q1) = P 00.'(Q1) SH and ff00= ' ff' -1 2 Aut F(Q00). Since *
* is
a p-group and |ff00| is prime to p, (v) implies that (ff00) = (IdQ), and henc*
*e that
1 2 (ff00) and hence ff002 AutFH (Q00). But then P 00=2QH , and hence P =2QH .
It remains to show that Fo1 Op*(F); i.e., to show that
Aut Fo1(P ) Op(Aut F(P )) (3)
for each P S. If P S is F-quasicentric, then (3) holds by (v): Aut Fo1(P ) *
*is the
kernel of a homomorphism from Aut F(P ) to a p-group. If P is not F-quasicentr*
*ic
but is fully centralized in F, then_every automorphism ff 2 AutF (P ) of order *
*prime
to p extends to an automorphism ff2 Aut F(P .CS(P )), which (after replacing it*
* by
an appropriate power) can also be assumed to have order prime to p, and hence in
Op(Aut F(P .CS(P ))). Thus (3) holds for P , since it holds for the F-centric s*
*ubgroup
P .CS(P ). Finally, by (1), every subgroup of S is Fo1-conjugate to a subgroup *
*which is
fully centralized in F, and thus (3) holds for all P S.
(c) This point holds, in fact, without assuming that the fusion system F be sat*
*urated.
Assume is a p0-group; then ` is the trivial homomorphism, and so SH = `-1(H) *
*= S
for all H. Fix H , and let P be any subgroup of S. Since each FH -conjugacy
class is contained in some F-conjugacy class, any subgroup which is fully centr*
*alized
(fully normalized) in F is also fully centralized (fully normalized) in FH . By*
* the same
reasoning, any F-centric subgroup P S is also FH -centric.
EXTENSIONS OF P-LOCAL FINITE GROUPS 25
Conversely, assume P is not fully centralized in F, and let P 0be a subgroup *
*in the F-
conjugacy class of P such that |CS(P 0)| > |CS(P )|. Fix some ' 2 IsoF(P, P 0).*
* By (1),
there are a subgroup P 00 S and isomorphisms ff 2 AutF (S) and '02 IsoF1(P, P *
*00),
such that ff-1(P 00) = P 0and ' = (ff|P00) O'0. Thus P 00is FH -conjugate to P *
*, and is
F-conjugate to P 0via a restriction of an F-automorphism of S. Hence |CS(P 00)*
*| =
|CS(P 0)| > |CS(P )|, and this shows that P is not fully centralized in FH . A*
* similar
argument shows that if P is not fully normalized in F (or not F-centric), then *
*it is not
fully normalized in FH (or not FH -centric).
0
The proof that Fo1 Op*(F) is identical to the proof of the corresponding res*
*ult in
(b).
(a) Fix H . Clearly, FH is a fusion system over SH ; we must prove it is sa*
*turated.
By (b) or (c), each FH -centric subgroup of SH is F-quasicentric. Hence by Theo*
*rem
1.5(b), it suffices to prove conditions (I) and (II) in Definition 1.2 for F-qu*
*asicentric
subgroups P SH . Thus we will be working only with subgroups P SH for which
is defined on Hom F(P, S).
Proof of (I): Assume is a p0-group; thus SH = S. If P is fully normalized in*
* FH ,
then by (c), P is fully normalized in F, hence it is fully centralized in F and*
* in FH .
Also, Aut S(P ) 2 Sylp(Aut F(P )), and hence Aut S(P ) is also a Sylow p-subgro*
*up of
AutFH (P ). This proves (I) in this case.
Now assume is a p-group. Fix P SH which is F-quasicentric and fully norma*
*lized
in FH , and let ff 2 Hom Fo1(NS(P ), NS(P 0)) be as in (2). In particular, 1 2 *
* (ff) by
definition of Fo1; so by (iv), `(ff(NSH(P ))) = `(NSH(P )) H. Hence ff(NSH(P*
* ))
SH \ NS(P 0) = NSH(P 0), and this is an equality since P is fully normalized in*
* FH . In
particular, this shows that P 0 SH . The conclusion of (I) holds for P (i.e., *
*P is fully
centralized in FH and AutSH (P ) 2 Sylp(Aut FH(P ))) if the conclusion of (I) h*
*olds for
P 0. So we can assume that P = P 0is fully normalized in F and in FH .
Fix Q SH which is FH -conjugate to P and fully centralized in FH , and choo*
*se
an isomorphism _ 2 IsoFH(P, Q). After applying (2) again, we can assume that Q *
*is
also fully normalized in F, and hence also fully centralized. Hence _ extends t*
*o some
_1 2 Hom F(P .CS(P ), Q.CS(Q)), which is an isomorphism since P and Q are both *
*fully
centralized in F. Fix h 2 (_-1) \ H and g 2 (_1). Then gh 2 (IdQ) = `(CS(Q))
by (i) and (ii). Let a 2 CS(Q) be such that `(a) = gh, and set
_2 = c-1aO_1 2 IsoF(P .CS(P ), Q.CS(Q)).
Thus _2|P = _1|P = _ since a 2 CS(Q), and h-1 = `(a)-1g 2 (_2) by (iii) and (i*
*).
By (iv), for all g 2 P .CS(P ), `(_2(g)) = h-1`(g)h, and hence _2(g) 2 H if and*
* only if
g 2 H. Thus _2 sends CSH(P ) onto CSH(Q). Since Q is fully centralized in FH , *
*so is
P .
It remains to show that AutSH(P ) 2 Sylp(Aut FH(P )). Set P = (IdP) = `(CS(*
*P ))
for short (see (ii)). By (v), restricts to a homomorphism
P :Aut F(P ) ------! N (P )=P .
Set fi
bH= hP fih 2 H \ N (P ) N (P )=P ;
then AutFH (P ) = P-1(Hb) by definition of FH . For any ' = ca 2 AutS(P )\Aut *
*FH(P )
(where a 2 NS(P )), (') = `(a).`(CS(P )) by (ii) and (iii); and thus (') cont*
*ains
some h 2 H where h = `(b) for some b 2 a.CS(P ). Hence b 2 SH , and ' = cb 2
26 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
AutSH(P ). This shows that
AutSH(P ) = AutS(P ) \ AutFH (P ) = AutS(P ) \ -1P(Hb).
Also, P(Aut S(P )) = P(Aut F(P )), since Aut S(P ) 2 Sylp(Aut F(P )) and is*
* a p-
group, and hence
[Aut S(P ) : AutSH(P )] = [Im ( P) : bH] = [Aut F(P ) : AutFH (P )].
Since AutS(P ) 2 Sylp(Aut F(P )) (i.e., [Aut F(P ) : AutS(P )] is prime to p), *
*this implies
that AutSH (P ) 2 Sylp(Aut FH(P )).
Proof of (II): Fix a morphism ' 2 IsoFH(P, Q), for some F-quasicentric P, Q *
*SH
such that Q = '(P ) is fully centralized in FH , and set N' = {g 2 NSH(P ) | 'c*
*g'-1 2
AutSH(Q)}. By (2), there is a subgroup Q0 fully normalized in F, and a morphism
_ 2 Hom Fo1(NS(Q), NS(Q0)), such that _(Q) = Q0. By condition (II) for the satu*
*rated
_ 0 _
fusion system F, there is '1 2 Hom F(N', NS(Q )) such that '1|P = _ O'. Fix some
_
x 2 (' 1) (_') = (')
(where the last equality holds since 1 2 (_)). Since ' 2 Mor (FH ), there is h*
* 2 H
such that h 2 ('), and thus
hx-1 2 (IdQ0) = `(CS(Q0)).
Fix a 2 CS(Q0) such that hx-1 = `(a), and set
_ _ 0
'2= ca O'1 2 Hom F(N', NS(Q )).
_ _ 0 _ _
Then by (i), h 2 (' 2), so ' 2 2 Hom FH(N', NSH(Q )); and ' 2|P = ' 1|P sin*
*ce
a 2 CS(Q0). Since Q is fully centralized in FH , _ sends CSH(Q) isomorphically *
*onto
_ _ def-1 _
CSH(Q0); and hence (by definition of N') '2(N') Im(_). So ' = _ O'2 sends N'
into NSH(Q) and extends '.
We next extend Proposition 3.8 to a result about linking systems and p-local *
*finite
groups. The main point of the following theorem is that for any p-local finite*
* group
`b
(S, F, L) and any epimorphism ss1(|L|) --i , where is a finite p-group or p0*
*-group,
there is another p-local finite group (SH , FH , LH ) for each subgroup H , *
*such that
|LH | is homotopy equivalent to the covering space of |L| whose fundamental gro*
*up is
b`-1(H).
Recall that for any p-local finite group (S, F, L) with associated quasicentr*
*ic link-
ing system Lq, j :S ___! ss1(|L|) ~=ss1(|Lq|) denotes the homomorphism induced*
* by
the distinguished monomorphism ffiS :S ___! AutL (S), and J :Lq ___! B(ss1(|L*
*q|))
is the functor which sends morphisms to loops as defined in Section 1. Let b`b*
*e a
homomorphism from ss1(|Lq|) to a group , and set
` = b`Oj 2 Hom (S, ) and b = B(b`) OJ :Lq ------! B( ).
Note that these depend on a choice of a compatible set of inclusions {'QP} for *
*Lq (since
J depends on such a choice). For any subgroup H , let LoH Lq be the subcate*
*gory
with the same objects and with morphism set b -1(H); and let LqH LoHbe the full
subcategory obtained by restricting to subgroups of SH def=`-1(H). Finally, let*
* FH be
the fusion system over SH generated by ss(LqH) Fq and restrictions of morphis*
*ms,
and let LH LqHbe the full subcategory on those objects which are FH -centric.
EXTENSIONS OF P-LOCAL FINITE GROUPS 27
Theorem 3.9. Let (S, F, L) be a p-local finite group, let Lq be its associated *
*quasicen-
tric linking system, and let ss :Lq ___! F be the projection. Assume a compati*
*ble set
of inclusions {'QP} has been chosen for Lq. Fix a finite group which is a p-g*
*roup or
a p0-group, and a surjective homomorphism
`b:ss1(|Lq|) -----i .
Set ` = b`Oj :S --! . Fix H , and set SH = `-1(H). Then (SH , FH , LH ) is *
*also a
p-local finite group, and (via the inclusion of LH into Lq) |LH | is homotopy e*
*quivalent
to the covering space of |Lq| ' |L| with fundamental group b`-1(H).
Proof.Define : Mor (Fq) ___! Sub ( ) by setting (ff) = b (ss-1(ff)), where b*
* =
B(b`) OJ as above. Then and ` satisfy hypotheses (i)-(iv) of Definition 3.6: *
*points (i)
and (ii) follow from (A)q, while (iii) follows from Proposition 1.13 and (iv) f*
*rom (C)q.
Thus ( , `, ) is a fusion mapping triple on Fq, and FH is a saturated fusion s*
*ystem
over SH by Proposition 3.8.
0 p0 q *
* p0
Let Op*(F) and Op*(F) be the categories of Definition 3.3, and let O* (F) *
*O* (F)
and Op*(F)q Op*(F) be the full subcategories whose objects are the F-quasicen*
*tric
0 q 0
subgroups of S. By Proposition 3.8(b,c), ss(Lo1) contains Op*(F) (if is a p *
*-group)
or Op*(F)q (if is a p-group). Hence in either case, by Lemma 3.4(b), all morp*
*hisms
in Lq are composites of morphisms in Lo1and restrictions of morphisms in Aut L(*
*S).
Since Lo1= b -1(1), by definition, and since b (ff) = b (fi) whenever ff is a r*
*estriction
of fi, this shows that b restricts to a surjection of AutL(S) onto . In partic*
*ular, this
implies that
8 P 2 Ob (Lq) and 8 g 2 , 9 P 0 S and ff 2 IsoLq(P, P 0) such that b (ff)(=*
*1g;)
where in fact, ff can always be chosen to be the restriction of an automorphism*
* of S.
We start by proving that LqH is a linking system associated to FH (for its s*
*et of
objects), and hence that LH is a centric linking system. Let P SH be a F-quas*
*icentric
subgroup, and choose g 2 P .CS(P ). By construction, b(ffiS(g)) = `(g), and b('*
*P) = 1.
In particular, the inclusion morphisms are in LH . Also, 'P OffiP(g) = ffiS(g)*
* O'P by
definition of an inclusion morphism (Definition 1.11). Hence b (ffiP(g)) = `(g)*
* in this
situation; and in particular ffiP(g) 2 AutLqH(P ) if and only if g 2 SH . Thus *
*ffiP restricts
to a distinguished monomorphism P .CSH(P ) --! AutLqH(P ) for LqHand axiom (D)*
*q is
satisfied. Moreover, if f, f0 2 Mor LqH(P, Q) are such that ss(f) = ss(f0) in H*
*om F(P, Q),
and g 2 CS(P ) is the unique element such that f0 = f OffiP(g) (using axiom (A)*
*q for
Lq), then b (ffiP(g)) = 1, and hence g 2 SH . This shows that axiom (A)q for Lq*
*Hholds.
Axioms (B)q and (C)q for LqHalso follow immediately from the same properties fo*
*r Lq.
We have now shown that (SH , FH , LH ) is a p-local finite group. It remains *
*to show
that |LH | is homotopy equivalent to a certain covering space of |Lq| ' |L|. We*
* show
this by first choosing certain full subcategories Lx Lq and LxH LqH such th*
*at
|Lx| ' |Lq| ' |L| and |LxH| ' |LqH| ' |LH |, and then proving directly that |Lx*
*H| is a
covering space of |Lx|.
If is a p0-group, then for any P SH = S, P is FH -centric if and only if*
* it is
F-centric (Proposition 3.8(c)). By the above remarks, LH is a centric linking s*
*ystem
associated to FH . Set Lx = L and LxH= LH in this case.
If is a p-group, then for any P SH , P is FH -quasicentric if and only if*
* it is
F-quasicentric (Proposition 3.8(b)). So by what was just shown, LqHis a quasice*
*ntric
28 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
linking system associated to FH which extends LH . Let Lx Lq be the full subc*
*ategory
whose objects are those subgroups P S such that P \ SH is F-quasicentric. S*
*et
S1 = Ker (`), and let F1 F be the saturated fusion system over S1 defined in
Proposition 3.8. The definition of ` as a restriction of b ensures that `(g) an*
*d `(g0)
are -conjugate whenever g, g0 are F-conjugate; in particular, no element of S1*
* is F-
conjugate to any element of Sr S1. Hence by Lemma 3.5, for each P S which is
F-centric and F-radical, P \ S1 is F1-centric, hence F-quasicentric, so P \ SH *
*is F-
quasicentric, and thus P 2 Ob (Lx). So the inclusion of |Lx| in |Lq| is a homo*
*topy
equivalence by Proposition 1.12.
Still assuming is a p-group, let LxH Lx be the subcategory with the same o*
*bjects,
where Mor(LxH) = b -1(H). For each P 2 Ob (LxH) = Ob (Lx), P \SH is F-quasicent*
*ric
by assumption, hence FH -quasicentric; and by Proposition 1.13, each ' 2 Mor Lx*
*H(P, Q)
restricts to a unique morphism 'H 2 Mor LxH(P \SH , Q\SH ). These restrictions *
*define
a deformation retraction from |LxH| to |LqH|, and thus the inclusion of categor*
*ies induces
a homotopy equivalence |LxH| ' |LqH| ' |LH |.
Thus in both cases, we have chosen categories LxH Lx with the same objects,
where Lx is a full subcategory of Lq and Mor (LxH) = Mor (Lx) \ b -1(H), and wh*
*ere
|Lx| ' |L| and |LxH| ' |LH |. Let E ( =H) be the category with object set =H,
and with a morphism ~gfrom aH to gaH for each g 2 and aH 2 =H. Thus
AutE ( =H)(1.H) ~=H, and |E ( =H)| = EG=H ' BH.
Let eLbe the pullback category in the following square:
eL_______! E ( =H)
| |
| |
# #
Lx ________!B( ) ,
Thus Ob (Le) = Ob (Lx) x =H, and Mor (Le) is the set of pairs of morphisms in *
*Lx and
E ( =H) which get sent to the same morphism in B( ). Then LxHcan be identified
with the full subcategory of eLwith objects the pairs (P, 1.H) for P 2 Ob (Lx).*
* By (1),
each object in eLis isomorphic to an object in LxH, and so |LxH| ' |Le|. By con*
*struction,
|Le| is the covering space over |Lx| with fundamental group `-1(H). Since |Le| *
*' |LH |
and |Lx| ' |L|, this finishes the proof of the last statement.
The following is an immediate corollary to Theorem 3.9.
Corollary 3.10. For any p-local finite group (S, F, L), any finite p-solvable g*
*roup ,
and any homomorphism
`b:ss1(|L|) ------! ,
there is a p-local finite group (S0, F0, L0) such that |L0| is homotopy equival*
*ent to the
covering space of |L| with fundamental group Ker(b`). Furthermore, this can be *
*chosen
such that S0 is a subgroup of S and F0 is a subcategory of F.
For any p-local finite group (S, F, L), since F is a finite category, Corolla*
*ry 3.10
implies that there is a unique maximal p-solvable quotient group of ss1(|L|), w*
*hich is
finite. In contrast, if we look at arbitrary finite quotient groups of the fun*
*damental
group, they can be arbitrarily large.
As one example, consider the case where p = 2, S 2 Syl2(A6) (so S ~= D8), F =
FS(A6), and L = LcS(A6). It is not hard to show directly, using Van Kampen's th*
*eorem,
EXTENSIONS OF P-LOCAL FINITE GROUPS 29
that ss1(|L|) ~= 4* 4: the amalgamated free product of two copies of 4 inters*
*ecting
S
in S, where each of the two subgroups C22in D8 is normalized by one of the 4.
Thus ss1(|L|) surjects onto any finite group which is generated by two copies*
* of 4
intersecting in the same way. This is the case when = A6, and also when = P*
* SL2(q)
for any q 9 (mod 16). However, the kernel of any such homomorphism defined on
ss1(|L|) is torsion free (and infinite), and hence cannot be the fundamental gr*
*oup of
the geometric realization of any centric linking system. In fact, in this case,*
* there is
no nontrivial homomorphism from ss1(|L|) to a finite 2-solvable group.
