CONSTRUCTION OF 2-LOCAL FINITE GROUPS OF A TYPE
STUDIED BY SOLOMON AND BENSON
RAN LEVI AND BOB OLIVER
Abstract. A p-local finite group is an algebraic structure with a classif*
*ying space
which has many of the properties of p-completed classifying spaces of fin*
*ite groups.
In this paper, we construct a family of 2-local finite groups, which are *
*exotic in the
following sense: they are based on certain fusion systems over the Sylow *
*2-subgroup
of Spin7(q) (q an odd prime power) shown by Solomon not to occur as the 2*
*-fusion
in any actual finite group. Thus, the resulting classifying spaces are no*
*t homotopy
equivalent to the 2-completed classifying space of any finite group. As p*
*redicted by
Benson, these classifying spaces are also very closely related to the Dwy*
*er-Wilkerson
space BDI(4).
As one step in the classification of finite simple groups, Ron Solomon [So] c*
*onsidered
the problem of classifying all finite simple groups whose Sylow 2-subgroups are*
* isomor-
phic to those of the Conway group Co3. The end result of his paper was that Co3*
* is
the only such group. In the process of proving this, he needed to consider grou*
*ps G
in which all involutions are conjugate, and such that the centralizer of each i*
*nvolution
contains a normal subgroup isomorphic to Spin7(q) with odd index, where q is an*
* odd
prime power. Solomon showed that such a group G does not exist. The proof of th*
*is
statement was also interesting, in the sense that the 2-local structure of the *
*group in
question appeared to be internally consistent, and it was only by analyzing its*
* inter-
action with the p-local structure (where p is the prime of which q is a power) *
*that he
found a contradiction.
In a later paper [Be ], Dave Benson, inspired by Solomon's work, constructed *
*certain
spaces which can be thought of as the 2-completed classifying spaces which the *
*groups
studied by Solomon would have if they existed. He started with the spaces BDI(*
*4)
constructed by Dwyer and Wilkerson having the property that
H*(BDI(4); F2) ~=F2[x1, x2, x3, x4]GL4(2)
(the rank four Dickson algebra at the prime 2). Benson then considered, for eac*
*h odd
prime power q, the homotopy fixed point set of the Z-action on BDI(4) generated*
* by
an Ä dams operation" _q constructed by Dwyer and Wilkerson. This homotopy fixed
point set is denoted here BDI4(q).
In this paper, we construct a family of 2-local finite groups, in the sense o*
*f [BLO2 ],
which have the 2-local structure considered by Solomon, and whose classifying s*
*paces
are homotopy equivalent to Benson's spaces BDI4(q). The results of [BLO2 ] com*
*bined
with those here allow us to make much more precise the statement that these spa*
*ces
have many of the properties which the 2-completed classifying spaces of the gro*
*ups
studied by Solomon would have had if they existed. To explain what this means, *
*we
first recall some definitions.
___________
1991 Mathematics Subject Classification. Primary 55R35. Secondary 55R37, 20D0*
*6, 20D20.
Key words and phrases. Classifying space, p-completion, finite groups, fusion.
B. Oliver is partially supported by UMR 7539 of the CNRS.
1
2 Construction of 2-local finite groups
A fusion system over a finite p-group S is a category whose objects are the s*
*ubgroups
of S, and whose morphisms are monomorphisms of groups which include all those
induced by conjugation by elements of S. A fusion system is saturated if it sa*
*tisfies
certain axioms formulated by Puig [Pu ], and also listed in [BLO2 , Definition*
* 1.2] as well
as at the beginning of Section 1 in this paper. In particular, for any finite g*
*roup G and
any S 2 Sylp(G), the category FS(G) whose objects are the subgroups of S and wh*
*ose
morphisms are those monomorphisms between subgroups induced by conjugation in G
is a saturated fusion system over S.
If F is a saturated fusion system over S, then a subgroup P S is called F-c*
*entric if
CS(P 0) = Z(P 0) for all P 0isomorphic to P in the category F. A centric linkin*
*g system
associated to F consists of a category L whose objects are the F-centric subgro*
*ups of
S, together with a functor L --! F which is the inclusion on objects, is surjec*
*tive on
all morphism sets and which satisfies certain additional axioms (see [BLO2 , D*
*efinition
1.6]). These axioms suffice to ensure that the p-completed nerve |L|^phas all *
*of the
properties needed to regard it as a "classifying spaceö f the fusion system F.*
* A p-local
finite group consists of a triple (S, F, L), where S is a finite p-group, F is *
*a saturated
fusion system over S, and L is a linking system associated to F. The classifyin*
*g space of
a p-local finite group (S, F, L) is the p-completed nerve |L|^p(which is p-comp*
*lete since
|L| is always p-good [BLO2 , Proposition 1.11]). For example, if G is a finit*
*e group
and S 2 Sylp(G), then there is an explicitly defined centric linking system LcS*
*(G)
associated to FS(G), and the classifying space of the triple (S, FS(G), LcS(G))*
* is the
space |LcS(G)|^p' BG^p.
Exotic examples of p-local finite groups for odd primes p _ i.e., examples wh*
*ich do
not represent actual groups _ have already been constructed in [BLO2 ], but us*
*ing ad
hoc methods which seemed to work only at odd primes.