4. Fusion subsystems and extensions of p-power index
Recall that for any saturated fusion system F over a p-group S, we defined Op*
*F(S) C
S to be the subgroup generated by all elements of the form g-1ff(g), for g 2 P *
* S,
and ff 2 AutF (P ) of order prime to p. In this section, we classify all satura*
*ted fusion
subsystems of p-power index in a given saturated fusion system F over S, and sh*
*ow
that there is a one-to-one correspondence between such subsystems and the subgr*
*oups
of S which contain OpF(S). In particular, there is a unique minimal subsystem O*
*p(F)
of this type, which is a fusion system over OpF(S). We then look at extensions*
*, and
describe the procedure for finding all larger saturated fusion systems of which*
* F is a
fusion subsystem of p-power index.
4.1___Subsystems_of_p-power_index__ In this subsection, we classify all satura*
*ted
fusion subsystems of p-power index in a given saturated fusion system F, and sh*
*ow
that there is a unique minimal subsystem Op(F) of this type. We also show that *
*there
is a bijective correspondence between subgroups T of the finite p-group
p(F) def=S=OpF(S),
and fusion subsystems FT of p-power index in F. We have already seen that p(F)*
* =
S=OpF(S) is isomorphic to ss1(|L|^p) for any centric linking system L associate*
*d to F,
and our result also shows that all (connected) covering spaces of |L|^pare real*
*ized as
classifying spaces of subsystems of p-power index. The index of FT in F can th*
*en
be defined to be the index of T in p(F), or equivalently the covering degree o*
*f the
covering space.
The main step in doing this is to construct a fusion mapping triple ( p(F), `*
*, ) for
Fq, where ` is the canonical surjection of S onto p(F). This construction par*
*allels
very closely the construction in Proposition 2.4 of a functor from Lq to B( p(F*
*)), when
L is a linking system associated to F. In fact, we could in principal state and*
* prove
the two results simultaneously, but the extra terminology which that would requ*
*ire
seemed to add more complications than would be saved by combining the two.
The following lemma provides a very general, inductive tool for constructing *
*explicit
fusion mapping triples.
Lemma 4.1. Fix a saturated fusion system F over a p-group S. Let H0 be a set of
F-quasicentric subgroups of S which is closed under F-conjugacy and overgroups.*
* Let
P be an F-conjugacy class of F-quasicentric subgroups maximal among those not in
H0, set H = H0 [ P, and let FH0 FH Fq be the full subcategories with these
30 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
objects. Fix a group and a homomorphism ` :S ___! , and let
0: Mor (FH0 ) -----! Sub ( )
be such that ( , `, 0) is a fusion mapping triple for FH0 . Let P 2 P be fully*
* normalized
in F, and fix a homomorphism
P :Aut F(P ) ------! N (`(CS(P )))=`(CS(P ))
such that the following two conditions hold:
(+)x`(g)x-1 = `(ff(g)) for all g 2 P , ff 2 AutF (P ), and x 2 P(ff).
(*)For all P Q S such that P C Q and Q is fully normalized in NF (P ), and *
*for
all ff 2 AutF (P ) and fi 2 AutF (Q) such that ff = fi|P, P(ff) 0(fi).
Then there is a unique extension of 0 to a fusion mapping triple ( , `, ) for*
* FH such
that (ff) = P(ff) for all ff 2 AutF (P ).
Proof.Note that (+) is just point (v) of Lemma 3.7 applied to the subgroup P , *
*while
(*) is just point (vi) applied to restrictions to P . So both of these conditi*
*ons are
necessary if we want to be able to extend 0 and P to a fusion mapping triple *
*for
FH .
The uniqueness of the extension is an immediate consequence of Alperin's fusi*
*on
theorem, in the form of Theorem 1.5(a). The proof of existence is almost identi*
*cal to
the proof of Lemma 2.3, so we just sketch it here briefly.
We first show that we can replace (*) by the following (a priori stronger) st*
*atement:
(**)for all Q, Q0 S which strictly contain P , and for all fi 2 Hom F (Q, Q0)*
* and
ff 2 AutF (P ) such that ff = fi|P, P(ff) 0(fi).
It suffices to show this when P is normal in Q and Q0, since otherwise we can r*
*eplace
Q and Q0by NQ(P ) and NQ0(P ). In this case, fi 2 Hom NF(P)(Q, Q0), and by Theo*
*rem
1.5(a) (Alperin's fusion theorem), it is a composite of restrictions of automor*
*phisms of
subgroups fully normalized in NF (P ). So it suffices to prove (**) when fi is *
*such an
automorphism, and this is what is assumed in (*).
Now fix any morphism ' 2 Hom F(P1, Q) which lies in FH but not in FH0 ; thus *
*P1 2
P. Set P2 = '(P1) Q, and let_'02 IsoF(P1, P2) be the "restriction" of '. By L*
*emma
1.3, there are isomorphisms 'i 2 IsoFq(NS(Pi), Ni), for some Ni NS(P ) contai*
*ning_
P , which restrict to isomorphisms 'i 2 IsoFq(Pi, P ). Fix elements xi 2 0(' *
*i). Set
_ = '2 O'0O'-112 AutFq(P ). Thus '0= '-12O_ O'1, and we define
(') = ('0) = x-12. P(_).x1.
_
This is independent of the choice of xi, since 0(' i) `(CS(P )).xi by axioms*
* (i) and
(ii) in the definition of a fusion mapping triple. It is independent of the cho*
*ice of '1
and '2 by the same argument as was used in the proof of Lemma 2.3 (and this is *
*where
we need point (**)). Conditions (i)-(iv) are easily checked. For example, (iv) *
*_ the
condition that x`(g)x-1 = `(ff(g)) whenever g 2 P1, ' 2 Hom F(P1, P2), and x 2 *
* (')
_ holds when ' can be extended to a larger subgroup since ( , `, 0) is already*
* a
fusion mapping triple, holds for ' 2 Aut F(P ) by (+), and thus holds in the ge*
*neral
case since (') was defined via a composition of such morphisms. Thus ( , `, )*
* is a
fusion mapping triple on FH .
EXTENSIONS OF P-LOCAL FINITE GROUPS 31
The construction of a fusion mapping triple to p(F) in the following lemma i*
*s a
first application of Lemma 4.1. Another application will be given in the next s*
*ection.
Lemma 4.2. Let F be a saturated fusion system over a p-group S, and let
` :S ------! p(F) = S=OpF(S)
be the projection. Then there is a fusion mapping triple ( p(F), `, ) on Fq.
Proof.The function will be constructed inductively, using Lemma 4.1. Let H0
Ob (Fq) be a subset (possibly empty) which is closed under F-conjugacy and over-
groups. Let P be an F-conjugacy class of F-quasicentric subgroups maximal among
those not in H0, set H = H0 [ P, and let FH0 FH Fq be the full subcate-
gories with these objects. Assume we have already constructed a fusion mapping *
*triple
( p(F), `, 0) for FH0 .
We recall the notation of Lemma 2.2. If G is any finite group, and S 2 Sylp(G*
*), then
fi ff
OpG(S) def=[g, x] fig 2 P S, x 2 NG(P ) of order prime to p.
By Lemma 2.2, OpG(S) = S \ Op(G), and hence G=Op(G) ~=S=OpG(S).
Fix P 2 P which is fully normalized in F. Let N0 be the subgroup generated by
commutators [g, x] for g 2 NS(P ) and x 2 NAutF(P)(NS(P )) of order prime to p.*
* Then
AutS(P ) 2 Sylp(Aut F(P )) and AutN0(P ) = OpAutF(P)(Aut S(P )), and by Lemma 2*
*.2,
ffip ffi
AutF (P ) O (Aut F(P )) ~=AutS(P )=AutN0(P ) ~=NS(P ) .
Also, N0 OpF(S), and so the inclusion of NS(P ) into S induces a homomorphism
P :Aut F(P ) ----i Aut F(P )=Op(Aut F(P ))-----!N p(F)(`(CS(P )))=`(CS(P )).
~=NS(P)= ~=NS(CS(P).S0)=(CS(P).S0)
Here, we write S0 = OpF(S) for short; thus p(F) = S=S0 and `(CS(P )) = CS(P ).*
*S0=S0.
Point (+) in Lemma 4.1 holds by the construction of P. So it remains only to*
* prove
that condition (*) in Lemma 4.1 holds.
To see this, fix P Q S such that P C Q and Q is fully normalized in NF (P*
* ),
and fix ff 2 Aut Fq(P ) and fi 2 Aut Fq(Q) such that ff = fi|P. We must show t*
*hat
P(ff) 0(fi). Upon replacing ff by ffk and fi by fik for some appropriate k*
* 1
(mod p), we can assume that both automorphisms have order a power of p. Since Q*
* is
fully normalized,_AutNS(P)(Q) is a Sylow subgroup of AutNF(P)(Q); and hence the*
*re_are
automorphisms_fl2_AutF (Q) and fl 2 AutF (P ) of order prime to p such that fl *
*= fl|P
and flfifl-1=_cg|Q for some g 2 NS(Q) \ NS(P ). Then flfffl-1 = cg|P. Also, 1 2*
* P(fl)
and 1 2 0(fl), since both automorphisms have order prime to p. So
0(fi) = (cg|Q) = g.`(CS(Q)) g.`(CS(P )) = P(cg|P) = P(ff).
Thus, by Lemma 2.3, we can extend 0 to a fusion mapping triple on FH . Upon
continuing this procedure, we obtain a fusion mapping triple defined on all of *
*Fq.
We now apply Lemma 4.2 to classify fusion subsystems of p-power index. Recall
that by definition, if F is a saturated fusion system over S and F0 F is a fu*
*sion
subsystem over S0 S, then F has p-power index if and only if S0 OpF(S), and
AutF0(P ) Op(Aut F(P )) for all P S0.
Theorem 4.3. Fix a saturated fusion system F over a p-group S. Then for each
subgroup T S containing OpF(S), there is a unique saturated fusion system FT *
* F
over T with p-power index. For each such T , FT has the properties:
32 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
(a)a subgroup P T is FT-quasicentric if and only if it is F-quasicentric; and
(b)for each pair P, Q T of F-quasicentric subgroups,
fi p
Hom FT(P, Q) = ' 2 Hom F(P, Q) fi (') \ (T=OF (S)) 6= ; .
Here,
: Mor (Fq) ------! Sub ( p(F)) = Sub (S=OpF(S))
is the map of Lemma 4.2.
Proof.Let FT F be the fusion system over T defined on F-quasicentric subgroups
by the formula in (b), and then extended to arbitrary subgroups by taking restr*
*ictions
and composites. (This is the fusion system denoted FT=OpF(S)in Proposition 3.8,*
* but
we simplify the notation here.) By Proposition 3.8(a,b) (applied with = p(F)*
* and
H = T=OpF(S)), FT is saturated, a subgroup P T is FT-quasicentric if and only*
* if it
is F-quasicentric, and AutFT (P ) Op(Aut F(P )) for all P T .
Now let F0T F be another saturated subsystem over the same subgroup T which
also has p-power index. We claim that F0T= FT, and thus that FT is the unique
subsystem with these properties. By assumption, for each P T , Aut FT(P ) a*
*nd
AutF0T(P ) both contain Op(Aut F(P )), and hence each is generated by Op(Aut F(*
*P ))
and any one of its Sylow p-subgroups. So if P is fully normalized in FT and F0T*
*both,
then
Aut FT(P ) = = AutF0T(P ). (1)
In particular, Aut FT(T ) = Aut F0T(T ). Set pk = |T |, fix 0 m < k, and a*
*ssume
inductively that Hom FT(P, Q) = Hom F0T(P, Q) for all P, Q T of order > pm . *
* By
Alperin's fusion theorem for saturated fusion systems (Theorem 1.5(a)), if |P |*
* = pm ,
|Q| pm , and P 6= Q, then all morphisms in Hom FT (P, Q) and Hom F0T(P, Q) a*
*re
composites of restrictions of morphisms between subgroups of order > pm , and h*
*ence
Hom FT(P, Q) = Hom F0T(P, Q) by the induction hypothesis. In particular, two s*
*ub-
groups of order pm are FT-conjugate if and only if they are F0T-conjugate. So f*
*or any
P T of order pm , P is fully normalized in FT if and only if it is fully norm*
*alized
in F0T. In either case, AutFT (P ) = AutF0T(P ): by (1) if P is fully normalize*
*d, and by
Alperin's fusion theorem again (and the induction hypothesis) if it is not.
We can now define Op(F) as the minimal fusion subsystem of F of p-power index:
the unique fusion subsystem over OpF(S) of p-power index. The next theorem will*
* show
that when F has an associated linking system L, then Op(F) has an associated li*
*nking
system Op(L), and that |Op(L)|^pis the universal cover of |L|^p.
Theorem 4.4. Fix a p-local finite group (S, F, L). Then for each subgroup T S
containing OpF(S), there is a unique p-local finite subgroup (T, FT, LT) such t*
*hat FT has
p-power index in F, and such that LqT= ss-1(FqT) where ss is the usual projecti*
*on of Lq
onto Fq. Furthermore, |LT| is homotopy equivalent, via the inclusion of |LqT| '*
* |LT|
into |Lq| ' |L|, to a covering space of |L| of degree [S : T ]. Hence the clas*
*sifying
space |LT|^pof (T, FT, LT) is homotopy equivalent to the covering space of |L|^*
*pwith
fundamental group T=OpF(S).
Proof.Assume a compatible set of inclusions {'QP} has been chosen for Lq. By Pr*
*oposi-
tion 2.4, there is a functor ~: Lq ___! B( p(F)) which sends inclusions to the*
* identity,
and such that ~(ffiS(g)) = g for all g 2 S. Hence by Theorem 3.9, (T, FT, LT) *
*is a
EXTENSIONS OF P-LOCAL FINITE GROUPS 33
p-local finite group, and |LT| is a covering space of |L|. Also, if we write L*
*1 for the
linking system over OpF(S), then the fibration sequences
|L1| ___! |L| ___! B p(F) and |L1| ___! |LT| ___! B(T=OpF(S))
are still fibration sequences after p-completion [BK , II.5.2(iv)], and hence |*
*LT|^pis the
covering space of |L|^pwith fundamental group T=OpF(S). The uniqueness follows *
*from
Theorem 4.3.
Thus there is a bijective correspondence between fusion subsystems of (S, F),*
* or
p-local finite subgroups of (S, F, L), of p-power index, and subgroups of S=OpF*
*(S) ~=
ss1(|L|^p). The classifying spaces of the p-local finite subgroups of (S, F, L)*
* of p-power
index are (up to homotopy) just the covering spaces of the classifying space of*
* (S, F, L).
4.2___Extensions_of_p-power_index__ We next consider the opposite problem: how
to construct extensions of p-power index of a given p-local finite group. In th*
*e course
of this construction, we will see that the linking system really is needed to c*
*onstruct
an extension of the fusion system. The following definition will be useful.
Definition 4.5. Fix a saturated fusion system F over a p-group S. An automorphi*
*sm
ff 2 Aut(S) is fusion preserving if it normalizes F; i.e., if it induces an aut*
*omorphism
of the category F by sending P to ff(P ) and ' 2 Mor (F) to ff'ff-1 2 Mor (F). *
* Let
Autfus(S, F) Aut(S) denote the group of all fusion preserving automorphisms, *
*and
set
Out fus(S, F) = Autfus(S, F)= AutF(S).
We first describe the algebraic data needed to determine extensions of p-powe*
*r index.
Fix a p-local finite group (S, F, L), let Lq be the associated quasicentric lin*
*king system,
and let {'QP} be a compatible set of inclusions. Then for any g 2 S, g acts on *
*the set
Mor (Lq) by composing on the left or right with ffiS(g) and its restrictions. T*
*hus for any
' 2 Mor Lq(P, Q), we set
g' = ffiQ,gQg-1(g) O' 2 Mor Lq(P, gQg-1)
and
'g = ' Offig-1Pg,P(g) 2 Mor Lq(g-1P g, Q).
This defines natural left and right actions of S on the set Mor (Lq). The resu*
*lting
conjugation action ' 7! g'g-1 extends to an action on the category, where g sen*
*ds
an object P to gP g-1. The functor ss :Lq ___! Fq is equivariant with respect *
*to the
conjugation action of S on Lq and the action of Inn(S) Autfus(S, F) on F.
If (S0, F0, L0) is contained in (S, F, L) with p-power index, and S0 C S, the*
*n the S
action on L clearly restricts to an S-action on L0. The following theorem provi*
*des a
converse to this. Given a p-local finite group (S0, F0, L0), an extension S of *
*S0, and an
S-action on L which satisfies certain obvious compatibility conditions, this da*
*ta always
determines a p-local finite group which contains (S0, F0, L0) with p-power inde*
*x.
Theorem 4.6. Fix a p-local finite group (S0, F0, L0), and assume that a compati*
*ble
set of inclusions {'QP} has been chosen for L0. Fix a p-group S such that S0 C *
*S and
AutS(S0) Autfus(S0, F0), and an action of S on L0 which:
(a)extends the conjugation action of S0 on L0;
(b)makes the canonical monomorphism ffiS0:S0 ---! Aut L0(S0) S-equivariant;
34 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
(c)makes the projection ss :L0 ---! F0 S-equivariant with respect to the Aut S(*
*S0)-
action on F0; and
(d)sends inclusion morphisms in L0 to inclusion morphisms.
Then there is a p-local finite group (S, F, L) such that F F0, Lq L0, the c*
*onju-
gation action of S on Lq restricts to the given S-action on L0, and (S0, F0, L0*
*) is a
subgroup of p-power index in (S, F, L).
Proof.Set
H0 = Ob (L0) = {P S0| P is F0-centric} and H = {P S | P \ S0 2 H0}.
To simplify notation, for any P S, we write P0 = P \ S0. For g 2 S and ' 2
Mor L0(P, Q), we write g'g-1 2 Mor L0(gP g-1, gQg-1) for the given action of g *
*on '.
By (a), when g 2 S0, this agrees with the morphism g'g-1 already defined.
Step 1: We first define categories L1 L0 and F1 F0, where Ob (F1) = Ob (F*
*0)
and Ob (L1) = H0. Set
ffi
Mor(L1) = S xS0Mor (L0) = S x Mor(L0) ~ ,
where (gg0, ') ~ (g, g0') for g 2 S, g0 2 S0, and ' 2 Mor (L0). If ' 2 Mor L0(P*
*, Q),
then [[g, ']] 2 Mor L1(P, gQg-1) denotes the equivalence class of the pair (g, *
*'). Com-
position is defined by
[[g, ']] O[[h, _]] = [[gh, h-1'h O_]].