In this paper, we first construct a fusion system FSol(q) (for any odd prime *
*power
q) over a 2-Sylow subgroup S of Spin7(q), with the properties that all elements*
* of
order 2 in S are conjugate (i.e., the subgroups they generated are all isomorph*
*ic in the
category), and the "centralizer fusion system" (see the beginning of Section 1)*
* of each
such element is isomorphic to the fusion system of Spin7(q). We then show that *
*FSol(q)
is saturated, and has a unique associated linking system LcSol(q). We thus obta*
*in a 2-
local finite group (S, FSol(q), LcSol(q)) where by Solomon's theorem [So] (as e*
*xplained
in more detail in Proposition 3.4), FSol(q) is not the fusion system of any fin*
*ite group.
Let BSol(q) def=|LcSol(q)|^2denote the classifying space of (S, FSol(q), LcSol(*
*q)). Thus,
BSol(q) does not have the homotopy type of BG^2for any finite group G, but does
have many of the nice properties of the 2-completed classifying space of a fini*
*te group
(as described in [BLO2 ]).
Relating BSol(q) to BDI4(q) requires taking the ü nionö f the categories LcS*
*ol(qn)
for all n 1. This however is complicated by the fact that an inclusion of fi*
*elds
Fpm Fpn (i.e., m|n) does not induce an inclusion of cenric linking systems. H*
*ence we
have to replace the centric linking systems LcSol(qn) by the full subcategories*
*SLccSol(qn)
whose objects are those 2-subgroups which are centric in FcSol(q1 ) = n 1FcSo*
*l(qn),
and show that the inclusion induces a homotopy equivalence BSol0(qn) def=|LccSo*
*l(qn)|^2'
BSol(qn). Inclusions of fields do induce inclusions of these categories, so we *
*can then
S
define LcSol(q1 ) def=n 1LccSol(qn), and spaces
i[ j
BSol(q1 ) = |LcSol(q1 )|^2' BSol0(qn) ^2.
n 1
Ran Levi and Bob Oliver *
* 3
The category LcSol(q1_) has an Ä dams map" _q induced by the Frobenius automor-
phism x 7! xq of Fq. We then show that BSol(q1 ) ' BDI(4), the space of Dwyer a*
*nd
Wilkerson mentioned above; and also that BSol(q) is equivalent to the homotopy *
*fixed
point set of the Z-action on BSol(q1 ) generated by B_q. The space BSol(q) is t*
*hus
equivalent to Benson's spaces BDI4(q) for any odd prime power q.
The paper is organized as follows. Two propositions used for constructing sat*
*urated
fusion systems, one very general and one more specialized, are proven in Sectio*
*n 1.
These are then applied in Section 2 to construct the fusion systems FSol(q), an*
*d to
prove that they are saturated. In Section 3 we prove the existence and uniquene*
*ss of
a centric linking systems associated to FSol(q) and study their automorphisms. *
*Also
in Section 3 is the proof that FSol(q) is not the fusion system of any finite g*
*roup. The
connections with the space BDI(4) of Dwyer and Wilkerson is shown in Section 4.
Some background material on the spinor groups Spin(V, b) over fields of charact*
*eristic
6= 2 is collected in an appendix.
We would like to thank Dave Benson, Ron Solomon, and Carles Broto for their h*
*elp
while working on this paper.
1.Constructing saturated fusion systems
In this section, we first prove a general result which is useful for construc*
*ting satu-
rated fusion systems. This is then followed by a more technical result, which i*
*s designed
to handle the specific construction in Section 2.
We first recall some definitions from [BLO2 ]. A fusion system over a p-grou*
*p S is a
category F whose objects are the subgroups of F, such that
Hom S(P, Q) Mor F(P, Q) Inj(P, Q)
for all P, Q S, and such that each morphism in F factors as the composite of *
*an
F-isomorphism followed by an inclusion. We write Hom F (P, Q) = Mor F(P, Q) to
emphasize that the morphisms are all homomorphisms of groups. We say that two
subgroups P, Q S are F-conjugate if they are isomorphic in F. A subgroup P S
is fully centralized (fully normalized) in F if |CS(P )| |CS(P 0)| (|NS(P )| *
* |NS(P 0)|)
for all P 0 S which is F-conjugate to P . A saturated fusion system is a fusio*
*n system
F over S which satisfies the following two additional conditions:
(I)For each fully normalized subgroup P S, P is fully centralized and AutS (P*
* ) 2
Sylp(Aut F(P )).
(II)For each P S and each ' 2 Hom F(P, S) such that '(P ) is fully centralize*
*d in
F, if we set
fi -1
N' = g 2 NS(P ) fi'cg' 2 AutS('(P )) ,
_
then ' extends to a homomorphism ' 2 Hom F(N', S).
For example, if G is a finite group and S 2 Sylp(G), then the category FS(G) *
*whose
objects are the subgroups of S and whose morphisms are the homomorphisms induced
by conjugation in G is a saturated fusion system over S. A subgroup P S is fu*
*lly
centralized in FS(G) if and only if CS(P ) 2 Sylp(CG(P )), and P is fully norma*
*lized in
FS(G) if and only if NS(P ) 2 Sylp(NG(P )).
4 Construction of 2-local finite groups
For any fusion system F over a p-group S, and any subgroup P S, the "centra*
*lizer
fusion system" CF (P ) over CS(P ) is defined by setting
fi 0 0
Hom CF(P)(Q, Q0) = ('|Q) fi' 2 Hom F(P Q, P Q ), '(Q) Q , '|P = IdP
for all Q, Q0 CS(P ) (see [BLO2 , Definition A.3] or [Pu ] for more detail).*
* We also
write CF (g) = CF () for g 2 S. If F is a saturated fusion system and P is *
*fully
centralized in F, then CF (P ) is saturated by [BLO2 , Proposition A.6] (or [P*
*u ]).
Proposition 1.1. Let F be any fusion system over a p-group S. Then F is saturat*
*ed
if and only if there is a set X of elements of order p in S such that the follo*
*wing
conditions hold:
(a)Each x 2 S of order p is F-conjugate to some element of X.
(b)If x and y are F-conjugate and y 2 X, then there is some _ 2 Hom F(CS(x), CS*
*(y))
such that _(x) = y.
(c)For each x 2 X, CF (x) is a saturated fusion system over CS(x).
Proof.Throughout the proof, conditions (I) and (II) always refer to the conditi*
*ons in
the definition of a saturated fusion system, as stated above or in [BLO2 , Def*
*inition
1.2].
Assume first that F is saturated, and let X be the set of all x 2 S of order *
*p such
that is fully centralized. Then condition (a) holds by definition, (b) foll*
*ows from
condition (II), and (c) holds by [BLO2 , Proposition A.6] or [Pu ].
Assume conversely that X is chosen such that conditions (a-c) hold for F. Def*
*ine
fi T
U = (P, x) fiP S, |x| = p, x 2 Z(P ) , some T 2 Sylp(AutF (P )) containing *
*AutS(P,)
where Z(P )T is the subgroup of elements of Z(P ) fixed by the action of T . Le*
*t U0 U
be the set of pairs (P, x) such that x 2 X. For each 1 6= P S, there is some *
*x such
that (P, x) 2 U (since every action of a p-group on Z(P ) has nontrivial fixed *
*set); but
x need not be unique.
We first check that
(P, x) 2 U0, P fully centralized in CF (x)=) P fully centralized in(F.1)
Assume otherwise: that (P, x) 2 U0 and P is fully centralized in CF (x), but P *
*is not
fully centralized in F. Let P 0 S and ' 2 IsoF(P, P 0) be such that |CS(P )| <*
* |CS(P 0)|.
Set x0 = '(x) Z(P 0). By (b), there exists _ 2 Hom F (CS(x0), CS(x)) such th*
*at
_(x0) = x. Set P 00= _(P 0). Then _ O' 2 IsoCF(x)(P, P 00), and in particular*
* P 00is
CF (x)-conjugate to P . Also, since CS(P 0) CS(x0), _ sends CS(P 0) injectiv*
*ely into
CS(P 00), and |CS(P )| < |CS(P 0)| |CS(P 00)|. Since CS(P ) = CCS(x)(P ) and *
*CS(P 00) =
CCS(x)(P 00), this contradicts the original assumption that P is fully centrali*
*zed in CF (x).
By definition of U, for each (P, x) 2 U, NS(P ) CS(x) and hence AutCS(x)(P *
*) =
AutS(P ). By assumption, there is T 2 Sylp(Aut F(P )) such that ø(x) = x for al*
*l ø 2 T ;
i.e., such that T AutCF(x)(P ). In particular, it follows that
8(P, x) 2 U : Aut S(P ) 2 Sylp(Aut F(P )) () AutCS(x)(P ) 2 Sylp(Aut CF(x)(P(*
*)).2)
We are now ready to prove condition (I) for F; namely, to show for each P S
fully normalized in F that P is fully centralized and AutS (P ) 2 Sylp(Aut F(P *
*)). By
definition, |NS(P )| |NS(P 0)| for all P 0F-conjugate to P . Choose x 2 Z(P )*
* such that
(P, x) 2 U; and let T 2 Sylp(Aut F(P )) be such that T AutS(P ) and x 2 Z(P )*
*T. By
(a) and (b), there is an element y 2 X and a homomorphism _ 2 Hom F(CS(x), CS(y*
*))
Ran Levi and Bob Oliver *
* 5
such that _(x) = y. Set P 0= _P , and set T 0= _T _-1 2 Sylp(Aut F(T 0)). Since*
* T
AutS(P ) by definition of U, and _(NS(P )) = NS(P 0) by the maximality assumpti*
*on,
0 0
we see that T 0 AutS(P 0). Also, y 2 Z(P 0)T (T y = y since T x = x), and thi*
*s shows
that (P 0, y) 2 U0. The maximality of |NS(P 0)| = |NCS(y)(P 0)| implies that P *
*0is fully
normalized in CF (y). Hence by condition (I) for the saturated fusion system C*
*F (y),
together with (1) and (2), P fully centralized in F and AutS(P ) 2 Sylp(Aut F(P*
* )).
It remains to prove condition (II) for F. Fix 1 6= P S and ' 2 Hom F(P, S) *
*such
that P 0def='P is fully centralized in F, and set
fi -1 0
N' = g 2 NS(P ) fi'cg' 2 AutS(P ) .
_ 0 0
We must show that ' extends to some ' 2 Hom F(N', S). Choose some x 2 Z(P ) of
order p which is fixed under the action of AutS(P 0), and set x = '-1(x0) 2 Z(P*
* ). For
all g 2 N', 'cg'-1 2 AutS(P 0) fixes x0, and hence cg(x) = x. Thus
x 2 Z(N') and hence N' CS(x); and NS(P 0) CS(x0). (3)
Fix y 2 X which is F-conjugate to x and x0, and choose
_ 2 Hom F(CS(x), CS(y)) and _02 Hom F(CS(x0), CS(y))
such that _(x) = _0(x0) = y. Set Q = _(P ) and Q0 = _0(P 0). Since P 0is fully
centralized in F, _0(P 0) = Q0, and CS(P 0) CS(x0), we have
_0(CCS(x0)(P 0)) = _0(CS(P 0)) = CS(Q0) = CCS(y)(Q0). (4)
Set ø = _0'_-1 2 IsoF(Q, Q0). By construction, ø(y) = y, and thus ø 2 IsoCF(y)(*
*Q, Q0).