Note that if ' 2 Mor L0(P, Q), then h-1'h 2 Mor L0(h-1P h, h-1Qh). To show that*
* this
is well defined, we note that for all g, h 2 S, g0, h0 2 S0, and ', _ 2 Mor (L0*
*) with
appropriate domain and range,
[[gg0, ']] O[[hh0, _]] = [[gg0hh0, h-10(h-1'h)h0 O_]] = [[gh.(h-1g0h), (h-1'h*
*) Oh0_]]
= [[gh, (h-1g0h).(h-1'h) Oh0_]] = [[gh, h-1(g0')h Oh0_]] = [[g, g0']] O[*
*[h, h0_]] .
Here, the second to last equality follows from assumptions (b) and (d).
Let F1 be the smallest fusion system over S0 which contains F0 and AutS(S0). *
*By
assumption, AutS(S0) Autfus(S0, F0). Thus for each g 2 S, cg normalizes the f*
*usion
system F0: for each ' 2 Mor (F0) there is '02 Mor (F0) such that 'Ocg = cgO'0. *
*Hence
each morphism in F1 has the form cg O' for some g 2 S and ' 2 Aut(F0). Define
ssL1: L1 -----! F1
by sending ssL1([[g, ']]) = cg Oss0('), where ss0 denotes the natural projectio*
*n from L0
to F0. This is a functor by (c).
For all P, Q 2 H0, define
bffiP,Q:NS(P, Q) ------! Mor L1(P, Q)
-1Qg
by setting bffiP,Q(g) = [[g, 'gP ]]. This extends the canonical monomorphism *
*ffiP,Qdefined
from NS0(P, Q) to Mor L0(P, Q). To simplify the notation below, we sometimes wr*
*ite
bx= bffiP,Q(x) for x 2 NS(P, Q).
Step 2: We next construct categories L2 and F2, both of which have object sets*
* H,
and which contain L1 and the restriction of F1 to H0, respectively. Afterwards,*
* we let
F be the fusion system over S generated by F2 and restrictions of morphisms.
EXTENSIONS OF P-LOCAL FINITE GROUPS 35
Before doing this, we need to know that the following holds for each P, Q 2 H*
*0 and
each _ 2 Mor L1(P, Q):
8 x 2 NS(P ) there is at most oney 2 NS(Q) such thatbyO_ = _ Obx. (1)
Since _ is the composite of an isomorphism and an inclusion (and the claim clea*
*rly
holds if _ is an isomorphism), it suffices to prove this when P Q and _ is t*
*he
inclusion. We will show that we must have y = x in that case. By definition,
-1Qx Q Q
'QPObx= [[1, 'QP]] O[[x, IdP]] = [[x, 'xP ]] and byO'P = [[y, '*
*P]]
(since conjugation sends inclusions to inclusions). So if byO'QP= 'QPObx, then *
*there exists
-1Qx
g0 2 S0 such that y = xg0 and 'QP= g-10'xP , and thus
-1Qx -1 *
*-1
ffiP,Q(1) = 'QP= g-10'xP = ffix-1Qx,Q(g0 ) OffiP,x-1Qx(1) = ffiP,Q(g*
*0 ).
Hence g0 = 1 by the injectivity of ffiP,Q in Proposition 1.13; and so byO'QP= '*
*QPObxonly
if x = y 2 NS(Q).
Now let L2 be the category with Ob (L2) = H, and where for all P, Q 2 H,
fi
Mor L2(P, Q) = _ 2 Mor L1(P0, Q0) fi8 x 2 P 9 y 2 Q such that_ Obx= byO_ .
Let
fbfiP,Q:NS(P, Q)------!MorL2(P, Q)
NS(P0,Q0) MorL1(P0,Q0)
be the restriction of bffiP0,Q0. Let F2 be the category with Ob (F2) = H, and w*
*here
fi
Mor F2(P, Q) = ' 2 Hom (P, Q) fi9 _ 2 MorL2(P, Q) such that_ Obx= ['(x)O_ 8 x*
* 2 P .
Let ss :L2 ___! F2 be the functor which sends _ 2 Mor L2(P, Q) to the homomorp*
*hism
ss(_)(x) = y if _ Obx= byO_ (uniquely defined by (1)). Let F be the fusion syst*
*em over
S generated by F2 and restriction of homomorphisms.
Set = S=S0. Let
b`:L2 --! B( )
be the functor defined by setting b`([[g, ']]) = gS0. In particular, `-1(1) |H*
*0 = L0.
Step 3: We next show that each P 2 H is F-conjugate to a subgroup P 0such that
P00is fully normalized in F0. Moreover, we show that P 0can be chosen so that *
*the
following holds:
8 g 2 S such thatgP0g-1 is F0-conjugate to P0, g.S0 \ NS(P00) 6= ;.(2)
To see this, let Pfnbe the set of all S0-conjugacy classes [P00] of subgroups*
* P00 S0
which are F0-conjugate to P0 and fully normalized in F0. (If P00is fully normal*
*ized in
F0, then so is every subgroup in [P00].) Let S0 S be the subset of elements *
*g 2 S
such that gP0g-1 is F0-conjugate to P0. In particular, S0 NS(P0) P . Since *
*each
g 2 S acts on F0 _ two subgroups Q, Q0 S0 are F0-conjugate if and only if gQg-1
and gQ0g-1 are F0-conjugate _ S0 is a subgroup of S.
For all g 2 S0 and [P00] 2 Pfn, gP00g-1 is F0-conjugate to gP0g-1 and hence t*
*o P0,
and is fully normalized since g normalizes S0. Thus S0=S0 acts on Pfn, and this*
* set has
order prime to p by Proposition 1.16. So we can thus choose a subgroup P00 S0 *
*such
that [P00] 2 Pfnand is fixed by S0. In particular, P00is fully normalized in F0*
*. Also, for
each g 2 S0, some element of g.S0 normalizes P00(since [P00] = [gP00g-1]) _ and*
* this
proves (2).
36 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
Now consider the set Rep F0(P0, S0) = Hom F0(P0, S0)=Inn(S0). Since P S0, *
*the
group P=P0 acts on this set by conjugation (i.e., gP0 2 P=P0 acts on ['], for '*
* 2
Hom F0(P0, S0), by sending it to [cg'c-1g]). In particular, since the S0-conjug*
*acy class
[P00] is invariant under conjugation by P=P0, this group leaves invariant the s*
*ubset X
RepF0(P0, S0) of all conjugacy classes ['] of homomorphisms such that [Im (')] *
*= [P00].
Fix any ' 2 IsoF0(P0, P00) (recall that subgroups in Pfnare F0-conjugate to P0)*
*. Every
element of X has the form [ff'] for some ff 2fAutF0(P00),fandi[ff'] = [fi'] if *
*and only if
fffi-1 2 AutS0(P00). Thus |X| = | AutF0(P00)| | AutS0(P00)|, and is prime to p *
*since P00is
fully normalized in F0. We can thus choose '0 2 Hom F0(P0, S0) such that '0(P0)*
* = P00
and ['0] is invariant under the P=P0-action.
Fix _ 2 IsoL0(P0, P00) such that ss(_) = '0. Since ['0] is P=P0-invariant, f*
*or each
x 2 P , there is some y 2 x.S0 such that cy O'0 = '0 Ocx. By (A)q (and since _ *
*is an
isomorphism), there is y02 y.CS0(P00) such that by0O_ = _ Obxin L1. This elemen*
*t y0is
unique by (1); and upon setting '(x) = y0 we get a homomorphism ' 2 Hom F(P, S)
which extends '0. Set P 0= '(P ); then P 0is F-conjugate to P and P00= P 0\ S0.
Step 4: In Step 5, we will prove that F is saturated, using [5A1 , Theorem 2.*
*2].
Before that theorem can be applied, a certain technical condition must be check*
*ed.
Assume that P is F-centric, but not in H. By Step 3, P is F-conjugate to some
P 0such that P00is fully normalized in F0. Thus P00is fully centralized in F0 a*
*nd not
F0-centric, which implies that CS0(P00) P00. Then P 0acts on CS0(P00).P00=P00*
*with fixed
subgroup QP00=P006= 1 for some Q CS0(P00), and [Q, P 0] P00since P 0=P00cen*
*tralizes
QP00=P00. Hence Q P00, and Q NS(P 0) since [Q, P 0] P 0. For any x 2 Qr*
* P00,
[cx] 6= 1 2 Out(P 0) (CS(P 0) P 0since P is F-centric), but cx induces the id*
*entity on
P00(since Q CS0(P00)) and on P 0=P00(since x 2 S0). Hence [cx] 2 Op(Out F(P 0*
*)) by
Lemma 1.15. This shows that
P F-centric,P =2H =) 9 P 0F-conjugate to P ,OutS(P 0)\Op(Out F(P 0)) 6= 1.(3)
Step 5: We next show that F is saturated, and also (since it will be needed in*
* the
proof of (II)) that axiom (A)q holds for L2. By [5A1 , Theorem 2.2] (the strong*
*er form
of Theorem 1.5(b)), it suffices to prove that the subgroups in H satisfy the ax*
*ioms for
saturation. Note in particular that condition (*) in [5A1 , Theorem 2.2] is pr*
*ecisely
what is shown in (3).
Proof of (I): Fix a subgroup P 2 H which is fully normalized in F. Let S0=S0
S=S0 be the stabilizer of the F0-conjugacy class of P0. By Step 3, P is F-conju*
*gate to
a subgroup P 0such that P00is fully normalized in F0; and such that for each g *
*2 S0,
some element of g.S0 normalizes P00(see (2)). Hence there are short exact seque*
*nces
1 ----! AutL0(P00) -----! Aut L1(P00) -----! S0=S0 --! 1
1 ----! NS0(P00) -----! NS(P00) -----! S0=S0 --! 1.
We consider P 0 NS(P00) as subgroups of AutL1(P00) via bffiP00,P00. Then
[Aut L1(P00) : NS(P00)] = [Aut L0(P00) : NS0(P00)]
is prime to p since P00is fully normalized in F0, and hence NS(P00) 2 Sylp(Aut *
*L1(P00)).
Fix _ 2 AutL1(P00) such that _-1NS(P00)_ contains a Sylow p-subgroup of the gro*
*up
NAutL1(P00)(P 0) (in particular, P 0 _-1NS(P00)_), and set P 00= _P 0_-1 NS(*
*P00).
Then _ 2 IsoL2(P 0, P 00) by definition of L2. In particular, P 00is F-conjugat*
*e to P , and
EXTENSIONS OF P-LOCAL FINITE GROUPS 37
P000= P00. Also, NS(P00) contains a Sylow p-subgroup of NAutL1(P00)(P 00),
AutL2(P 00) = NAutL1(P00)(P 00) and NS(P 00) = NNS(P00)(P 00),
and it follows that NS(P 00) 2 Sylp(Aut L2(P 00)).
Now, AutL2(P ) ~=Aut L2(P 00) since they are F-conjugate, and |NS(P )| |NS(*
*P 00)|
since P is fully normalized. Thus NS(P ) 2 Sylp(Aut L2(P )), and hence Aut S(P*
* ) 2
Sylp(Aut F(P )). Also, NS(P ) contains the kernel of the projection from AutL2(*
*P ) to
AutF (P ); i.e., CS(P ) is isomorphic to this kernel. For all Q which is F-conj*
*ugate to
P , CS(Q) is isomorphic to a subgroup of the same kernel, so |CS(Q)| |CS(P )|*
*, and
thus P is fully centralized in F.
Proof of (A)q for L2: It is clear from the construction that for any P, Q 2 H,
CS(P ) acts freely on Mor L2(P, Q) via bffiP,P. So it remains to show that when*
* P is fully
centralized in F, then for all _, _0 2 Mor L2(P, Q) such that ss(_) = ss(_0), t*
*here is
some x 2 CS(P ) such that _0 = _ Obx. Since every morphism in L2 is the composi*
*te
of an isomorphism followed by an inclusion, it suffices to show this when _ and*
* _0
are isomorphisms. But in this case, _-1_0 2 AutL2(P ) lies in the kernel of th*
*e map
to Aut F(P ). We have just seen, in the proof of (I), that this implies there *
*is some
x 2 CS(P ) such that bx= _-1_0, so _0= _ Obx, and this is what we wanted to pro*
*ve.
Proof of (II): Fix ' 2 Hom F (P, S), where '(P ) is fully centralized in F. *
* Set
P 0= '(P ). By definition of F2 F and of L2, there is some _ 2 IsoL2(P, P 0)
IsoL1(P0, P00) such that _ Obg= d'(g)O_ for all g 2 P . Upon replacing _ by bxO*
*_ for
some appropriate x 2 S (and replacing ' by cx O' and P 0by xP 0x-1), we can ass*
*ume
that _ 2 IsoL0(P0, P00) and '|P0 2 Hom F0(P0, S0).
Consider the subgroups
N' = {x 2 NS(P ) | 'cx'-1 2 AutS(P 0)}
fi -1 0
N0 = N'|P0\ N' = x 2 N' \ S0fi('cx' )|P0 2 AutS0(P0) .
We will see shortly that N0 = N' \ S0. By (II) applied to the saturated fusion *
*system
_ 0 _ *
* 0
F0, there is '0 2 Hom F0(N , S) which extends '|P0, and it_lifts to _ 2 Hom L0(*
*N , S)._
By (A)_(applied to L0), there is z 2 Z(P0) such that _ = (_ |P0) Obz. Upon repl*
*acing _
_ _ _
by _ Obz(and '0 by '0 Ocz), we can assume that _ = _|P0.
For any x 2 N' \ S0, 'cx'-1 = cy 2 Aut F(P 0) for some y 2 NS(P 0). Hence by
(A)q (for L2), _bx_-1 = cyzfor some unique z 2 CS(P 0). Thus cyz2 AutL0(P00), a*
*nd by
definition of the distinguished monomorphisms for L1, this is possible only if *
*yz 2 S0.
Thus 'cx'-1 = cyzwhere yz 2 NS0(P 0), and so x 2 N0. This shows that N0 = N' \S*
*0.
_ -1
Define ' 2 Hom (N', S) by the relation '(x) = y if _bx_ = by. In particular*
*, this
implies_that [yx-1= _O(x_x-1)-1 is a morphism in L0, and hence that_yx-1 2 S0. *
*Also,
'|P = ' by the original assumption on__. It remains_to show that ' 2 Hom F(N', *
*S).
To do_this, it suffices to show that _ O bx= byO_ in Mor L2(N0, S) for all x, *
*where
y = '(x). Equivalently, we must show that
_ _
_ = [yx-1O(x_ x-1). (4)
Since yx-1 2 S0, both sides in (4) are in L0, and they are equal after restrict*
*ion to P0.
Hence they are equal as morphisms defined on N' \ S0 by [5A1 , Lemma 3.9], and *
*this
finishes the proof.
38 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
Step 6: We next check that F0 has p-power index in F. For any P S and any
ff 2 Aut F(P ) of order prime to P , ff induces the identity on P=P0 by constru*
*ction,
and hence x-1ff(x) 2 S0 for all x 2 P . This shows that S0 OpF(S) (see Defini*
*tion
2.1). Also, by construction, for all P S0, Aut F0(P ) is normal of p-power i*
*ndex in
AutF (P ), and thus contains Op(Aut F(P )). This proves that the fusion subsyst*
*em F0
has p-power index in F in the sense of Definition 3.1.
Step 7: It remains to construct a quasicentric linking system Lq which contai*
*ns
L2 as a full subcategory, and which is associated to F. Note first that the axi*
*oms of
Definition 1.9 are all satisfied by L2: axiom (A)q holds by Step 5, while axiom*
*s (B)q,
(C)q, and (D)q follow directly from the construction in Steps 1 and 2.
Let Lc2 L2 be the full subcategory whose objects are the set Hc H of subgr*
*oups
in H which are F-centric. We first construct a centric linking system L Lc2as*
*soci-
ated to F. For any set K of F-centric subgroups of S, let Oc(F) be the orbit ca*
*tegory
of F, let OK (F) Oc(F) be the full subcategory with object set K, and let ZKF*
*be
the functor P 7! Z(P ) on OK (F) which sends (see Definition 1.7 for more detai*
*l).
By [BLO2 , Proposition 3.1], when K is closed under F-conjugacy and overgroups*
*, the
obstruction to the existence of a linking system with object set K lies in lim-*
*3(ZKF), and
the obstruction to its uniqueness lies in lim-2(ZKF). Furthermore, by [BLO2 , *
*Proposi-
tion 3.2], if P is an F-conjugacy class of F-centric subgroups maximal among th*
*ose
not in K, and K0 = K [ P, then the inclusion0of functors induces an isomorphism
between the higher limits of ZKFand ZKF if certain groups *(Out F(P ); Z(P )) *
*vanish
for P 2 P. By (3), Op(Out F(P )) 6= 1 for any F0-centric subgroup P =2Hc, and h*
*ence
*(Out F(P ); Z(P )) = 0 for such P by [JMO , Proposition 6.1(ii)]. So these o*
*bstructions
all vanish, and there is a centric linking system L Lc2associated to F.
Now let Lq be the quasicentric linking system associated to (S, F, L). For e*
*ach
P 2 H = Ob (L2), P \ S0 is F0-centric by definition, hence is F-quasicentric by
Theorem 4.3(a), and thus P is also F-quasicentric. Also, H contains all F-cent*
*ric
F-radical subgroups by (3), and H is closed under F-conjugacy and overgroups (by
definition of F and H). Hence by [5A1 , Proposition 3.12], since Lc2is a full s*
*ubcategory
of L by construction, L2 is isomorphic to a full subcategory of Lq in a way whi*
*ch
preserves the projection functors and distinguished monomorphisms. So L0 can a*
*lso
be identified with a linking subsystem of Lq, and this finishes the proof.
We now prove a topological version of Theorem 4.6. By Theorem 4.4, if (S0, F0*
*, L0)
(S, F, L) is an inclusion of p-local finite groups of p-power index, where S0 C*
* S and
= S=S0, then there is a fibration sequence |L0|^p---! |L|^p---! B . So it is *
*natural
to ask whether the opposite is true: given a fibration sequence whose base is *
*the
classifying space of a finite p-group, and whose fiber is the classifying space*
* of a p-local
finite group, is the total space also the classifying space of a p-local finite*
* group? The
next proposition shows that this is, in fact, the case.
Before stating the proposition, we first define some categories which will be*
* needed in
its proof. Fix a space Y , a p-group S, and a map f :BS ---! Y . For P S, we *
*regard
BP as a subspace of BS; all of these subspaces contain the basepoint * 2 BS. We
define three categories in this situation, FS,f(Y ), LS,f(Y ), and MS,f(Y ), al*
*l of which
have as objects the subgroups of S. Of these, the first two are discrete catego*
*ries, while
MS,f(Y ) has a topology on its morphism sets. Morphisms in FS,f(Y ) are defined*
* by
setting
fi
Mor FS,f(Y()P, Q) = ' 2 Hom (P, Q) fif|BP ' f|BQ OB' ;
EXTENSIONS OF P-LOCAL FINITE GROUPS 39
we think of this as the fusion category of Y (with respect to S and f).