Since P 0is fully centralized in F, (4) implies that Q0is fully centralized in *
*CF (y). Hence
condition (II), when_applied to the saturated fusion system CF (y), shows that *
*ø extends
to a homomorphism ø 2 Hom CF(y)(Nfi, CS(y)), where
fi -1 0
Nfi= g 2 NCS(y)(Q) fiøcgø 2 AutCS(y)(Q ) .
Also, for all g 2 N' CS(x) (see (3)),
c_fi(_(g))= øc_(g)ø-1 = (ø_)cg(ø_)-1 = (_0')cg(_0')-1 = c_0(h)2 AutCS(y)(Q0)
for some h 2 NS(P 0) such that 'cg'-1 = ch. This shows that _(N') Nfi; and al*
*so
(since CS(Q0) = _0(CS(P 0)) by (4)) that
_ 0 0
ø(_(N')) _ (NCS(x0)(P )).
We can now define
_def 0 -1 _
' = (_ ) O(ø O_)|N' 2 Hom F(N', S),
_
and '|P = '.
Proposition 1.1 will also be applied in a separate paper of Carles Broto & Je*
*sper
Møller [BM ] to give a construction of some "exotic" p-local finite groups at *
*certain odd
primes.
Our goal now is to construct certain saturated fusion systems, by starting wi*
*th the
fusion system of Spin7(q) for some odd prime power q, and then adding to that t*
*he
automorphisms of some subgroup of Spin7(q). This is a special case of the gene*
*ral
problem of studying fusion systems generated by fusion subsystems, and then sho*
*wing
that they are saturated. We first fix some notation. If F1 and F2 are two fusio*
*n systems
over the same p-group S, then denotes the fusion system over S generat*
*ed by
F1 and F2: the smallest fusion system over S which contains both F1 and F2. More
generally, if F is a fusion system over S, and F0 is a fusion system over a sub*
*group
6 Construction of 2-local finite groups
S0 S, then denotes the fusion system over S generated by the morphism*
*s in
F between subgroups of S, together with morphisms in F0 between subgroups of S0
only. In other words, a morphism in is a composite
'1 '2 'k
P0 ---! P1 ---! P2 ---! . . .---!Pk-1 ---! Pk,
where for each i, either 'i2 Hom F(Pi-1, Pi), or 'i2 Hom F0(Pi-1, Pi) (and Pi-1*
*, Pi
S0).
As usual, when G is a finite group and S 2 Sylp(G), then FS(G) denotes the fu*
*sion
system of G over S. If Aut (G) is a group of automorphisms which contains
Inn(G), then FS( ) will denote the fusion system over S whose morphisms consist*
* of
all restrictions of automorphisms in to monomorphisms between subgroups of S.
The next proposition provides some fairly specialized conditions which imply *
*that
the fusion system generated by the fusion system of a group G together with cer*
*tain
automorphisms of a subgroup of G is saturated.
Proposition 1.2. Fix a finite group G, a prime p dividing |G|, and a Sylow p-su*
*bgroup
S 2 Sylp(G). Fix a normal subgroup Z C G of order p, an elementary abelian subg*
*roup
U C S of rank two containing Z such that CS(U) 2 Sylp(CG(U)), and a subgroup
Aut(CG(U)) containing Inn(CG(U)) such that fl(U) = U for all fl 2 . Set
S0 = CS(U) and F def=,
and assume the following hold.
(a)All subgroups of order p in S different from Z are G-conjugate.
(b) permutes transitively the subgroups of order p in U.
(c){' 2 | '(Z) = Z} = AutNG(U)(CG(U)).
(d)For each E S which is elementary abelian of rank three, contains U, and is*
* fully
centralized in FS(G),
{ff 2 AutF (CS(E)) | ff(Z) = Z} = AutG (CS(E)).
(e)For all E, E0 S which are elementary abelian of rank three and contain U, *
*if E
and E0 are -conjugate, then they are G-conjugate.
Then F is a saturated fusion system over S. Also, for any P S such that Z P*
* ,
{' 2 Hom F(P, S) | '(Z) = Z} = Hom G(P, S). (1)
Proposition 1.2 follows from the following three lemmas. Throughout the proof*
*s of
these lemmas, references to points (a-e) mean to those points in the hypotheses*
* of the
proposition, unless otherwise stated.
Lemma 1.3. Under the hypotheses of Proposition 1.2, for any P S and any centr*
*al
subgroup Z0 Z(P ) of order p,
Z 6= Z0 U =) 9 ' 2 Hom (P, S0) such that '(Z0) = Z (1)
and
Z0 U =) 9 _ 2 Hom G(P, S0) such that _(Z0) U. (2)
Proof.Note first that Z Z(S), since it is a normal subgroup of order p in a p*
*-group.
Assume Z 6= Z0 U. Then U = ZZ0, and
P CS(Z0) = CS(ZZ0) = CS(U) = S0
Ran Levi and Bob Oliver *
* 7
since Z0 Z(P ) by assumption. By (b), there is ff 2 such that ff(Z0) = Z. S*
*ince
S0 2 Sylp(CG(U)), there is h 2 CG(U) such that h.ff(P ).h-1 S0; and since
ch 2 AutNG(U)(CG(U))
by (c), ' def=ch Off 2 Hom (P, S0) and sends Z0 to Z.