Next define
fi
Mor MS,f(Y()P, Q) = (', H) fi' 2 Hom (P, Q), H :BP x [0, t] ___! Y ,
t 0, H|BPx0 = f|BP , H|BPxt = f|BQ OB'.
Thus a morphism in MS,f(Y ) has the form (', H) where H is a Moore homotopy in
Y . Composition is defined by
(_, K) O(', H) = (_', (K O(B' x Id)).H),
where if H and K are homotopies parameterized by [0, t] and [0, s], respectivel*
*y, then
(K O(B' x Id)).H is the composite homotopy parameterized by [0, t + s].
Let P(Y ) be the category of Moore paths in Y , and let Res*:MS,f(Y ) ___! P*
*(Y )
be the functor which sends each object to f(*), and sends a morphism (', H) to *
*the
path obtained by restricting H to the basepoint * 2 BS. Define a map evfrom |P(*
*Y )|
to Y as follows. For any n-simplex n in |P(Y )|, indexed by a composable seque*
*nce
of paths OE1, . .,.OEn where OEiis defined on the interval [0, ti], let ev| n b*
*e the composite
~(t1,...,tn) OEn...OE1
n ------! [0, t1 + . .+.tn] ------! Y,
where ~(t0, . .,.tn) is the affine map which sends the i-th vertex to t1 + . .+*
*.ti. The
category MS,f(Y ) thus comes equipped with an "evaluation function"
|Res*| ev
eval:|MS,f(Y )| -----! |P(Y )| -----! Y.
Now set
MorLS,f(Y()P, Q) = ss0 Mor MS,f(Y()P, Q) .
We think of LS,f(Y ) as the linking category of Y . Also, for any set H of subg*
*roups of
S, we let LHS,f(Y ) and MHS,f(Y ) denote the full subcategories of LS,f(Y ) and*
* MS,f(Y )
with object set H. For more about the fusion and linking categories of a space*
*, see
[BLO2 , x7].
Theorem 4.7. Fix a p-local finite group (S0, F0, L0), a p-group , and a fibrat*
*ion
X --v-!B with fiber X0 ' |L0|^p. Then there is a p-local finite group (S, F, L*
*) such
that S0 C S, F0 F is a fusion subsystem of p-power index, S=S0 ~= , and X ' |*
*L|^p.
Proof.Let * denote the base point of B , and assume X0 = v-1(*). Fix a homo-
topy equivalence f :|L0|^p--! X0, regard BS0 as a subspace of |L0|, and set f0 =
f|BS0: BS0 --! X0, also regarded as a map to X. Let H0 be the set of F0-centric
subgroups of S0.
Step 1: By [BLO2 , Proposition 7.3],
F0 ~=FS0,f0(X0) and L0 ~=LH0S0,f0(X0).
We choose the inclusions 'QP2 Mor L0(P, Q) (for P Q in H0) to correspond to t*
*he
morphisms (inclQP, [c]) in LS0,f0(X0), where c is the constant homotopy f0|BP .*
* Set
F1 = FS0,f0(X) and L1 = LH0S0,f0(X),
where f0 is now being regarded as a map BS0 ---! X. The inclusion X0 X makes
F0 into a subcategory of F1 and L0 into a subcategory of L1.
For all P S0 which is fully centralized in F0, Map (BP, X0)f0|BP' |CL0(P )|*
*^pby
[BLO2 , Theorem 6.3], where CL0(P ) is a linking system over the centralizer C*
*S(P ).
40 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
Since P 2 H0 (i.e., P is F0-centric) if and only if it is fully centralized and*
* CS0(P ) =
Z(P ), this shows that
P 2 H0 () Map (BP, X0)f0|BP' BZ(P ). (1)
If P and P 0are F1-conjugate, and ' 2 IsoF1(P, P 0), then the homotopy between *
*f0|BP
and f0|BP0OB' as maps from BP to X induces, using the homotopy lifting property*
* for
the fibration v, a homotopy between wOf0|BP and f0|BP0OB' (as maps from BP to X*
*0),
'
where w :X0 ___! X0 is the homotopy equivalence induced by lifting some loop i*
*n B .
Since w is a homotopy equivalence and B' is a homeomorphism, this shows that the
mapping spaces Map (BP, X0)f0|BPand Map (BP 0, X0)f0|BP0are homotopy equivalent,
and hence (by (1)) that P 02 H0 if P 02 H0. Thus for all P, P 0 S0,
P F1-conjugate to P 0and P 2 H0 =) P 02 H0. (2)
Fix S 2 Sylp(Aut L1(S0)). We identify S0 as a subgroup of S via the distingui*
*shed
monomorphism ffiS0 from S0 to AutL0(S0) AutL1(S0).
Step 2: For all P S, and for Y = X0 or X, we define
fi
Map (BP, Y ) = f :BP ___! Y fif ' f0 OB', ' 2 Hom (P, S), '(P ) 2 H0 .
Using (2), we see that the fibration sequence X0 ---! X ---! B induces a fibra*
*tion
sequence of mapping spaces
Map (BP, X0) -----! Map (BP, X) -----! Map (BP, B )ct' B , (3)
where Map (BP, B )ctis the space of null homotopic maps, and the last equivalen*
*ce
is induced by evaluation at the basepoint. By (1), each connected component of
Map (BP, X0) is homotopy equivalent to BZ(P ), and hence the connected compo-
nents of Map (BP, X) are also aspherical.
For any morphism (', [H]) 2 Mor L1(P, Q), where ' 2 Hom F1(P, Q) and [H] is t*
*he
homotopy class of the path H in Map (BP, X) , restricting v OH to the basepoint*
* of
BP defines a loop in B , and thus an element of . This defines a map from Mor *
*(L1)
to which sends composites to products, and thus a functor
b`:L1 -----! B( ).
By construction (and the fibration sequence (3)), for any _ 2 Mor (L1), we have*
* _ 2
Mor (L0) if and only if b`(_) = 1.
Using the homotopy lifting property in (3), we see that b`restricts to a surj*
*ection
of AutL1(S0) onto , with kernel AutL0(S0). Since S0 2 Sylp(Aut L0(S0)), this *
*shows
that b`induces an isomorphism S=S0 ~= , where S is a Sylow p-subgroup of AutL1(*
*S0)
which contains S0. Hence for any _ 2 Mor (L1), there is g 2 S such that b`(_) =*
* b`(g);
and for any such g there is a unique morphism _0 2 Mor (L0) such that _ = ffi(g*
*) O_0
where ffi(g) 2 Mor (L1) denotes the appropriate restriction of g 2 AutL1(S0). I*
*n other
words,
Mor (L1) ~=S xS0Mor (L0). (4)
Step 3: The conjugation action of S on Mor(L0) Mor (L1) defines an action of*
* S on
L0, which satisfies the hypotheses of Theorem 4.6. (Note in particular that thi*
*s action
sends inclusions to inclusions, since they are assumed to be represented by con*
*stant
homotopies.) So we can now apply that theorem to construct a p-local finite gr*
*oup
(S, F, L) which contains (S0, F0, L0) with p-power index. Let Lq be the quasice*
*ntric
linking system associated to (S, F, L). By (4), the category L1 defined here is*
* equal
EXTENSIONS OF P-LOCAL FINITE GROUPS 41
to the category L1 defined in the proof of Theorem 4.6; i.e., the full subcateg*
*ory of Lq
with object set H0.
Let H be the set of subgroups P S such that P \ S0 2 H0, and let L2 Lq be*
* the
full subcategory with object set H. By Step 4 in the proof of Theorem 4.6, all *
*F-centric
F-radical subgroups of S lie in H, and so |L| ' |Lq| ' |L2| by Proposition 1.12*
*(a).
Also, |L1| is a deformation retract of |L2|, where the retraction is defined by*
* sending
P 2 H to P \ S0 2 H0 (and morphisms are sent to their restrictions, uniquely de*
*fined
by Proposition 1.12(b)). Thus |L| ' |L1|. So by Theorem 4.4, we have a homoto*
*py
fibration sequence |L0|^p___! |L1|^p___! B .
Step 4: It remains to construct a homotopy equivalence |L1|^p--! X, which exte*
*nds
to a homotopy equivalence between the fibration sequences. This is where we nee*
*d to
use the topological linking categories defined above. Set
M0 = MH0S0,f0(X0) and M1 = MH0S0,f0(X)
for short, and consider the following commutative diagram:
proj0 ^ eval0
|L0|^p ______'|M0|p _______! X0
| | |
| | |
# # #
proj1 ^ eval1
|L1|^p ______'|M1|p ________!X .
The vertical maps in the diagram are all inclusions. Also, X0 is p-complete by *
*definition
and X by [BK , II.5.2(iv)], so the evaluation maps defined above extend to the *
*p-
completed nerves |Mi|^p. The maps proj1and eval1both commute up to homotopy
with the projections to B . The projection maps proj0and proj1are both homotopy
equivalences: the connected components of the morphism spaces in Miare contract*
*ible
since the connected components of the fiber and total space in (3) are aspheric*
*al (see
Step 2).
We claim that eval0is homotopic to f Oproj0as maps to X0. By naturality, it s*
*uffices
to check this on the uncompleted nerve |M0|, and in the case where X0 = |L|^pand
f = Id. But in this case, the only real difference between the maps is that eva*
*l0sends
all vertices of |M0| to the base point of X0 = |L|^p, while proj0sends the vert*
*ex for a
subgroup P S to the corresponding vertex in |L|^p. So the maps are homotopic *
*via a
homotopy which sends vertices to the base point along the edges of |L| correspo*
*nding
to the inclusion morphisms. In particular, this shows that eval0is also a homo*
*topy
equivalence. We thus have an equivalence between the fibration sequences
|L0|^p__! |L1|^p___! Bw
| | ww
' | | w
# # w
|L0|^p____! X _____! B ,
and this proves that the two sequences are equivalent.
Recall (Definition 4.5) that for any saturated fusion system F over a p-group*
* S,
Autfus(S, F) denotes the group of all fusion preserving automorphisms of S. The*
* fol-
lowing corollary to Proposition 4.6 describes how "exotic" fusion systems could*
* poten-
tially arise as extensions of p-power index; we still do not know whether the s*
*ituation
it describes can occur.
42 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
Corollary 4.8. Fix a finite group G, with Sylow subgroup S 2 Sylp(G). Assume
_
there is an automorphism ff 2 Aut fus(S, FS(G)) of p-power order, which is not *
*the
restriction to S of an automorphism of G, and which moreover is not the restric*
*tion of
an automorphism of G0for any finite group G0with S 2 Sylp(G0) and FS(G0) = FS(G*
*).
Then there is a saturated fusion system (Sb, bF) (S, FS(G)), such that FS(G) *
*has p-
power index in bF, and such that bFis not the fusion system of any finite group.
Proof.By [BLO1 , Theorem E], together with [O1 , Theorem A] and [O2 , Theorem *
*A],
there is a short exact sequence
0 ---! lim-1(ZG) ------! Out typ(LcS(G)) ------! Out fus(S, FS(G)) ---! 0
OcS(G)
where Out fus(S, F) = Autfus(S, F)= AutF(S), where OcS(G) and ZG are the catego*
*ry
and functor of Definition 1.7(b), and where Out typ(LcS(G)) is the group of "is*
*otypi-
cal" automorphisms of LcS(G) modulo natural isomorphism (see the introduction_o*
*f _
[BLO1 ], or [BLO1 , Definition_3.2], for the definition). Let [ff] be the cl*
*ass of ffin
Outfus(S, FS(G)); then [ff] lifts to an automorphism ff of the linking system L*
*cS(G).
Upon replacing ff by some appropriate power, we can assume that it still has p-*
*power
order. We can also assume, upon replacing ff by another automorphism in the same
conjugacy class if necessary, that the ff-action on AutL(S) leaves invariant th*
*e subgroup
ffiS(S); i.e., that the action of ff on LcS(G) restricts to an action on S.
Set bS= So , where |x| = |ff| and x acts on S via ff. Then bShas an action*
* on
LcS(G) induced by the actions of S and of ff, and this action satisfies conditi*
*ons (a)-(d)
in Theorem 4.6. Let bF FS(G) be the saturated fusion system over bSconstructed*
* by
the theorem.
We claim that Fb is not the fusion system of any finite group. Assume otherw*
*ise:
assume Fb is the fusion system of a group bG. Since F has p-power index in Fb,*
* S
OpbF(Sb), and so S bS\ Op(Gb) by the hyperfocal subgroup theorem (Lemma 2.2).*
* Set
G0= S.Op(Gb). Then G0C bGsince S C bS; bG=G0~= bS=S0, and thus G0C bGhas p-power
index and S 2 Sylp(G0). By Theorem 4.3, there is a unique saturated fusion subs*
*ystem
over S of p-power index in bF, and thus F = FS(G0). Also, x 2 bS bGacts on G0v*
*ia an
automorphism_whose restriction to S is ff, and this contradicts the original as*
*sumption
about ff.
5.Fusion subsystems and extensions of index prime to p.
In this section, we classify all saturated fusion subsystems of index prime t*
*o p in a
given0saturated fusion system F, and show that there is a unique minimal subsys*
*tem
Op (F) of this type. More precisely, we show that there is a certain finite gro*
*up of order
prime to p associated to F, denoted below p0(F), and a one-to-one corresponden*
*ce
between subgroups T p0(F) and fusion subsystems FT of index prime to p in F.
The index of FT in F can then be defined to be the index of T in p0(F).
Conversely, we also describe extensions of saturated fusion systems of index *
*prime
to p. Once more the terminology requires motivation. Roughly speaking, an exten*
*sion
of index prime to p of a given saturated fusion system F is a saturated fusion *
*system
F0 over the same p-group S, but where the morphism set has been "extended" by an
EXTENSIONS OF P-LOCAL FINITE GROUPS 43
action of a group of automorphisms whose order is prime to p. Here again, a one*
*-to-one
correspondence statement is obtained, thus providing a full classification.
5.1___Subsystems_of_index_prime_to_p__ We first classify all saturated fusion *
*sub-
systems of index prime to p in a given saturated fusion system F. The fusion s*
*ub-
systems and associated linking systems will be constructed using Proposition 3.*
*8 and
Theorem 3.9, respectively. More precisely, Theorem 3.9 has already told us tha*
*t for
any p-local finite group (S, F, L), any surjection ` of ss1(|L|) onto a finite *
*p0-group ,
and any subgroup H , there is a p-local finite subgroup (S0, F0, L0) such th*
*at |L0|
is homotopy equivalent to the covering space of |L| with fundamental group `-1(*
*H).
What is new in this section is first, that we describe the "universal" p0-group*
* quotient
of ss1(|L|) as a certain quotient group of OutF (S); and second, we show that a*
*ll fusion
subsystems of index prime to p in F (in the sense of Definition 3.1) are obtain*
*ed in
this way.
When applying Proposition 3.8 to this situation, we need to consider fusion m*
*apping
triples ( , `, ) on Fq, where is finite of order prime to p. Since ` 2 Hom (*
*S, ), it
must then be the trivial homomorphism. In this case, conditions (i)-(iii) in De*
*finition
3.6 are equivalent to requiring that there is some functor b :Fq ___! B( ) suc*
*h that
(') = {b (')} for each ' 2 Mor (Fq) (and condition (iv) is redundant). So inst*
*ead
of explicitly constructing fusion mapping triples, we instead construct functor*
*s of this
form.
0
We start with some definitions. For a finite group G, one defines Op (G) to *
*be
the smallest normal subgroup of G of index prime to p, or equivalently the subg*
*roup
generated by elements of p-power order in G. These two definitions are not, in *
*general,
equivalent in the case of an infinite group, (the case G = Z being an obvious e*
*xample).
We do, however, need to deal with such subgroups here. The following definitio*
*n is
most suitable for our purposes, and it is a generalization of the finite case.
0
Definition 5.1. For any group G (possibly infinite), let Op (G) be the intersec*
*tion of
all normal subgroups in G of finite index prime to p.
In particular, under this definition, an epimorphism ff: G --i H with Ker(ff)
0 p0 p0
Op (G) induces an isomorphism G=O (G) ~=H=O (H). Thus by Proposition 2.6, for
any p-local finite group (S, F, L), the projections of |L| ' |Lq| onto |Fc| and*
* |Fq|
induce isomorphisms
ffip0 c ffip0 c q ffip0 q
ss1(|L|) O (ss1(|L|)) ~=ss1(|F |) O (ss1(|F |)) ~=ss1(|F |) O (ss1(|F *
*|)).
Fix a saturated fusion system F over a p-group S, and define
0 c
p0(F) = ss1(|Fc|)=Op (ss1(|F |)).
We will show that the natural functor
"Fc: Fc ------! B( p0(F))
induces a bijective correspondence between subgroups of p0(F) and fusion subsy*
*stems
of F of index prime to p.
Recall (Definition 3.3) that for any saturated fusion system F over a p-group*
* S,
0 p0
Op*(F) F is the smallest fusion subsystem which contains the groups O (Aut F*
*(P ))
44 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
for all P S; i.e., the smallest fusion subsystem which contains all automorph*
*isms in
F of p-power order. Define
fi *
*ff
Out0F(S) = ff 2 OutF (S) fiff|P 2 Mor Op0*(F)(P, S), some F-centric P S*
* .
0
Then Out0F(S) C OutF (S), since Op*(F) is normalized by AutF (S) (Lemma 3.4(a)).
Proposition 5.2. There is a unique functor
b`:Fc --------! B(Out F(S)= Out0F(S))
with the following properties:
(a)b`(ff) = ff (modulo Out0F(S)) for all ff 2 AutF (S).
0 c
(b)`b(') = 1 if ' 2 Mor (Op*(F) ). In particular, b`sends inclusion morphisms t*
*o the
identity.
Furthermore, there is an isomorphism
_ 0 ~=
`: p0(F) = ss1(|Fc|)=Op (ss1(|Fc|)) ------! Out F(S)= Out0F(S)
_
such that b`= B` O"cF.
Proof.By Lemma 3.4(c), each morphism in Fc factors as the composite of the rest*
*ric-
0 c
tion of a morphism in AutF (S) followed by a morphism in Op*(F) . If
' = '1 Off1|P = '2 Off2|P,
where ' 2 Hom F (P, Q), ffi 2 AutF (S), and 'i 2 Hom Op0*(F)c(ffi(P ), Q), then*
* we can
assume (after factoring out inclusions) that all of these are isomorphisms, and*
* hence
(ff2 Off-11)|P = '-12O'1 2 IsoOp0*(F)c(ff1(P ), ff2(P )).
Thus ff2 Off-112 Out0F(S); and so we can define
b`(') = [ff1] = [ff2] 2 OutF (S)= Out0F(S).
This shows that b`is well defined on morphisms (and sends each object in Fc to *
*the
unique object of B(Out F(S)= Out0F(S))). By Lemma 3.4(c) again, b`preserves com*
*po-
sitions, and hence is a well defined functor. It satisfies conditions (a) and (*
*b) above by
construction. The uniqueness of b`is clear.
It remains to prove the last statement. Since Out F(S)= Out0F(S) is finite o*
*f order
prime to p, ss1(|b`|) factors through a homomorphism
~`:ss1(|Fc|)=Op0(ss1(|Fc|)) ------! Out F(S)= Out0F(S).
The inclusion of B AutF(S) into |Fc| (as the subcomplex with one vertex S) indu*
*ces
a homomorphism
0 c
o :Out F(S) ------! ss1(|Fc|)=Op (ss1(|F |)).
0
Furthermore, o is surjective since F = (Lemma 3.4(b)), and si*
*nce
0
any automorphism in Op*(F) is a composite of restrictions of automorphisms of p*
*-power
order. By (a),_and since ` restricted to AutF (S) is the projection onto Out F(*
*S), the
composite `Oo is the projection of OutF (S) onto the quotient group OutF (S)=_O*
*ut0F(S).
Finally, Out 0F(S) Ker (o) by definition of Out 0F(S), and this shows that ` *
*is an
isomorphism.
EXTENSIONS OF P-LOCAL FINITE GROUPS 45
The following lemma shows that any fusion mapping triple on Fc can be extended
uniquely to Fq. This will allow us later to apply Proposition 3.8 in order to p*
*roduce
saturated fusion subsystems of index prime to p.
Lemma 5.3. Let F be a saturated fusion system over a p-group S, and let ( , `, *
* 0)
be a fusion mapping triple on Fc. Then there is a unique extension
: Mor (Fq) ___! Sub ( )
of 0, such that ( , `, ) is a fusion mapping triple on Fq.
Proof.We construct the extension one F-conjugacy class at a time. Thus, assume
has been defined on a set H0 of F-quasicentric subgroups of S which is a unio*
*n of
F-conjugacy classes, contains all F-centric subgroups, and is closed under over*
*groups.
Let P be maximal among F-conjugacy classes of F-quasicentric subgroups not in H*
*0;
we show that can be extended to H = H0 [ P.
Fix P 2_P which is fully normalized in F. For each ff 2 Aut F(P ), there is*
* an
extension ff2 AutF (P .CS(P )) (axiom (II)), and we define a map
P :Aut F(P ) ___! Sub (N (`(CS(P ))))
_ _
by P(ff) = (ff).`(CS(P )). By Definition 3.6 ((i) and (ii)) (ff) is a left *
*coset of
`(CS(P .CS(P ))), and by (iv), it is also a right coset (where the right coset *
*representa-
tive can be taken to be the same as the one representing the left coset); hence*
* P(ff)
is a left and a right_coset of `(CS(P )) (again with the same coset representat*
*ive on
both sides). If ff02 AutF (P .CS(P )) is any_other_extension of ff, then by [5A*
*1 , Lemma
3.8],_there is some_g 2 CS(P_) such that ff0= cg Off. Thus, by Definition 3.6 *
*again,
(ff0) = (cg Off) = `(g) (ff), and
_ 0 _ _ _ _
(ff) . `(CS(P )) = `(g) (ff) . `(CS(P )) = (ff)`(ff(g)) . `(CS(P )) = (ff) .*
* `(CS(P )),
_
and so the definition of P(ff) is independent of the choice of the extension f*
*f. This
shows that P is well defined.
Notice also that P clearly respects compositions, and since P(ff) = x.`(CS(*
*P )) =
`(CS(P )) . x, for some x 2 , we conclude that x 2 N (`(CS(P ))). Thus P indu*
*ces
a homomorphism
P :Aut F(P ) ___! N (`(CS(P )))=`(CS(P )).
We now make use of Lemma 4.1, which gives sufficient conditions to the existenc*
*e of
an extension of a fusion mapping triple.
*
* _
If ff_2 AutF (P ), and x 2 P(ff), then x = y .`(h) for some h 2 CS(P ) and y*
* 2 (ff),
where ffis an extension of ff to P . CS(P ). Hence for any g 2 P ,
_
x`(g)x-1 = y . `(hgh-1)y-1 = y`(g)y-1 = `(ff(g)) = `(ff(g)).
This shows that point (+) of Lemma 4.1 holds, and so it remains to check (*).
Assume P Q S, P C Q, and let ff 2 AutF (P ) and fi 2 AutF (Q) be such that
ff = fi|P. Then Q.CS(P ) Nffin the terminology of axiom (II), so ff extends t*
*o another
automorphism fl 2 AutF (Q.CS(P )), and P(ff) = (fl).`(CS(P )) by definition o*
*f P.
By [5A1 , Lemma 3.8] again, fl|Q = cg Ofi for some g 2 CS(P ). Hence, by Defini*
*tion
3.6, (fl) = (cg Ofi) = `(g) . (fi), and so
P(ff) = `(g) . (fi) . `(CS(P )) = (fi)`(fi(g)) . `(CS(P )) = (fi) . `(C*
*S(P )).
In particular, P(ff) (fi). This shows that point (*) of Lemma 4.1 is satisf*
*ied as
well, and thus, by the lemma, can be extended to a fusion mapping triple on F*
*H .
46 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
Recall that a fusion subsystem of index prime to p in a saturated fusion syst*
*em F
over S is a saturated subsystem F0 F over the same p-group S, such that AutF0*
*(P )
0
Op (Aut F(P )) for all P S. Equivalently, F0 F has index prime to p if and *
*only
0
if it is saturated and contains the subcategory Op*(F) of Definition 3.3. We ar*
*e now
ready to prove our main result about these subsystems.
Theorem 5.4. For any saturated fusion system F over a p-group S, there is a bij*
*ective
correspondence between subgroups
H p0(F) = OutF (S)= Out0F(S),
and saturated fusion subsystems FH of F over S of index prime to p in F. The
correspondence is given by associating to H the fusion system generated by b`-1*
*(B(H)),
where b`is the functor of Proposition 5.2.
0
Proof.Let F0 F be any saturated fusion subsystem over S which contains Op*(F).
Then Out0F(S) C OutF0(S), and one can set H = OutF0(S)= Out0F(S). We first show
that a morphism ' of Fc is in F0 if and only if b`(') 2 H. Clearly it suffices *
*to prove
this for isomorphisms in Fc.
Let P, Q S be F-centric, F-conjugate subgroups, and fix an isomorphism ' 2
IsoF(P, Q). By Lemma 3.4, we can write ' = _ O(ff|P), where ff 2 Aut F(S) and
_ 2 IsoOp0*(F)(ff(P ), Q). Then ' is in F0 if and only if ff|P is in F0. Also, *
*by definition
of b`(and of H), b`(') 2 H if and only if ff 2 AutF0(S). So it remains to prove*
* that
ff|P 2 Mor (F0) if and only if ff 2 AutF0(S).
If ff 2 Aut F0(S), then ff|P is also in F0 by definition of a fusion system. *
* So it
remains to prove the converse. Assume ff|P is in F0. The same argument as that *
*used
to prove Proposition 3.8(c) shows that ff(P ) is F0-centric, and hence fully ce*
*ntralized
in F0. Since ff|P extends to an (abstract) automorphism of S, axiom (II) implie*
*s that
it extends to some ff1 2 Hom F0(NS(P ), S). Since P is F-centric, [BLO2 , Prop*
*osition
A.8] applies to show that ff1 = (ff|NS(P))Ocg for some g 2 Z(P ), and thus that*
* ff|NS(P)2
Hom F0(NS(P ), S). Also, NS(P ) P whenever P S, and so we can continue this
process to show that ff 2 AutF0(S). This finishes the proof that F0 = b`-1(H).
Now fix a subgroup H Out F(S)= Out0F(S), and let FH be the smallest fusion
system over S which contains b`-1(B(H)). We must show that FH is a saturated fu*
*sion
subsystem of index prime to p in F. For F-centric subgroups P, Q S, Hom FH(P,*
* Q)
is the set of all morphisms ' in Hom F (P, Q) such that b`(') 2 H. In particul*
*ar,
0 p0
FH Op*(F), since morphisms in O* (F) are sent by b`to the identity element.
Define : Mor (Fc) ___! Sub ( p0(F)) by setting (') = {b`(')}; i.e., the im*
*age
consists of subsets with one element. Let ` 2 Hom (S, p0(F)) be the trivial *
*(and
unique) homomorphism. Then ( p0(F), `, ) is a fusion mapping triple on Fc. By
Lemma 5.3, this can be extended to a fusion mapping triple on Fq; and hence FH *
*is
saturated by Proposition 3.8.
By Theorem 1.5(a) (Alperin's fusion theorem), FH is the unique saturated fusi*
*on
subsystem of F with the property that a morphism ' 2 Hom F(P, Q) between F-cent*
*ric
subgroups of S lies in FH if and only if b`(') 2 H. This shows that the corresp*
*ondence
is indeed bijective.
The next theorem describes the relationship between subgroups of index prime *
*to p
in a p-local finite group (S, F, L) and certain covering spaces of |L|.
EXTENSIONS OF P-LOCAL FINITE GROUPS 47
Theorem 5.5. Fix a p-local finite group (S, F, L). Then for each subgroup H
OutF (S) containing Out 0F(S), there is a unique p-local finite subgroup (S, FH*
* , LH ),
such that FH has index prime to p in F, Out FH(S) = H, and LH = ss-1(FH ) (where
ss is the usual functor from L to F). Furthermore, |LH | is homotopy equivalen*
*t, via
its inclusion into |L|, to the covering space of |L| with fundamental group He:*
* where
eH ss1(|L|) is the subgroup such that ~`(He=Op0(ss1(|L|))) corresponds to H= O*
*ut0F(S)
under the isomorphism
0 c p0 c _` 0
ss1(|L|)=Op (ss1(|L|)) ~=ss1(|F |)=O (ss1(|F |)) ------!~ Out F(S)= OutF(S)
=
of Proposition 5.2.
Proof.By Theorem 3.9, applied to the composite functor
b` 0
L ---ss---!Fc ------! B(Out F(S)= OutF(S)),
(S, FH , LH ) is a p-local finite group, and |LH | is homotopy equivalent to th*
*e covering
space of |L| with fundamental group He defined above. The uniqueness follows f*
*rom
Theorem 5.4.
Theorem 5.4 shows, in particular, that0any saturated fusion system F over S c*
*ontains
a unique minimal saturated subsystem Op (F) of index prime to p: the subsystem
F0 F with OutF0(S) = Out0F(S). Furthermore, if F has an associated centric li*
*nking
0 *
*p0
system L, then Theorem 5.5 shows that Op (F) has an associated linking system O*
* (L),
0
whose geometric realization |Op (L)| is homotopy equivalent to the covering spa*
*ce of
0
|L| with fundamental group Op (ss1(|L|)).
5.2___Extensions_of_index_prime_to_p_ It remains to consider the opposite prob-
lem: describing the extensions of a given saturated fusion system of index prim*
*e to p.
As before, for a saturated fusion system F over a p-group S, Autfus(S, F) denot*
*es the
group of fusion preserving automorphisms of S (Definition 4.5). Theorem 5.7 be*
*low
states that each subgroup of Outfus(S, F) = Autfus(S, F)= AutF(S) of order prim*
*e to p
gives rise to an extension of F.
The following lemma will be needed to compare the obstructions to the existen*
*ce and
uniqueness of linking systems, in a fusion system and in a fusion subsystem of *
*index
prime to p. Recall the definition of the orbit category of a fusion system in D*
*efinition
1.7.
Lemma 5.6. Fix a saturated fusion system F over a p-group S, and let F0 F be
a saturated subsystem of index prime to p. Assume Out F0(S) C Out F(S), and set
ss = OutF (S)= OutF0(S). Then for any F : Oc(F) ___! Z(p)-mod, there is a natu*
*ral
action of ss on the higher limits of F |Oc(F0), and
h iss
lim-*(F ) ~= lim-*F |Oc(F0) .
Oc(F) Oc(F0)
Proof.Let Oc(F)-mod denote the categoryQof functors Oc(F)op ___! Ab . For any
F in Oc(F)-mod , Out F(S) acts on P2Ob(Oc(F))F (P ) by letting ff 2 Out F(S) *
*send
F (ff(P )) to F (P ) via the induced map ff*. This restricts to an action of ss*
* on ~(F ) def=
48 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
lim-(F |Oc(F0)); and by definition of inverse limits,
Oc(F0)
h iss ss
lim-(F ) ~= lim-(F |Oc(F0)) = ~(F ) .
Oc(F) Oc(F0)
Since F0 is a subsystem of index prime to p, Out F0(S) is a subgroup of Out 0*
*F(S).
Hence, by a slight abuse of notation, one has a functor b`:F ___! B(ss) given *
*as the
composite of the functor b`of Proposition 5.2 with projection to ss. For any Z*
*[ss]-
module M, regarded as a functor on B(ss), we let fi(M) denote the composite fun*
*ctor
fi c
M Ob`. Then Z[ss]-mod ___! O (F)-mod is an exact functor, and a left adjoin*
*t to
~
Oc(F)-mod ___! Z[ss]-mod . In particular, the existence of a left adjoint sho*
*ws that ~
sends injective objects in Oc(F)-mod to injective Z[ss]-modules.
Thus, if 0 ! F ! I0 ! I1 ! . .i.s an injective resolution of F in Oc(F)-mod ,*
* and
F takes values in Z(p)-modules, then
ss ss
lim-*(F )~=H* 0 ! ~(I0)(p)___! ~(I1)(p)___! . . .
Oc(F)
h iss
~= H* 0 ___! ~(I0)(p)! ~(I1)(p)___! . . .ss~= lim*(F | c 0) .
- O (F )
Oc(F0)
We are now ready to examine extensions of index prime to p.
Theorem 5.7. Fix a saturated fusion system F over a p-group S. Let
ss Outfus(S, F) def=Autfus(S, F)= AutF(S)
be any subgroup of order prime to p, and let essdenote the inverse image of ss *
*in
Autfus(S, F).
(a)Let F.ss be the fusion system over S generated (as a category) by F together*
* with
restrictions of automorphisms in ess. Then F.ss is a saturated fusion system*
*, which
contains F as a fusion subsystem of index prime to p.
(b)If F has an associated centric linking system, then so does F.ss.
(c)Let L be a centric linking system associated to F. Assume that for each ff 2*
* ss, the
action of ff on F lifts to an action on L. Then there is a unique centric l*
*inking
system L.ss associated to F.ss whose restriction to F is L.
Proof.(a) By definition, every morphism in F.ss is the composite of morphisms *
*in F
and restrictions of automorphisms of S which normalize F. If _ 2 Hom F (P, Q) a*
*nd
' 2 Autfus(S, F), then one has
'|Q O_ = '|Q O_ O'-1|'(P)O'|P = _0O'|P, (1)
where _02 Hom F('(P ), '(Q)) (since ' is fusion preserving). Hence each morphis*
*m in
F.ss is the composite of the restriction of a morphism in essfollowed by a morp*
*hism in
F.
We next claim F is a fusion subsystem of F.ss of index prime to p. More preci*
*sely,
we will show, for all P S, that AutF (P ) C AutF.ss(P ) with index prime to p*
*. To see
this, let ss1 essbe the set of automorphisms ' in ess Autfus(S, F) such that*
* '(P ) is
F-conjugate to P , and let ss0 ss1 be the set of classes ' such that '|P 2 Ho*
*m F(P, S).
If '(P ) and _(P ) are both F-conjugate to P , then _('(P )) is F-conjugate to *
*_(P )
since _ is fusion preserving, and thus is F-conjugate to P . This shows that s*
*s1 is
EXTENSIONS OF P-LOCAL FINITE GROUPS 49
a subgroup of ess, and an argument using (1) shows that ss0 is also a subgroup.*
* By
definition, ss0 AutF (S), so ss1=ss0 has order prime to p since ss does. Usin*
*g (1), define
`P :Aut F.ss(P ) ------! ss1=ss0
by setting `P(ff) = [_] if ff = _|fi(P)Ofi for some fi 2 Hom F(P, S) and some _*
*.Aut F(S) 2
ss. By definition of ss0, this is well defined, and Ker(`P) = AutF (P ). Since *
*ss1=ss0 has
order prime to p, this shows that AutF (P ) C AutF.ss(P ) with index prime to p.
We next claim that F and F.ss have the same fully centralized, fully normaliz*
*ed and
centric subgroups (compare with the proof of Proposition 3.8(c)). By definitio*
*n, F
and F.ss are fusion systems over the same p-group S. Since each F-conjugacy cla*
*ss is
contained in some F.ss-conjugacy class, any subgroup P S which is fully centr*
*alized
(fully normalized) in F.ss is fully centralized (fully normalized) in F. By th*
*e same
argument, any F.ss-centric subgroup is also F-centric.
Conversely, assume that P is not fully centralized in F.ss, and let P 0 S be*
* a
subgroup F.ss-conjugate to P such that |CS(P 0)| > |CS(P )|. Let _ :P 0___! P *
*be an
F.ss-isomorphism between them. Then, by the argument above _ = _0O', where _0is
a morphism in F and ' 2 essis the restriction to P 0of an automorphism of S. He*
*nce
|CS('(P 0))| = |CS(P 0)| > |CS(P )|, and since '(P 0) is F-conjugate to P , thi*
*s shows
that P is not fully centralized in F. A similar argument shows that if P is not*
* fully
normalized (or not centric) in F.ss, then it is not fully normalized (or not ce*
*ntric) in
F.
We prove that F.ss is saturated using Theorem 1.5(b). Thus, we must show that
conditions (I) and (II) of Definition 1.2 are satisfied for all F.ss-centric su*
*bgroups, and
that F.ss is generated by restrictions of morphisms between its centric subgrou*
*ps. Let
P S be a subgroup which is fully normalized in F.ss. Then it is fully normali*
*zed in F,
and since F is saturated, it is fully centralized there and AutS(P ) 2 Sylp(Aut*
* F(P )) =
Sylp(Aut F.ss). Hence, condition (I) holds for any P in (F.ss)c.