If Z0 U, then by (a), there is g 2 G such that gZ0g-1 Ur Z. Since Z is
central in S, gZ0g-1 is central in gP g-1, and U is generated by Z and gZ0g-1, *
*it
follows that gP g-1 CG(U). Since S0 2 Sylp(CG(U)), there is h 2 CG(U) such th*
*at
h(gP g-1)h-1 S0; and we can take _ = chg 2 Hom G(P, S0).
We are now ready to prove point (1) in Proposition 1.2.
Lemma 1.4. Assume the hypotheses of Proposition 1.2, and let F =
be the fusion system generated by G and . Then for all P, P 0 S which contain*
* Z,
{' 2 Hom F(P, P 0) | '(Z) = Z} = Hom G(P, P 0).
Proof.Upon replacing P 0by '(P ) P 0, we can assume that ' is an isomorphism,*
* and
thus that it factors as a composite of isomorphisms
'1 '2 '3 'k-1 'k 0
P = P0 ---!~ P1 ---! P2 ---! . . .---! Pk-1 ---! Pk = P ,
= ~= ~= ~= ~=
where for each i, 'i2 Hom G(Pi-1, Pi) or 'i2 Hom (Pi-1, Pi). Let Zi Z(Pi) be *
*the
subgroups of order p such that Z0 = Zk = Z and Zi= 'i(Zi-1).
To simplify the discussion, we say that a morphism in F is of type (G) if it *
*is
given by conjugation by an element of G, and of type ( ) if it is the restricti*
*on of
an automorphism in . More generally, we say that a morphism is of type (G, )*
* if
it is the composite of a morphism of type (G) followed by one of type ( ), etc.*
* We
regard IdP, for all P S, to be of both types, even if P S0. By definition, *
*if any
nonidentity isomorphism is of type ( ), then its source and image are both cont*
*ained
in S0 = CS(U).
For each i, using Lemma 1.3, choose some _i2 Hom F(PiU, S) such that _i(Zi) =*
* Z.
More precisely, using points (1) and (2) in Lemma 1.3, we can choose _i to be o*
*f type
( ) if Zi U (the inclusion if Zi = Z), and to be of type (G, ) if Z U. Set
Pi0= _i(Pi). To keep track of the effect of morphisms on the subgroups Zi, we w*
*rite
them as morphisms between pairs, as shown below. Thus, ' factors as a composite*
* of
isomorphisms
_-1i-1 'i _i 0
(Pi0-1, Z) -----! (Pi-1, Zi-1) -----! (Pi, Zi) -----! (Pi, Z).
If 'iis of type (G), then this composite (after replacing adjacent morphisms of*
* the same
type by their composite) is of type ( , G, ). If 'i is of type ( ), then the c*
*omposite
is again of type ( , G, ) if either Zi-1 U or Zi U, and is of type ( , G, ,*
* G, ) if
neither Zi-1nor Zi is contained in U. So we are reduced to assuming that ' is o*
*f one
of these two forms.
Case 1: Assume first that ' is of type ( , G, ); i.e., a composite of isomorph*
*isms of
the form
'1 '2 '3
(P0, Z) ----! (P1, Z1) ----! (P2, Z2) ----! (P3, Z).
( ) (G) ( )
Then Z1 = Z if and only if Z2 = Z because '2 is of type (G). If Z1 = Z2 = Z, th*
*en
'1 and '3 are of type (G) by (c), and the result follows.
8 Construction of 2-local finite groups
If Z1 6= Z 6= Z2, then U = ZZ1 = ZZ2, and thus '2(U) = U. Neither '1 nor '3 c*
*an
be the identity, so Pi S0 = CS(U) for all i by definition of Hom (-, -), and*
* hence
'2 is of type ( ) by (c). It follows that ' 2 Iso (P0, P3) sends Z to itself, *
*and is of
type (G) by (c) again.
Case 2: Assume now that ' is of type ( , G, , G, ); more precisely, that it *
*is a
composite of the form
'1 '2 '3 '4 '5
(P0, Z) ---! (P1, Z1) ---! (P2, Z2) ---! (P3, Z3) ---! (P4, Z4) ---! (P5,*
* Z),
( ) (G) ( ) (G) ( )
where Z2, Z3 U. Then Z1, Z4 U and are distinct from Z, and the groups
P0, P1, P4, P5 all contain U since '1 and '5 (being of type ( )) leave U invari*
*ant.
In particular, P2 and P3 contain Z, since P1 and P4 do and '2, '4 are of type (*
*G). We
can also assume that U P2, P3, since otherwise P2\U = Z or P3\U = Z, '3(Z) = *
*Z,
and hence '3 is of type (G) by (c) again. Finally, we assume that P2, P3 S0 =*
* CS(U),
since otherwise '3 = Id.
Let Ei Pibe the rank three elementary abelian subgroups defined by the requi*
*re-
ments that E2 = UZ2, E3 = UZ3, and 'i(Ei-1) = Ei. In particular, Ei Z(Pi)
for i = 2, 3 (since Zi Z(Pi), and U Z(Pi) by the above remarks); and hence
Ei Z(Pi) for all i. Also, U = ZZ4 '4(E3) = E4 since '4(Z) = Z, and thus
U = '5(U) E5. Via similar considerations for E0 and E1, we see that U Eifor*
* all
i.
Set H = CG(U) for short. Let E3 be the set of all elementary abelian subgroup*
*s E
S of rank three which contain U, and with the property that CS(E) 2 Sylp(CH (E)*
*).