Let _ :P ___! Q be a morphism in F.ss, and write _ = _0O', as before. Set
N_ = {g 2 NS(P ) | _ Ocg O_-1 2 AutS(_(P ))};
then '(N_) = N_0 since 'cg'-1 = c'(g)for all g 2 S. Since condition (II) holds*
* for
F, the morphism _0 can be extended to N_0. Hence _ = _0O ' can be extended to
N_ = '(N_0), and so condition (II) holds for F.ss.
That all morphisms in F.ss are composites of restrictions of F.ss-morphisms b*
*etween
F.ss-centric subgroups holds by construction. Thus Theorem 1.5(b)applies, and F*
*.ss is
saturated.
(b) Let ZF :Oc(F) ___! Ab be the functor ZF (P ) = Z(P ), and similarly for Z*
*F.ss
(see Definition 1.7). By Lemma 5.6, restriction of categories induces a monomor*
*phism
lim-*(ZF.ss) ------! lim-*(ZF ) (2)
Oc(F.ss) Oc(F)
whose image is the subgroup of ss-invariant elements.
By [BLO2 , Proposition 3.1], the obstruction j(F.ss) to the existence of a c*
*entric
linking system associated to F.ss lies in lim-3(ZF.ss). From the construction i*
*n [BLO2 ]
of these obstructions (and the fact that F.ss and F have the same centric subgr*
*oups),
it is clear that the restriction map (2) sends j(F.ss) to j(F). So if there is *
*a linking
system L associated to F, then j(F) = 0, so j(F.ss) = 0 by Lemma 5.6, and there*
* is
a linking system L.ss associated to F.ss.
50 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
(c) Let L.ss be a centric linking system associated to F.ss, as constructed in*
* (b), and
let L0 be its restriction to F (i.e., the inverse image of F under the the proj*
*ection
L.ss ___! (F.ss)c). By [BLO2 , Proposition 3.1] again, the group lim-2(ZF ) a*
*cts freely
and transitively on the set of all centric linking systems associated to F. If *
*the action
on F of each ff 2 ss lifts to an action on L, then the element of lim-2(ZF ) wh*
*ich measures
the difference between L and L0is ss-invariant, and hence (by Lemma 5.6 again) *
*is the
restriction of an element of lim-2(ZF.ss). Upon modifying the equivalence class*
* of L.ss
by this element, if necessary, we get a centric linking system associated to F.*
*ss whose
restriction to F is L.
As was done in the last section for extensions with p-group quotient, we now *
*translate
this last result to a theorem stated in terms of fibration sequences.
Theorem 5.8. Fix a p-local finite group (S, F, L), a finite group of order pr*
*ime to
p, and a fibration E --v-!B with fiber X ' |L|^p. Then there is a p-local fini*
*te group
(S, F0, L0) such that F F0 is normal of index prime to p, Aut F0(S)= AutF(S) *
*is a
quotient group of , and E^p' |L0|^p.
Proof.For any space Y , let Aut(Y ) denote the topological monoid of homotopy e*
*quiv-
alences Y --'-! Y . Fibrations with fiber Y and base B are classified by homo*
*topy
classes of maps B ___! B Aut(Y ). This follows, for example, as a special case*
* of the
main theorem in [DKS ].
We are thus interested in the classifying space B Aut(|L|^p), whose homotopy *
*groups
were determined in [BLO2 , x8]. To describe these, let Aut typ(L) be the mono*
*id of
isotypical self equivalences of the category L; i.e., the monoid of all equival*
*ences of
categories _ 2 Aut (L) such that for all P 2 Ob (L), _P,P(Im (ffiP)) = Im(ffi_(*
*P)). Let
Outtyp(L) be the group of all isotypical self equivalences modulo natural isomo*
*rphisms
of functors. By [BLO2 , Theorem 8.1], ssi(B Aut(|L|^p)) is a finite p-group fo*
*r i = 2 and
vanishes for i > 2, and
ss1(B Aut(|L|^p)) ~=Out typ(L).
Each isotypical self equivalence of L is naturally isomorphic to one which se*
*nds
inclusions to inclusions (this was shown in [BLO1 , Lemma 5.1] for linking sys*
*tems of
a group, and the general case follows by the same argument). Thus each element*
* of
Outtyp(L) is represented by some fi which sends inclusions to inclusions. This *
*in turn
implies that fiS _ the restriction of fiS,Sto ffiS(S) AutL (S) _ lies in Autf*
*us(S, F),
and that for every F-centric subgroup P of S, the functor fi sends P to the sub*
*group
fiS(P ) S. In particular, fi is an automorphism of L, since it induces a bije*
*ction on
the set of objects.
fv ^ v
Now let B ___! B Aut(|L|p) be the map which classifies the fibration E ---!*
* B .
Since is a p0-group, the fibration E --v-!B is uniquely determined by the ac*
*tion
of on |L|^p; more precisely, by the map induced by fv on fundamental groups:
ss1(fv): -----! Out typ(L) .
By an argument identical to that used to prove [BLO1 , Theorem 6.2], there i*
*s an
exact sequence
~L
0 ----! lim-1(ZF ) -----! Out typ(L) -----! Out fus(S, F),
Oc(F)
EXTENSIONS OF P-LOCAL FINITE GROUPS 51
where ~L is defined by restricting a functor L ! L to ffiS(S) AutL(S). Also, *
*Oc(F)
and ZF are as in Definition 1.7, but all that we need to know here is that Ker(*
*~L) is
a p-group. Set
_v = ~L Oss1(fv): -----! Out fus(S, F).
Set ss = Im(_v) Outfus(S, F). By its definition, _v comes equipped with a l*
*ift to
Outtyp(L). So by the above remarks, every element of ss lifts to an automorphis*
*m of
L. Let (S, F.ss, L.ss) be the p-local finite group constructed in Theorem 5.7(a*
*,c) as an
extension of (S, F, L). This induces a fibration
|L| -----! |L.ss| -----! Bss.
By [BK , Corollary I.8.3], there is a fiberwise completion Ew of |L.ss| which s*
*its in a
fibration sequence
|L|^p-----! Ew ---w--!Bss,
and this fibration is classified by a map
fw :Bss -----! B Aut(|L|^p).
As was the case for fv, the induced map between fundamental groups ss1(fw) dete*
*rmines
the fibration.
Consider the diagram
ae
___________! ss
| |
ss1(fv)ss1(fw)| |incl (1)
# #
ss ~L
Out typ(L)__! Outfus(S, F) ,
where ae: ! ss is the restriction of _v to its image, and where the square co*
*mmutes
since each composite is equal to _v. The composite ~L Oss1(fw) 2 Hom (ss, Outfu*
*s(S, F))
is the homomorphism induced by the homotopy action of ss = ss1(Bss) on the fiber
|L|^p. By the construction in Theorem 5.7, this is just the inclusion map. Henc*
*e the
lower right triangle in the above diagram commutes. Since has order prime to *
*p and
Ker(~L) ~= lim-1(ZF ) is a p-group, any homomorphism from to Out fus(S, F) ha*
*s a
unique conjugacy class of lifting to Out typ(L) by the Schur-Zassenhaus theorem*
* (cf.
[Go , Theorem 6.2.1]). In particular, ss1(fw) Oae and ss1(fv) are conjugate as *
*homomor-
phisms from to Out typ(L). Thus, the upper left triangle in (1) commutes up *
*to
conjugacy.
Since ss2(B Aut(|L|^p)) is a finite p-group and the higher homotopy groups al*
*l vanish,
we have shown that fw OBae is homotopic to fv. In other words, we have a map of
fibrations
v
|L|^p_____! E ______! B
ww
ww ~ || |Bae|
# #
w
|L|^p_____!Ew _____!Bss,
A comparison of the spectral sequences for these two fibrations shows that ~ is*
* an Fp-
homology equivalence. Since |L| is p-good [BLO2 , Proposition 1.12], the natur*
*al map
from |L.ss| to its fiberwise completion Ew is also an Fp-homology equivalence. *
*Hence
these maps induce homotopy equivalences E^p' (Ew)^p' |L.ss|^p. This finishes t*
*he
proof of the theorem, with F0 = F.ss and L0= L.ss.
52 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
Note that we are not assuming that injects into Outfus(S, F) in the hypothe*
*ses of
Theorem 5.8. Thus, for example, if we start with a product fibration E = |L|^px*
* B ,
then we end up with (S, F0, L0) = (S, F, L) (and E^p' |L|^p).
We do not know whether the index prime to p analog of Corollary 4.8 holds. The
key point in the proof of Corollary 4.8 is the observation that if G is a finit*
*e group,
whose fusion system at p contains a normal subsystem of index pm , then G conta*
*ins
a normal subgroup of index pm . This is not true for subsystems of index prime *
*to p.
For example, the fusion system at the prime 3 of the simple groups J4 and Ru ha*
*s a
subsystem of index 2.
6.Central extensions of fusion systems and linking systems
Let F be a fusion system over a p-group S. We say that a subgroup A S is
central in F if CF (A) = F, where CF (A) is the centralizer fusion system defin*
*ed in
[BLO2 , Definition A.3]. Thus A is central_in F if A Z(S), and each_morphism*
* ' 2
Hom F(P, Q) in F extends to a morphism ' 2 Hom F(P A, QA) such that '|A = IdA.
In this section, we first study quotients of fusion systems, and of p-local f*
*inite groups,
by central subgroups. Afterwards, we will invert this procedure, and study cen*
*tral
extensions of fusion systems and p-local finite groups.
6.1___Central_quotients_of_fusion_and_linking_systems_ We first note that ev-
ery saturated fusion system F contains a unique maximal central subgroup, which*
* we
regard as the center of F.
Proposition 6.1. For any saturated fusion system F over a p-group S, define
fi c
ZF (S) = x 2 Z(S) fi'(x) = x, all' 2 Mor (F ) = lim-Z(-) :
Fc
the inverse limit of the centers of all F-centric subgroups of S. Then ZF (S) *
*is the
center of F: it is central in F, and contains all other central subgroups.
Proof.By definition, if A is central in F, then A Z(S), and any morphism in F
between subgroups containing A restricts to the identity on A. Since all F-cen*
*tric
subgroups contain Z(S), this shows that A ZF (S).
By Alperin's fusion theorem (Theorem 1.5(a)), each morphism in F is a composi*
*te of
restrictions of morphisms between F-centric subgroups. In particular, each morp*
*hism
is a restriction of a morphism between subgroups containing ZF (S) which is the*
* identity
on ZF (S), and thus ZF (S) is central in F.
The center of a fusion system F has already appeared when studying mapping sp*
*aces
of classifying spaces associated to F. By [BLO2 , Theorem 8.1], for any p-loca*
*l finite
group (S, F, L),
Map (|L|^p, |L|^p)Id' BZF (S).
We next define the quotient of a fusion system by a central subgroup.
Definition 6.2. Let F be a fusion system over a p-group S, and let A be a centr*
*al
subgroup. Define F=A to be the fusion system over S=A with morphism sets
Hom F=A(P=A, Q=A) = Im Hom F (P, Q) -----! Hom (P=A, Q=A) .
EXTENSIONS OF P-LOCAL FINITE GROUPS 53
By [BLO2 , Lemma 5.6], if F is a saturated fusion system over a p-group S, a*
*nd
A ZF (S), then F=A is also saturated as a fusion system over S=A. We now want*
* to
study the opposite question: if F=A is saturated, is F also saturated? The foll*
*owing
example shows that it is very easy to construct counterexamples to this. In fac*
*t, under
a mild hypothesis on S, if a fusion system over S=A has the form F=A for any fu*
*sion
system F over S, then it has that form for a fusion system F which is not satur*
*ated.
Example 6.3. Fix a p-group S and a central subgroup A Z(S). Assume there is *
* __
ff 2 Aut(S)r Inn(S) such that ff|A = IdA and ff induces the identity on S=A. Le*
*t F be
a saturated fusion system_over S=A, and assume there is some fusion system F0 o*
*ver
S such that F0=A = F . Let F F0 be the fusion system over S defined by setting
fi _
Hom F(P, Q) = ' 2 Hom (P, Q) fi9 ' 2 Hom (P A, QA),
_ _
'|P = ', '|A = IdA, ' =A 2 Hom _F(P A=A, QA=A) .
__
Then A is a central subgroup of F, and F=A = F . But F is not saturated, since
AutF (S) contains as normal subgroup the group of automorphisms of S which are *
*the
identity on A and on S=A (a p-group by Lemma 1.15), and this subgroup is by ass*
*ump-
tion not contained in Inn(S). Hence AutS(S) is not a Sylow subgroup of AutF (S).
The hypotheses of the above example are satisfied, for example, by any pair o*
*f p-
groups 1 6= A S with A Z(S), such that S is abelian, or more generally such*
* that
A \ [S, S] = 1.
We will now describe conditions which allow us to say when F is saturated. Be*
*fore
doing so, in the next two lemmas, we first compare properties of subgroups in F*
* with
those of subgroups of F=A, when F is a fusion system with central subgroup A. T*
*his
is done under varying assumptions as to whether F or F=A is saturated.
Lemma 6.4. The following hold for any fusion system F over a p-group S, and any
subgroup A Z(S) central in F.
(a)If A P S and P=A is F=A-centric, then P is F-centric.
(b)If F=A is saturated, and P S is F-quasicentric, then P A=A is F=A-quasicen*
*tric.
(c)If F and F=A are both saturated and P=A S=A is F=A-quasicentric, then P is
F-quasicentric.
Proof.For each P S containing A, let CeS(P ) S be the subgroup such that
eCS(P )=A = CS=A(P=A). Let
jP :eCS(P ) -----! Hom (P, A)
be the homomorphism jP(x)(g) = [x, g] for x 2 eCS(P ) and g 2 P . Thus Ker(jP) =
CS(P ).
For each P S containing A, set
P = Ker AutF (P ) ----! AutF=A(P=A) .
Since every ff 2 AutF (P ) is the identity on A, P is a p-group by Lemma 1.15.
(a) If g, g0 commute in S, then their images commute in S=A. Thus for all Q *
*S,
CS(Q)=A CS=A(Q=A).
54 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
Assume P=A is F=A-centric. Then for each P 0which is F-conjugate to P , P 0=A
is F=A-conjugate to P=A, and hence CS(P 0)=A CS=A(P 0=A) P 0=A. Thus P is
F-centric.
(b) Fix P S such that P is F-quasicentric. Since CS(P 0A) = CS(P 0) for all*
* P 0
which is F-conjugate to P , P 0A is fully centralized in F if and only if P 0is*
*. Also,
CF (P 0) = CF (P 0A) for such P 0, and this shows that P A is also F-quasicentr*
*ic. So
after replacing P by P A, if necessary, we can assume P A.
Choose P 0=A which is fully centralized in F=A and F=A-conjugate to P=A. Then
P 0is F-conjugate to P , hence still F-quasicentric. So upon replacing P by P 0*
*, we are
reduced to showing that P=A is F=A-quasicentric when P A, P is F-quasicentric,
and P=A is fully centralized.
For any P 0which is F-conjugate to P , there is a morphism
0 0
'=A 2 Hom F=A (P .eCS(P ))=A, (P .eCS(P ))=A
which sends P 0=A to P=A (by axiom (II) for the saturated fusion system F=A). T*
*hen
' restricts to a morphism from CS(P 0) to CS(P ). Thus |CS(P 0)| |CS(P )| for*
* all P 0
which is F-conjugate to P , and this proves that P is fully centralized in F.
If P=A is not F=A-quasicentric, then by Lemma 1.6 (and since F=A is saturated*
*),
there is some Q=A P=A.CS=A(P=A) containing P=A, and some ff 2 AutF (Q), such
that Id 6= ff=A 2 Aut F=A(Q=A) has order prime to p and ff=A is the identity on
P=A. Since ff is also the identity on A, Lemma 1.15 implies that ff|P = IdP. *
*Set
Q0= Q \ eCS(P ). Then ff(Q0) = Q0, and jP Off = jP since ff is the identity on *
*P A.
Thus ff induces the identity on Q0=CQ0(P ) since Ker(jP) = CS(P ). Since ff has*
* order
prime to p, Lemma 1.15 now implies that ff|CQ0(P)6= Id. Thus CQ0(P ) CS(P ) a*
*nd
ff|P.CQ0(P)is a nontrivial automorphism in CF (P ) of order prime to p. Since P*
* is fully
centralized in F, this implies (by definition) that CF (P ) is not the fusion s*
*ystem of
CS(P ), and hence that P is not F-quasicentric.
(c) Now assume that F and F=A are both saturated, and that P=A S=A is
F=A-quasicentric. If P is not F-quasicentric, and P 0is F-conjugate to P and f*
*ully
centralized in F, then by Lemma 1.6(b), there is some P 0 Q P 0.CS(P 0) and
some Id 6= ff 2 Aut F(Q) such that ff|P0 = IdP0 and ff has order prime to p. T*
*hen
Q=A (P 0=A).CS=A(P 0=A), ff=A 2 Aut F=A(Q=A) also has order prime to p, and so
ff=A 6= Id by Lemma 1.15 again. But by Lemma 1.6(a), this contradicts the assum*
*p-
tion that P=A is F=A-quasicentric.
In the next lemma, we compare conditions for being fully normalized in F and *
*in
F=A.
Lemma 6.5. The following hold for any fusion system F over a p-group S, and any
subgroup A Z(S) central in F.
(a)Assume F is saturated. Then for all P, Q S containing A such that P is ful*
*ly
normalized in F, if ', '02 Hom F(P, Q) are such that '=A = '0=A, then '0= ' *
*Ocx
for some x 2 NS(P ) such that xA 2 CS=A(P=A).
(b)Assume F=A is saturated, and let P, P 0 S be F-conjugate subgroups which
contain A. Then P is fully normalized in F if and only if P=A is fully nor-
malized in F=A. Moreover, if P is fully normalized in F, then there is _ 2
Hom F (NS(P 0), NS(P )) such that _(P 0) = P .
EXTENSIONS OF P-LOCAL FINITE GROUPS 55
Proof.(a) Assume F is saturated, and fix P, Q S containing A such that P is
fully normalized in F. Let ', '0 2 Hom F (P, Q) be such that '=A = '0=A. Then
Im(') = Im('0). Set ff = '-1 O'02 AutF (P ); then '0= ' Off, and ff=A = IdP=A. *
*Thus
ff 2 Ker[Aut F(P ) ---! AutF=A(P=A)],
which is a normal p-subgroup by Lemma 1.15. Since P is fully normalized, AutS(P*
* ) 2
Sylp(Aut F(P )) and so ff 2 Aut S(P ). Thus ff 2 cx for some x 2 NS(P ), and x*
*A 2
CS=A(P=A) since ff=A = IdP=A.
(b) We are now assuming that F=A is saturated. Assume first that P=A is fully
normalized in F=A. Then by Lemma 1.3, there is a morphism
'0 2 Hom F=A(NS=A(P 0=A), NS=A(P=A)) (1)
such that '0(P 0=A) = P=A, and this lifts to ' 2 Hom F (NS(P 0), NS(P )) such t*
*hat
'(P 0) = P . Therefore |NS(P 0)| |NS(P )| for any P 0which is F-conjugate to *
*P and
P is fully normalized in F. This also proves that the last statement in (b) hol*
*ds, once
we know that P=A is fully normalized.
Assume now that P is fully normalized in F; we want to show that P=A is fully
normalized in F=A. Fix P 0which is F-conjugate to P and such that P 0=A is ful*
*ly
normalized in F=A. By Lemma 1.3 again, there exists
'0 2 Hom F=A(NS=A(P=A), NS=A(P 0=A))
such that '0(P=A) = P 0=A. Then '0 = '=A for some ' 2 Hom F(NS(P ), NS(P 0)), a*
*nd
' is an isomorphism since P is fully normalized in F. Thus '0 is an isomorphism*
*, and
hence P=A is fully normalized in F=A.
Thus if P is fully normalized in F, then P=A is also fully normalized, and we*
* have
already seen that the last statement in (b) holds in this case.
We are now ready to give conditions under which we show that F is saturated if
F=A is saturated. As in [5A1 , x2], for any fusion system F over a p-group S, a*
*nd any
set H of subgroups of S, we say that F is H-generated if each morphism in F is a
composite of restrictions of morphisms between subgroups in H.
Proposition 6.6. Let A be a central subgroup of a fusion system F over a p-group
S, such that F=A is a saturated fusion system. Let H be any set of subgroups o*
*f S,
closed under F-conjugacy and overgroups, which contains all F-centric subgroups*
* of
S. Assume
(a)Ker Aut F(P ) ---! AutF=A(P=A) AutS(P ) for each P 2 H which is fully no*
*r-
malized in F; and
(b)F is H-generated.
Then F is saturated.
Proof.Let H0 be the set of all P 2 H such_that P A. Since A is central,_each
morphism ' 2 Hom F(P, Q) extends to some ' 2 Hom F(P A, QA) such that '|A = IdA.
Thus F is H0-generated if it is H-generated. So upon replacing H by H0, we can *
*assume
all subgroups in H contain A.
By assumption, H contains all F-centric subgroups of S. So by Theorem 1.5(b),*
* it
suffices to check that axioms (I) and (II) hold for all P 2 H. For all P S co*
*ntaining
56 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
A, we write
P def=KerAutF (P ) ---! AutF=A(P=A) .
(I) Assume P 2 H is fully normalized in F. Then P=A is fully normalized in F=A*
* by
Lemma 6.5(b). By assumption, P AutS(P ). Hence
[Aut F(P ) : AutS(P )] = [Aut F=A(P=A) : AutS=A(P=A)],
and AutS(P ) 2 Sylp(Aut F(P )) since AutS=A(P=A) 2 Sylp(Aut F=A(P=A)).
Assume P 0 S is F-conjugate to P and fully centralized in F. By Lemma 6.5(b)
again, there is _ 2 Hom F (NS(P 0), NS(P )) such that _(P 0) = P . Then _(CS(P *
*0))
CS(P ), so P is also fully centralized.
(II) Assume ' 2 Hom F(P, S) is such that P 2 H and '(P ) is fully centralized.*
* Set
N' = {g 2 NS(P ) | 'cg'-1 2 AutS('(P ))},
as usual. Then N'=A N'=A.
Assume first that '(P ) is fully normalized in F. Then '(P )=A is fully norma*
*lized
in F=A by Lemma 6.5(b), and hence also fully centralized. So by (II), applied t*
*o the
saturated_fusion system F=A, there is b'2 Hom_F(N'=A, S=A) such_that b'|(P=A)= *
*'=A.
Let ' 2 Hom F(N',_S) be a lift of b', then (' |P)=A = '=A_and '(P ) = '(P ). So*
* there
is ff = ' O(' |P)-1 2 '(P) Aut F('(P )) such_that ff O' |P = '. By (a), ther*
*e is
x 2 NS('(P )) such that ff = cx; then cx O' lies in Hom F(N', S) and extends '.
It remains to prove the general case. Choose P 0which is fully normalized in *
*F and
F-conjugate to P . By Lemma 6.5(b), there is _ 2 Hom F(NS('(P )), NS(P 0)) such*
* that
_('(P ))_= P 0. Then N' N_'. Since _'(P ) = P 0is fully_normalized, _' extends
to_some _ 2 Hom F(N', NS(P 0)). We will_show that Im(_ ) Im(_), and thus ther*
*e is
' 2 Hom F(N', NS('(P ))) such that '|P = '.
For each g 2 N', choose x 2 NS('(P_)) such that 'cg'-1 = cx; then we obtain
c_(x)= _cx_-1 = c__(g), and so _(x)_ (g)-1 2 CS(P 0). Since '(P ) is fully cent*
*ralized
_
in F, _(CS('(P ))) = CS(P 0), and thus _(g) 2 Im(_).
For example, one can take as the set H in Proposition 6.6 either the set of s*
*ubgroups
P S containing A such that P=A is F=A-quasicentric (by Lemma 6.4(b)), or the *
*set
of F-centric subgroups of S.
Now let (S, F, L) be a p-local finite group. One can also define a centralize*
*r linking
system CL(A) when A S is fully centralized [BLO2 , Definition 2.4]. Since t*
*his is
always a linking system associated to CF (A), A is central in L (i.e., CL(A) = *
*L) if and
only if A is central in F. So from now on, by a central subgroup of (S, F, L), *
*we just
mean a subgroup A Z(S) which is central in F.
Definition 6.7. Let (S, F, L) be a p-local finite group with a central subgroup*
* A. Define
L=A to be the category with objects the subgroups P=A for P 2 Ob (L) (i.e., suc*
*h that
P is F-centric), and with morphism sets
MorL=A(P=A, Q=A) = Mor L(P, Q)=ffiP(A).
Let (L=A)c L=A be the full subcategory with object set the F=A-centric subgro*
*ups in
S=A.
Similarly, if Lq is the associated quasicentric linking system, then define L*
*q=A to be
the category with objects the F=A-quasicentric subgroups of S=A _ equivalently,*
* the
EXTENSIONS OF P-LOCAL FINITE GROUPS 57
subgroups P A=A for P 2 Ob (Lq) _ and with morphisms
Mor Lq=A(P=A, Q=A) = Mor Lq(P, Q)=ffiP(A).
Note that by Lemma 6.4(a,c), for any P=A S=A, if P=A is F=A-centric or F=A-
quasicentric, then P is F-centric or F-quasicentric, respectively. Thus the cat*
*egories
(L=A)c L=A and Lq=A are well defined.
We are now ready to prove our main theorem about quotient fusion and linking
systems.
Theorem 6.8. Let A be a central subgroup of a p-local finite group (S, F, L) wi*
*th
associated quasicentric linking system Lq. Let Lq A Lq be the full subcategory*
* with
objects those F-quasicentric subgroups of S which contain A. Then the following*
* hold.
(a)(S=A, F=A, (L=A)c) is again a p-local finite group, and Lq=A is a quasicentr*
*ic link-
ing system associated to F=A.
(b)The inclusions of categories induce homotopy equivalences |L| ' |Lq A| ' |Lq*
*| and
|(L=A)c| ' |L=A| ' |Lq=A|.
(c)The functor o :Lq ___! Lq=A, defined by o(P ) = P A=A and with the obvious *
*maps
on morphisms, induces principal fibration sequences
|o| q |o| q
BA -----! |L| -----! |L=A| and BA -----! |L A | -----! |L =A|
which remain principal fibration sequences after p-completion.
Proof.(a) The first statement is shown in [BLO2 , Lemma 5.6] when |A| = p, and
the general case follows by iteration. So we need only prove that Lq=A is a qua*
*sicen-
tric linking system associated to F=A. Axioms (B)q, (C)q, and (D)q for Lq=A fol*
*low
immediately from those axioms applied to Lq, so it remains only to prove (A)q.
Let H be the set of subgroups P S such that A P and P=A is F=A-quasicentr*
*ic
and fix P, Q 2 H. By construction, CS=A(P=A) acts freely on Mor Lq=A(P=A, Q=A) *
*and
induces a surjection
ffi
Mor Lq=A(P=A, Q=A) CS=A(P=A) ----i Hom F=A(P=A, Q=A).
We must show that this is a bijection whenever P=A is fully centralized in F=A.*
* Since
any other fully centralized subgroup in the same F-conjugacy class has centrali*
*zer
of the same order, it suffices to show this when P=A is fully normalized in F=A*
*; or
equivalently (by Lemma 6.5(b)) when P is fully normalized in F.
Let eCS(P ) S be the subgroup such that eCS(P )=A = CS=A(P=A). Fix any F=A-
quasicentric subgroup Q=A, and consider the following sequence of maps
MorLq(P, Q) -----! Hom F(P, Q) -----! Hom F=A(P=A, Q=A).
Since Lq is the quasicentric linking system of F, by property (A)q the first ma*
*p is
the orbit map of the action of CS(P ). The second is the orbit map for the ac-
tion of Aut eCS(P)(P ) by Lemma 6.5(a). Thus the composite is the orbit map for
the free action of eCS(P ). It now follows that eCS(P )=A ~= CS=A(P=A) acts fr*
*eely on
Mor Lq=A(P=A, Q=A) = Mor Lq(P, Q)=A with orbit set Hom F=A(P=A, Q=A).
(b) These homotopy equivalences are all special cases of Proposition 1.12(a).
(c) For any category C and any n 0, let Cn denote the set of n-simplices in *
*the
nerve of C; i.e., the set of composable n-tuples of morphisms. For each n 0,*
* the
58 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
q
group B(A)n acts freely on (L A )n: an element (a1, . .,.an) acts by composing *
*the i-th
component with ffiP(ai) for appropriate P . This action commutes with the face*
* and
degeneracy maps, and its orbit set is (Lq=A)n. It follows that the projection o*
*f |Lq A|
onto |Lq=A| (and of |L| onto |L=A|) is a principal fibration with fiber the top*
*ological
group BA = |B(A)| (see, e.g., [May , xx18-20] or [GJ , Corollary V.2.7]).
By the principal fibration lemma of Bousfield and Kan [BK , II.2.2], these se*
*quences
are still principal fibration sequences after p-completion.
6.2___Central_extensions_of_p-local_finite_groups_We first make it more precise
what we mean by this.
Definition 6.9. A central extension of a (saturated) fusion system F over a p-g*
*roup
S by an abelian p-group A consists of a (saturated) fusion system eFover a p-gr*
*oup eS,
together with an inclusion A Z(Se), such that A is a central subgroup of eF, *
*eS=A ~=S,
and eF=A ~=F as fusion systems over S.
Similarly, a central extension of a p-local finite group (S, F, L) by an abel*
*ian group
A consists of a p-local finite group (Se, eF, eL), together with an inclusion A*
* ZFe(Se),
such that (Se=A, eF=A, (Le=A)c) ~=(S, F, L).
Extensions of categories were defined and studied by Georges Hoff in [Hf], wh*
*ere he
proves that they are classified by certain Ext-groups. We will deal with one p*
*artic-
ular case of this. Hoff's theorem implies that an extension of categories of t*
*he type
B(A) ---! Leq A--o-!Lq is classified by an element in lim-2(A). What this exten*
*sion
Lq
really means is that Lq is a quotient category of eLq A, where each morphism se*
*t in eLq A
admits a free action of A with orbit set the corresponding morphism set in Lq. *
*Also,
lim-2(A) means the second derived functor of the limit of the constant functor *
*which
sends each object of Lq to A and each morphism to IdA.
We regard A as an additive group. Fix an element [!] 2 lim-2(A), where ! is a
Lq
(reduced) 2-cocyle. Thus ! is a function from pairs of composable morphisms in *
*Lq to
A such that !(f, g) = 0 if f or g is an identity morphism, and such that for an*
*y triple
f, g, h of composable morphisms, the cocycle condition is satisfied
d!(f, g, h) def=!(g, h) - !(gf, h) + !(f, hg) - !(f, g) = 0.
Consider the composite i OffiS : BS ! Lq, where ffiS is induced by the distin*
*guished
morphism S ! AutLq(S) and i is induced by the inclusion of AutLq(S) in Lq as the
full subcategory with one object S. Then !S = (i OffiS)*(!) is a 2-cocycle defi*
*ned on S
which classifies a central extension of p-groups
1 ---! A -----! eS---o--!S ---! 1 .
Thus eS= S x A, with group multiplication defined by
(h, b).(g, a) = (hg, b + a + !S(g, h)),
and o(g, a) = g. For each F-quasicentric subgroup P S, set eP= o-1(P ) eS.
Using this cocycle !, we can define a new category eL0as follows. There is on*
*e object
eP= o-1(P ) of eL0for each object P of Lq. Morphism sets in eL0are defined by s*
*etting
Mor eL0(Pe, eQ) = Mor Lq(P, Q) x A.
EXTENSIONS OF P-LOCAL FINITE GROUPS 59
Composition in eL0is defined by
(g, b) O(f, a) = (gf, a + b + !(f, g)).
The associativity of this composition law follows since d! = 0. Furthermore, i*
*f we
chose another representative ! + d~ where ~ is a 1-cochain, the categories obta*
*ined are
isomorphic. This construction comes together with a projection functor o :eL0__*
*_! Lq.
Finally, we define
ffieS:S ------! Aut eL0(S)
by setting ffieS(g, a) = (ffiS(g), a). This is a group homomorphism by definiti*
*on of !S.
Proposition 6.10. Let (S, F, L) be a p-local finite group, and let Lq be the as*
*sociated
quasicentric linking system. Fix an abelian group A and a reduced 2-cocycle ! o*
*n Lq
with coefficients in A. Let
eL0---o---!Lq and eS---o---!S
be the induced extensions of categories and of groups, with distinguished monom*
*orphism
ffieSas defined above.
Then there is a unique saturated fusion system eFover eSand a functor ess:eL0*
*_! eF,
together with distinguished monomorphisms ffiPe:eP! AutLe0(P ).
If eL eL0denotes the full subcategory whose objects are the Fe-centric subgr*
*oups of
eSthen (Se, eF, eL) is a p-local finite group with central subgroup A Z(Se), *
*such that
(Se=A, eF=A, eL0=A) ~=(S, F, Lq). (1)
Also, eL0extends to a quasicentric linking system eLqassociated to eF.
Proof.Assume that inclusion morphisms 'P have been chosen in the quasicentric l*
*inking
system Lq associated to (S, F, L). For each F-quasicentric P S, define
'Pe= ('P, 0) 2 Mor eL0(Pe, eS).
Next, define the distinguished monomorphism
ffiPe:eP.CeS(Pe) ------! Aut eL0(Pe)
to be the unique monomorphism such that the following square commutes for each *
*eP
and each q 2 eP.CeS(Pe):
eP__________'Pe!eS
| |
#ffiPe(q,a)| #ffieS(q,a)| (2)
eP__________'Pe!eS.
More precisely, since ffieS(q, a) = (ffiS(q), a) by definition, this means that
ffiPe(q, a) = (ffiP(q), a + !('P, ffiS(q)) - !(ffiP(q), 'P)).
For each morphism (f, a) 2 Mor eL0(Pe, eQ), and each element (q, b) 2 Pe, the*
*re is a
unique element c 2 A such that the following square commutes:
eP__________(f,a)!eQ
| |
#ffiPe(q,b)| #ffiQe(ss(f)(q),c)| (3)
eP__________(f,a)!eQ.
60 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
In this situation, we set
ess(f, a)(q, b) = (ss(f)(q), c) 2 eQ.
By juxtaposing squares of the form (3), we see that ess(f, a) 2 Hom (Pe, eQ), a*
*nd that this
defines a functor essfrom eL0to the category of subgroups of eSwith monomorphis*
*ms.
Define eFto be the fusion system over eSgenerated by the image of essand rest*
*rictions.
By construction, the surjection o :eS---! S induces a functor o*: eF---! F betw*
*een
the fusion systems, which is surjective since F is generated by restrictions of*
* morphisms
between F-quasicentric subgroups (Theorem 1.5(a)). So we can identify F with eF*
*=A.
By Lemma 6.4(b), for each eF-quasicentric subgroup P eS, P A=A is F-quasicent*
*ric.
So we can extend Le0to a category Leqdefined on all Fe-quasicentric subgroups, *
*by
setting
fi
MorLq(P, Q) = f 2 Mor eL0(P A, QA) fiess(f)(P ) Q and ffiP = ffiPA
for each pair P, Q of eF-quasicentric subgroups. We extend essto eLqin the obvi*
*ous way.
It remains to prove that eFis saturated, and that eLqis a quasicentric linkin*
*g system
associated to eF. In this process of doing this, we will also prove the isomorp*
*hism (1).
Let H be the set of subgroups eP = o-1(P ) eSfor all F-quasicentric subgrou*
*ps
P S.
eFis saturated: We want to apply Proposition 6.6 to prove that eFis saturated.*
* By
Lemma 6.4(b), H contains all eF-quasicentric subgroups of eSwhich contain A, an*
*d in
particular, all eF-centric subgroups (since every eF-centric subgroup of eSmust*
* contain
A). Since eFis H-generated by construction, it remains only to prove condition *
*(a) in
Proposition 6.6.
Fix some eP= o-1(P ) in H, and let ' 2 AutFe(Pe) be such that o*(') = IdP. Ch*
*oose
(f, a) 2 ess-1('); thus
' = ess(f, a) 2 Ker AutFe(Pe) ----! AutF(P ) .
Then ss(f) = IdP, so f = ffiP(q) for some q 2 CS(P ), and (f, a) = ffiPe(q, c) *
*for some
c 2 A. Since ffiPeis a homomorphism, the definition of ' = ess(f, a) via (3) sh*
*ows that
' = ess(f, a) = effiPe(q, c) = c(q,c), and thus that ' 2 Aut eS(Pe). Thus cond*
*ition (a) in
Proposition 6.6 holds, and this finishes the proof that eFis saturated.
eLqis a quasicentric linking system associated to eF: The distinguished monomor-
phisms ffiPe, for eP2 Ob (Leq), were chosen so as to satisfy (D)q, and this was*
* independent
of the choice of inclusion morphisms which lift the chosen inclusion morphisms *
*in Lq.