Since CS(E) CS(U) = S0 2 Sylp(H), the last condition implies that E is fully
centralized in the fusion system FS0(H). If E S is any rank three elementary *
*abelian
subgroup which contains U, then there is some a 2 H such that E0 = aEa-1 2 E3,
since FS0(H) is saturated and U C H. Then ca 2 IsoG(E, E0) \ Iso(E, E0) by (c).*
* So
upon composing with such isomorphisms, we can assume that Ei 2 E3 for all i, and
also that 'i(CS(Ei-1)) = CS(Ei) for each i.
_
In this way, ' can be assumed to extend to an F-isomorphism ' from CS(E0)
to CS(E5) which sends Z to itself. By (e), the rank three subgroups Ei are all*
* G-
conjugate to each other. Choose g 2 G such that gE5g-1 = E0. Then g.CS(E5).g-1
and CS(E0) are both Sylow p-subgroups of_CG(E0), so there is h 2 CG(E0) such th*
*at
(hg)CS(E5)(hg)-1 = CS(E0). By (d), chgO '2 AutF (CS(E0)) is of type (G); and th*
*us
' 2 IsoG(P0, P5).
To finish the proof of Proposition 1.2, it remains only to show:
Lemma 1.5. Under the hypotheses of Proposition 1.2, the fusion system F generat*
*ed
by FS(G) and FS0( ) is saturated.
Proof.We apply Proposition 1.1, by letting X be the set of generators of Z. Con*
*dition
(a) of the proposition (every x 2 S of order p is F-conjugate to an element of *
*X) holds
by Lemma 1.3. Condition (c) holds since CF (Z) is the fusion system of the gro*
*up
CG(Z) by Lemma 1.4, and hence is saturated by [BLO2 , Proposition 1.3].
It remains to prove condition (b) of Proposition 1.1. We must show that if y,*
* z 2 S
are F-conjugate and = Z, then there is _ 2 Hom F(CS(y), CS(z)) such that _(*
*y) =
z. If y =2U, then by Lemma 1.3(2), there is ' 2 Hom F(CS(y), S0) such that '(y)*
* 2 U.
If y 2 Ur Z, then by Lemma 1.3(1), there is ' 2 Hom F(CS(y), S0) such that '(y)*
* 2 Z.
We are thus reduced to the case where y, z 2 Z (and are F-conjugate).
Ran Levi and Bob Oliver *
* 9
In this case, then by Lemma 1.4, there is g 2 G such that z = gyg-1. Since Z *
*C G,
[G:CG(Z)] is prime to p, so S and gSg-1 are both Sylow p-subgroups of CG(Z), and
hence are CG(Z)-conjugate. We can thus choose g such that z = gyg-1 and gSg-1 =*
* S.
Since CS(y) = CS(z) = S (Z Z(S) since it is a normal subgroup of order p), th*
*is
shows that cg 2 IsoG(CS(y), CS(z)), and finishes the proof of (b) in Propositio*
*n 1.1.
2. A fusion system of a type considered by Solomon
The main result of this section and the next is the following theorem:
Theorem 2.1. Let q be an odd prime power, and fix S 2 Syl2(Spin7(q)). Let z 2
Z(Spin7(q)) be the central element of order 2. Then there is a saturated fusion*
* system
F = FSol(q) which satisfies the following conditions:
(a)CF (z) = FS(Spin7(q)) as fusion systems over S.
(b)All involutions of S are F-conjugate.
Furthermore, there is a unique centric linking system L = LcSol(q) associated t*
*o F.
Theorem 2.1 will be proven in Propositions 2.11 and 3.3. Later, at the end of*
* Section
3, we explain why Solomon's theorem [So] implies that these fusion systems are *
*not
the fusion systems of any finite groups, and hence that the spaces BSol(q) are *
*not
homotopy equivalent to the 2-completed classifying spaces of any finite groups.
Background results needed for computations in Spin(V, b) have been collected *
*in
Appendix A. We focus attention here on SO7(q) and Spin7(q). In fact, since we w*
*ant
to compare the constructions over Fq with those over_its field extensions,_most*
* of the
constructions will first be made in the groups SO7(F q) and Spin7(F q).
We now fix, for the rest of the_section, an odd prime power q. It will be con*
*venient
to write Spin7(q1 ) def=Spin7(F q), etc. In order to make certain computations*
* more
explicit, we set
_ _ _
V1 = M2(F q) M02(F q) ~=(F q)7 and b(A, B) = det(A) + det(B)
(where M02(-) is the group of (2 x 2) matrices of trace zero), and for each n *
* 1 set
Vn = M2(Fqn) M02(Fqn) V1 . Then b is a nonsingular quadratic form on V1 and
on Vn. Identify SO7(q1 ) = SO(V1 , b) and SO7(qn)_= SO(Vn, b), and similarly_f*
*or
Spin7(qn) Spin7(q1 ). For all ff 2 Spin(M2(F q), det) and fi 2 Spin(M02(F q),*
* det), we
write ff fi for their image in Spin7(q1 ) under the natural homomorphism
'4,3:Spin4(q1 ) x Spin3(q1 ) -----! Spin7(q1 ).
There are isomorphisms
~= 1 1 ~= 1
eæ4:SL2(q1 ) x SL2(q1 ) --! Spin4(q ) and eæ3:SL2(q ) --! Spin3(q )
which are defined explicitly in Proposition A.5, and which restrict to isomorph*
*isms
SL2(qn) x SL2(qn) ~=Spin4(qn) and SL2(qn) ~=Spin3(qn)
for each n. Let
z = eæ4(-I, -I) 1 = 1 eæ3(-I) 2 Z(Spin7(q))
10 Construction of 2-local finite groups
denote the central element of order two, and set
z1 = eæ4(-I, I) 1 2 Spin7(q).
Here, 1 2 Spink(q) (k = 3, 4) denotes the identity element. Define U = .
Definition 2.2. Define
! :SL2(q1 )3 -----! Spin7(q1 )
by setting
!(A1, A2, A3) = eæ4(A1, A2) eæ3(A3)
for A1, A2, A3 2 SL2(q1 ). Set
H(q1 ) = !(SL2(q1 )3) and [[A1, A2, A3]] = !(A1, A2, A3) .
Since eæ3and eæ4are isomorphisms, Ker(!) = Ker('4,3), and thus
Ker(!) = <(-I, -I, -I)>.
In particular, H(q1 ) ~=(SL2(q1 )3)={ (I, I, I)}. Also,
z = [[I, I, -I]] and z1 = [[-I, I, I]],
and thus
U = [[ I, I, I]]
(with all combinations of signs).
For each 1 n < 1, the natural homomorphism
Spin7(qn) ------! SO7(qn)
has kernel and cokernel both of order 2. The image of this homomorphism is the
commutator subgroup 7(qn) C SO7(qn),_which is partly described by Lemma A.4(a).
In contrast, since all elements of F qare squares, the natural homomorphism from
Spin7(q1 ) to SO7(q1 ) is surjective.
Lemma 2.3. There is an element ø 2 NSpin7(q)(U) of order 2 such that
ø.[[A1, A2, A3]].ø-1 = [[A2, A1, A3]] (1)
for all A1, A2, A3 2 SL2(q1 ).
_
Proof.Let ø 2 SO7(q) be the involution defined by setting
_
ø(X, Y ) = (-`(X), -Y )
_ _
for (X, Y ) 2 V1 = M2(F q) M02(F q), where
` abcd= -d-bca.
_ _
Let ø 2 Spin7(q1 ) be a lifting of ø. The (-1)-eigenspace of ø on V1 has ortho*
*gonal
basis
(I, 0) , 0, 100-1 , 0, 0110 , 0, 0-110 ,
and in particular_has discriminant 1 with respect to this basis. Hence by Lemm*
*a _
A.4(a), ø 2 7(q), and so ø 2 Spin7(q). Since in addition, the (-1)-eigenspace *
*of ø is
4-dimensional, Lemma A.4(b) applies to show that ø2 = 1.
By definition of the isomorphisms eæ3and eæ4, for all Ai 2 SL2(q1 ) (i = 1, 2*
*, 3) and
all (X, Y ) 2 V1 ,
[[A1, A2, A3]](X, Y ) = (A1XA-12, A3Y A-13).
Ran Levi and Bob Oliver *
* 11
*
* _
Here, Spin7(q1 ) acts on V1 via its projection to SO7(q1 ). Also, for all X, Y *
*2 M2(F q),
t -1
`(X) = -0110.X . 0-110 and in particular `(XY ) = `(Y ).`(X);
and `(X) = X-1 if det(X) = 1. Hence for all A1, A2, A3 2 SL2(q1 ) and all (X, Y*
* ) 2
V1 ,
-1 -1 -1
ø.[[A1, A2, A3]].ø (X,=Yø)(-A1.`(X).A2 , -A3Y A3 )
= (A2XA-11, A3Y A-13) = [[A2, A1, A3]](X, Y ).
This shows that (1) holds modulo = Z(Spin7(q1 )). We thus have two automor-
phisms of H(q1 ) ~= (SL2(q1 )3)={ (I, I, I)} _ conjugation by ø and the permuta-
tion automorphism _ which are liftings of the same automorphism of H(q1 )=.
Since H(q1 ) is perfect, each automorphism of H(q1 )= has at most one lifting
to an automorphism of H(q1 ), and thus (1) holds. Also, since U is the subgroup
of all elements [[ I, I, I]] with all combinations of signs, formula (1) show*
*s that
ø 2 NSpin7(q)(U).
Definition 2.4. For each n 1, set
H(qn) = H(q1 ) \ Spin7(qn) and H0(qn) = !(SL2(qn)3) H(qn).
Define
n = Inn(H(qn)) o b 3 Aut(H(qn)),
where b 3denotes the group of permutation automorphisms
fi
b 3= [[A1, A2, A3]] 7! [[Aff1, Aff2, Aff3]] fioe 2 3 Aut(H(qn)) .
n 1
For each n, letn_q be the automorphism of Spin7(q ) induced by thenfield is*
*omor-
phism (q 7! qp ). By Lemma A.3, Spin7(qn) is the fixed subgroup of _q . Hence e*
*ach
element of H(qn) is of the form [[A1,nA2, A3]], where either Ai2 SL2(qn) for ea*
*ch i (and
the element lies in H0(qn)), or _q (Ai) = -Ai for each i. This shows that H0(qn*
*) has
index 2 in H(qn).
The goal is now to choose öc mpatible" Sylow subgroups S(qn) 2 Syl2(Spin7(qn))
(all n 1) contained in N(H(qn)), and let FSol(qn) be the fusion system over S*
*(qn)
generated by conjugation in Spin7(qn) and by restrictions of n.
Proposition 2.5. The following hold for each n 1.
(a)H(qn) = CSpin7(qn)(U).
(b)NSpin7(qn)(U) = NSpin7(qn)(H(qn)) = H(qn).<ø>, and contains a Sylow 2-subgro*
*up of
Spin7(qn).