Once the ffiP were determined, then esswas defined to satisfy (C)q, and eFwas d*
*efined
as the category generated by Im(ess) and restrictions. Thus all of these struct*
*ures were
uniquely determined by the starting data. Axiom (B)q follows from (C)q by Lemma
1.10.
It remains only to prove (A)q. We have already seen that the functor essis su*
*rjective
on all morphism sets. Also, since CeS(P ) = CeS(P A) for all P eS, it suffice*
*s to prove
(A)q for morphisms between subgroups eP = o-1(P ) and eQ = o-1(Q) containing A.
By construction, CeS(Pe) acts freely on each morphism set Mor eLq(Pe, eQ), and *
*it remains
to show that if ePis fully centralized, then essPe,Qeis the orbit map of this a*
*ction. As in
the proof of Theorem 6.8(a), it suffices to do this when ePand P are fully norm*
*alized.
EXTENSIONS OF P-LOCAL FINITE GROUPS 61
Fix two morphisms (f, a) and (g, b) from ePto eQsuch that ess(f, a) = ess(g, *
*b). Then
ss(f) = ss(g), so g = f OffiP(x) for some x 2 CS(P ), and
(g, b) = (f, a) O(ffiP(x), b - a - !(ffiP(x), f)) = (f, a) OffiPe(x,*
* c),
where c = b - a - !(ffiP(x), f) - (!(', ffiS(x)) - !(ffiS(x), 'P)). Also, ess(f*
*fiPe(x, c)) = IdeP
implies (via (3)) that (x, c) commutes with all elements of eP, so (x, c) 2 CeS*
*(Pe), and
this finishes the argument.
We now want to relate the obstruction theory for central extensions of linkin*
*g systems
with kernel A to those for central extensions of p-groups, and to those for pri*
*ncipal
fibrations with fiber BA. As a consequence of this, we will show (in Theorem 6.*
*13) that
when appropriate restrictions are added, these three types of extensions are eq*
*uivalent.
Given a central extension Fe of F by the central subgroup A, there is an indu*
*ced
central extension 1 ! A ! eS! S ! 1 of Sylow subgroups. Restriction to subgroups
P S produces corresponding central extensions 1 ! A ! Pe ! P ! 1. The
homology classes of these central extensions are all compatible with morphisms *
*from
the fusion system, and hence define an element in lim-H2(-; A). This, together *
*with
F
notation already used in [BLO2 ], motivates the following definition:
Definition 6.11. For any saturated fusion system over a p-group S, and any fini*
*te
abelian p-group A, define
H*(F; A) = lim-H*(-; A) ~=lim-H*(-; A).
F Fc
The following lemma describes the relation between the cohomology of F and the
cohomology of the geometric realization of any linking system associated to F.
Lemma 6.12. For any p-local finite group (S, F, L), and any finite abelian p-gr*
*oup A,
the natural homomorphism
H*(|L|; A) ------! H*(F; A), (1)
induced by the inclusion of BS into |L|, is an isomorphism. Furthermore, there*
* are
natural isomorphisms
lim-2(A) ~=H2(|Lq|; A) ~=H2(|L|; A). (2)
Lq
Proof.The second isomorphism in (2) follows from Proposition 1.12(a). The first
isomorphism holds for higher limits of any constant functor over any small, dis*
*crete
category C, since both groups are cohomology groups of the same cochain complex
Y Y Y
0 ---! A ----! A ----! A ----! . ...
c c0!c1 c0!c1!c2
This cochain complex for higher limits is shown in [GZ , Appendix II, Propositi*
*on 3.3]
(applied with M = Ab op).
To prove the isomorphism (1), it suffices to consider the case where A = Z=pn*
* for
some n. This was shown in [BLO2 , Theorem 5.8] when A = Z=p, so we can assume
that n 2, and that the lemma holds when A = Z=pn-1. Consider the following
62 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
diagram of Bockstein exact sequences
! Hi+1(|L|; Z=p)__! Hi(|L|; Z=pn-1)_! Hi(|L|; Z=pn)_! Hi(|L|; Z=p)__!
| | | |
| | | | (1)
# # # #
! Hi+1(BS; Z=p) _! Hi(BS; Z=pn-1) _! Hi(BS; Z=pn) _! Hi(BS; Z=p) __! .
We claim that the bottom row restricts to an exact sequence of groups H*(F; -).*
* Once
this is shown, the result follows by the 5-lemma.
By [BLO2 , Proposition 5.5], there is a certain (S, S)-biset which induces*
*, via a sum
of composites of transfer maps and maps induced by homomorphisms, an idempotent
endomorphism of H*(BS; Z=p) whose image is H*(F; Z=p). This biset also induces
endomorphisms [ ] of H*(BS; Z=pn) and H*(BS; Z=pn-1) which commute with the
bottom row in (1), since any exact sequence induced by a short exact sequence o*
*f coef-
ficient groups will commute with transfer maps and maps induced by homomorphism*
*s.
The same argument as that used in the proof of [BLO2 , Proposition 5.5] shows *
*that
in all of these cases, Im ([ ]) = H*(F; -), and the restriction of [ ] to its i*
*mage is
multiplication by | |=|S| 2 1 + pZ. Thus the sequence of the H*(F; -) splits a*
*s a
direct summand of the bottom row in (1), and hence is exact.
We can now collect the results about central extensions of (S, F, L) in the f*
*ollowing
theorem, which is the analog for p-local finite groups of the classical classif*
*ication of
central extensions of groups.
Theorem 6.13. Let (S, F, L) be a p-local finite group. For each finite abelian *
*p-group
A, the following three sets are in one-to-one correspondence:
(a)equivalence classes of central extensions of (S, F, L) by A;
(b)equivalence classes of principal fibrations BA ___! X ___! |L|^p; and
o
(c)isomorphism classes of central extensions 1 ____! A ____! eS____! S ____!*
* 1 for
which each morphism ' 2 Hom F(P, Q) lifts to some e'2 Hom (o-1(P ), o-1(Q)).
The equivalence between the first two is induced by taking classifying spaces, *
*and the
equivalence between (a) and (c) is induced by restriction to the underlying p-g*
*roup.
These sets are all in natural one-to-one correspondence with
lim-2(A) ~=H2(|L|; A) ~=H2(F; A). (1)
L
Proof.The three groups in (1) are isomorphic by Lemma 6.12. By Proposition 6.10,
central extensions of L are classified by lim-2(A). Principal fibrations over *
*|L|^pwith
L
fiber BA are classified by
^ 2 ^ 2
|L|p, B(BA) ~=H (|L|p; A) ~=H (|L|; A),
where the second isomorphism holds since |L| is p-good ([BLO2 , Proposition 1.*
*12])
and A is an abelian p-group. Central extensions of S by A are classified by H2(*
*S; A),
and the central extension satisfies the condition in (c) if and only if the cor*
*responding
element of H2(S; A) extends to an element in the inverse limit H2(F; A).
A central extension of p-local finite groups induces a principal fibration of*
* classifying
spaces by Theorem 6.8(c), and this principal fibration restricts to the princip*
*al fibration
over BS of classifying spaces of p-groups. Thus, if the fibration over |L| is c*
*lassified
EXTENSIONS OF P-LOCAL FINITE GROUPS 63
by O 2 H2(|L|; A), then the fibration over BS is classified by the restriction *
*of O to
H2(BS; A). It is well known that the invariant in H2(S; A) = H2(BS; A) for a ce*
*ntral
extension of S by A is the same as that which describes the principal fibration*
* over
BS with fiber BA (see [AM , Lemma IV.1.12]). Since H2(|L|; A) injects into H2(*
*F; A)
(Lemma 6.12), this shows that O is also the class of the 2-cocycle which define*
*s the
extension of categories. So the map between the sets in (a) and (b) defined by *
*taking
geometric realization is equal to the bijection defined by the obstruction theo*
*ry.
Since the isomorphism lim-2(A) ~=H2(F; A) is defined by restriction to S (as *
*a group
L
of automorphisms in L), the bijection between (a) and (c) induced by the biject*
*ion of
obstruction groups is the same as that induced by restriction to S.
The following corollary shows that all minimal examples of "exotic" fusion sy*
*stems
have trivial center.
Corollary 6.14. Let F be a saturated fusion system over a p-group S, and assume
there is a nontrivial subgroup 1 6= A Z(S) which is central in F. Then F is *
*the
fusion system of some finite group if and only if F=A is.
Proof.Assume F is the fusion system of the finite group G, with S 2 Sylp(G). By
assumption (A is central in F), each morphism in F extends to a morphism between
subgroups containing A which is the identity on A. Hence F is also the fusion s*
*ystem
of CG(A) over S, and so F=A is the fusion system of CG(A)=A.
It remains to prove_the converse. Assume F=A is isomorphic to the fusion_syst*
*em
of the finite group G , and identify_S=A with a Sylow p-subgroup of G . Since *
*by
Lemma 6.12, H2(F=A; A)_~=H2(BG ; A) and A is F-central, the cocycle classifying*
* the
extension is in H2(BG ; A), and hence there is an extension of finite groups
__
1 ----! A -----! G ---o--!G ----! 1
with the same obstruction invariant as the_extension F ___! F=A. In particular*
*, we
can identify S = o-1(S=A) 2 Sylp(G), and G = G=A.
We will prove the following two statements:
(a)F has an associated centric linking system L; and
(b)LcS=A(G=A) is the unique centric linking system associated to F=A = FS=A(G=A*
*).
Once these have been shown, then they imply that
(S=A, F=A, (L=A)c) ~=(S=A, FS=A(G=A), LcS=A(G=A))
as p-local finite groups. Hence (S, F, L) ~=(S, FS(G), LcS(G)) as p-local finit*
*e groups
by Theorem 6.13, and thus F is the fusion system of G.
It remains to prove (a) and (b). Let
ZF :Oc(F) ----! Z(p)-mod and ZG :OcS(G) ----! Z(p)-mod
be the categories and functors of Definition 1.7. By [O1 , Lemma 2.1], ZG can a*
*lso be
regarded as a functor on Oc(FS(G)), and
(c) lim-*(ZG) ~= lim-*(ZG).
OcS(G) Oc(FS(G))
64 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
By [BLO2 , Proposition 3.1], the existence and uniqueness of a centric linki*
*ng system
depends on the vanishing of certain obstruction classes: the obstruction to exi*
*stence
lies in lim-3(ZF ) and the obstruction to uniqueness in lim-2(ZF ). Thus (b)*
* follows
OcS(G) OcS(G)
from [BLO2 , Proposition 3.1] and (c), once we know that lim-2(ZG=A) = 0; and *
*this is
shown in [O1 , Theorem A] (if p is odd) or [O2 , Theorem A] (if p = 2).
It remains to prove point (a), and we will do this by showing that
lim-3(ZF ) ~= lim-3(ZG) = 0. (1)
Oc(F) OcS(G)
The last equality follows from [O1 , Theorem A] or [O2 , Theorem A] again, so i*
*t remains
only to prove the isomorphism.
Let H be the set of subgroups P S containing A such that P=A is F=A-centric;
or equivalently, p-centric in G=A. Let OH (F) Oc(F) and OH (FS(G)) Oc(FS(G))
be the full subcategories with object sets H.
We claim that
lim-*(ZF ) ~= lim-*(ZF ) and lim-*(ZG) ~= lim-*(ZG) . (2)
OH(F) Oc(F) OH(FS(G)) Oc(FS(G))
If P S is F-centric but not in H, then there is x 2 NS(P )r P such that cx 2 *
*AutF (P )
is the identity on P=A and on A, but is not in Inn(P ). Thus 1 6= [cx] 2 Op(Out*
* F(P )),
and so P is not F-radical. By [BLO2 , Proposition 3.2], if F :Oc(F)op ___! Z(*
*p)-mod
is any functor which vanishes except on the conjugacy class of P , then lim-*(F*
* ) ~=
*(Out F(P ); F (P )), where * is a certain functor defined in [JMO , x5]. B*
*y [JMO ,
Proposition 6.1(ii)], *(Out F(P ); F (P )) = 0 for any F , since Op(Out F(P ))*
* 6= 1. From
the long exact sequences of higher limits induced by extension of functors, it *
*now
follows that lim-*(F ) = 0 for any functor F on Oc(F) which vanishes on OH (F)*
*; and
Oc(F)
this proves the first isomorphism in (2). The second isomorphism follows by a s*
*imilar
argument.
The natural surjection of F onto F=A induces isomorphisms of categories
OH (F) ~=Oc(F=A) ~=OH (FS(G)) . (3)
By the definition of H, we clearly have bijections between the sets of objects *
*in these
categories, so it remains only to compare morphism sets. The result follows fr*
*om
Lemma 6.5(a) for sets of morphisms between pairs of fully normalized subgroups,*
* and
the general case follows since every object is isomorphic to one which is fully*
* normalized.
We next claim that for all i > 0,
lim-i(ZF ) ~= lim-i(ZF =A) and lim-i(ZG) ~= lim-i(ZG=A)(.4)
OH(F) Oc(F=A) OH(FS(G)) Oc(F=A)
To show this, by (3), together with the long exact sequence of higher limits in*
*duced by
the short exact sequence of functors
1 ___! A -----! ZF -----! ZF =A ___! 1,
we need only show that lim-i(A) = 0 for all i > 0. Here, A denotes the const*
*ant
Oc(F=A)
functor on Oc(F=A) which sends all objects to A and all morphisms to IdA. For e*
*ach
P 2 H, let FA,P be the functor on Oc(F=A) where FA,P(P 0=A) = A if P 0is F-conj*
*ugate
to P and FA,P(P 0=A) = 0 otherwise; and which sends isomorphisms between subgro*
*ups
conjugate to P to IdA. By [BLO2 , Proposition 3.2], lim-*(FA,P) ~= *(Out F=A(P*
*=A); A).
EXTENSIONS OF P-LOCAL FINITE GROUPS 65
Also, i(Out F=A(P=A); A) = 0 for i > 0 by [JMO , Proposition 6.1(i,ii)], sinc*
*e the action
of OutF=A(P=A) on A is trivial. From the long exact sequences of higher limits *
*induced
by extension of functors, we now see that lim-i(A) = 0 for all i > 0.
Finally, we claim that
ZF =A ~=ZG=A (5)
as functors on Oc(F=A) under the identifications in (3). To see this, note tha*
*t for
each P , (ZF =A)(P ) = Z(P )=A = (ZG=A)(P ); and since these are both identifie*
*d as
subgroups of P=A, any morphism in Oc(F=A) from P=A to Q=A induces the same map
(under the two functors) from Z(Q)=A to Z(P )=A. (This argument does not apply
to prove that ZF ~= ZG. These two functors send each object to the same subgroup
of S, but we do not know that they send each morphism to the same homomorphism
between the subgroups.)
Thus by (2), (4), and (5), for all i > 0,
lim-i(ZF ) ~= lim-i(ZF~)=lim-i(ZF =A)
Oc(F) OH(F) Oc(F=A)
~= limi (Z =A) ~= limi (Z ) ~= limi(Z ).
- G - G - G
Oc(F=A) OH(FS(G)) OcS(G)
This finishes the proof of (1), and hence finishes the proof of the corollary.
References
[AM] A. Adem & J. Milgram, Cohomology of finite groups, Springer-Verlag (1994)
[Al] J. Alperin, Sylow intersections and fusion, J. Algebra 6 (1967), 222-241
[BK] P. Bousfield & D. Kan, Homotopy limits, completions, and localizations, L*
*ecture notes in
math. 304, Springer-Verlag (1972)
[5A1] C. Broto, N. Castellana, J. Grodal, R. Levi and B. Oliver, Subgroup famil*
*ies controlling
p-local finite groups, Proc. London Math. Soc. (to appear)
[BLO1]C. Broto, R. Levi, & B. Oliver, Homotopy equivalences of p-completed clas*
*sifying spaces of
finite groups, Invent. math. 151 (2003), 611-664
[BLO2]C. Broto, R. Levi, & B. Oliver, The homotopy theory of fusion systems, J.*
* Amer. Math. Soc.
16 (2003), 779-856
[DKS] W. Dwyer, D. Kan, & J. Smith, Towers of fibrations and homotopical wreath*
* products, J.
Pure Appl. Algebra 56 (1989), 9-28
[GZ] P. Gabriel & M. Zisman, Calculus of fractions and homotopy theory, Spring*
*er-Verlag (1967)
[Gl] G. Glauberman, Central elements in core-free groups, J. Algebra 4 (1966),*
* 403-420
[GJ] P. Goerss & R. Jardine, Simplicial homotopy theory, Birkh"auser (1999)
[Go] D. Gorenstein, Finite groups, Harper & Row (1968)
[Hf] G. Hoff, Cohomologies et extensions de categories, Math. Scand. 74 (1994)*
*, 191-207.
[JMO] S. Jackowski, J. McClure, & B. Oliver, Homotopy classification of self-ma*
*ps of BG via G-
actions, Annals of Math. 135 (1992), 184-270
[May] P. May, Simplicial objects in algebraic topology, Van Nostrand (1967)
[O1] B. Oliver, Equivalences of classifying spaces completed at odd primes, Ma*
*th. Proc. Camb.
Phil. Soc. 137 (2004), 321-347
[O2] B. Oliver, Equivalences of classifying spaces completed at the prime two,*
* Amer. Math. Soc.
Memoirs (to appear)
[Pu2] Ll. Puig, Unpublished notes (ca. 1990)
[Pu3] Ll. Puig, The hyperfocal subalgebra of a block, Invent. math. 141 (2000),*
* 365-397
[Pu4] L. Puig, Full Frobenius systems and their localizing categories, preprint
[St] R. Stancu, Equivalent definitions of fusion systems, preprint
[Suz2]M. Suzuki, Group theory II, Springer-Verlag (1986)
66 C. BROTO, N. CASTELLANA, J. GRODAL, R. LEVI, & B. OLIVER
Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, E-08193 Bel-
laterra, Spain
E-mail address: broto@mat.uab.es
Departament de Matem`atiques, Universitat Aut`onoma de Barcelona, E-08193 Bel-
laterra, Spain
E-mail address: natalia@mat.uab.es
Department of Mathematics, University of Chicago, Chicago, IL 60637, USA
E-mail address: jg@math.uchicago.edu
Department of Mathematical Sciences, University of Aberdeen, Meston Building
339, Aberdeen AB24 3UE, U.K.
E-mail address: ran@maths.abdn.ac.uk
LAGA, Institut Galil'ee, Av. J-B Cl'ement, 93430 Villetaneuse, France
E-mail address: bob@math.univ-paris13.fr