_ _
Proof.Let_z1 2 SO7(q) be the image of z1 2 Spin7(q). Set V- = M2(F q) and V+ =
_
M02(F q): the eigenspaces of z1acting on V . By Lemma A.4(c),
CSpin7(q1()U) = CSpin7(q1()z1)
_ 1
is the group of all elements ff 2 Spin7(q1 ) whose image ff2 SO7(q ) has the f*
*orm
_
ff= ff- ff+ where ff 2 SO(V ).
In other words,
1 1 1 3 1
CSpin7(q1()U) = '4,3Spin4(q ) x Spin3(q ) = !(SL2(q ) ) = H(q ).
Furthermore, since
øz1ø-1 = ø[[-I, I, I]]ø-1 = [[I, -I, I]] = zz1
12 Construction of 2-local finite groups
by Lemma 2.3, and since any element of NSpin7(q1()U) centralizes z, conjugation*
* by ø
generates OutSpin7(q1()U). Hence
NSpin7(q1()U) = H(q1 ).<ø>.
Point (a), and the first part of point (b), now follow upon taking intersection*
*s with
Spin7(qn).
If NSpin7(qn)(U) did not contain a Sylow 2-subgroup of Spin7(qn), then since *
*every
noncentral involution of Spin7(qn) is conjugate to z1 (Proposition A.8), the Sy*
*low 2-
subgroups of Spin7(q) would have no normal subgroup isomorphic to C22. By a the*
*orem
of Hall (cf. [Go , Theorem 5.4.10]), this would imply that they are cyclic, di*
*hedral,
quaternion, or semidihedral. This is clearly not the case, so NSpin7(qn)(U) mus*
*t contain
a Sylow 2-subgroup of Spin7(q), and this finishes the proof of point (b).
Alternatively, point (b) follows from the standard formulas for the orders of*
* these
groups (cf. [Ta , pp.19,140]), which show that
|Spin7(qn)|_q9n(q6n - 1)(q4n - 1)(q2n - 1)6n 4n 2n i q2n + 1j
= ___________________________= q (q + q + 1) _______
|H(qn).<ø>| 2.[qn(q2n - 1)]3 2
is odd.
We next fix, for each n, a Sylow 2-subgroup of Spin7(qn) which is contained in
H(qn).<ø> = NSpin7(qn)(U).
Definition 2.6. Fix elements A, B 2 SL2(q) such that ~= Q8 (a quaternion
group of order 8), and set bA= [[A, A, A]] and bB= [[B, B, B]]. Let C(q1 ) CS*
*L2(q1()A)
be the subgroup of elements of 2-power order in the centralizer (which is abeli*
*an), and
set Q(q1 ) = . Define
S0(q1 ) = !(Q(q1 )3) H0(q1 ) and S(q1 ) = S0(q1 ).<ø> H(q1 ) Spin7(q1*
* ).
Here, ø 2 Spin7(q) is the element of Lemma 2.3. Finally, for each n 1, define
C(qn) = C(q1 ) \ SL2(qn), Q(qn) = Q(q1 ) \ SL2(qn),
S0(qn) = S0(q1 ) \ Spin7(qn), and S(qn) = S(q1 ) \ Spin7(qn).
Since the two eigenvalues of A are distinct, its centralizer in SL2(q1 ) is c*
*onjugate to
the subgroup of diagonal matrices, which is abelian. Thus C(q1 ) is conjugate t*
*o the
subgroup of diagonal matrices of 2-power order. This shows that each finite sub*
*group
of C(q1 ) is cyclic, and that each finite subgroup of Q(q1 ) is cyclic or quate*
*rnion.
Lemma 2.7. For all n, S(qn) 2 Syl2(Spin7(qn)).
Proof.By [Sz, 6.23], A is contained in a cyclic subgroup of order qn - 1 or qn *
*+ 1
(depending on which of them is divisible by 4). Also, the normalizer of this c*
*yclic
subgroup is a quaternion group of order 2(qn 1), and the formula |SL2(qn)| = qn*
*(q2n-
1) shows that this quaternion group has odd index. Thus by construction, Q(qn) *
*is a
Sylow 2-subgroup of SL2(qn). Hence !(Q(qn)3) is a Sylow 2-subgroup of H0(qn), *
*so
!(Q(q1 )3) \ Spin7(qn) is a Sylow 2-subgroup of H(qn). It follows that S(qn) is*
* a Sylow
2-subgroup of H(qn).<ø>, and hence also of Spin7(qn) by Proposition 2.5(b).
Following the notation of Definition A.7, we say that an elementary abelian 2-
subgroup E Spin7(qn) has type I if its eigenspaces all have square discrimina*
*nt,
and has type II otherwise. Let Er be the set of elementary abelian subgroups of*
* rank
r in Spin7(qn) which contain z, and let EIrand EIIrbe the sets of those of type*
* I or
Ran Levi and Bob Oliver *
* 13
II, respectively. In Proposition A.8, we show that there are two conjugacy clas*
*ses of
subgroups in EI4and one conjugacy class of subgroups in EII4. In Proposition A.*
*9, an
invariant xC(E) 2 E is defined, for all E 2 E4 (and where C is one of the conju*
*gacy
classes in EI4) as a tool for determining the conjugacy class of a subgroup. Mo*
*re pre-
cisely, E has type I if and only if xC(E) 2 , and E 2 C if and only if xC(E)*
* = 1. The
next lemma provides some more detailed information about the rank four subgroups
and these invariants.
Recall that we define bA= [[A, A, A]] and bB= [[B, B, B]].
Lemma 2.8. Fix n 1, set E* = S(qn), and let C be the Spin7(q*
*n)-
conjugacy class of E*. Let EU4be the set of all elementary abelian subgroups E *
* S(qn)
of rank 4 which contain U = . Fix a generator X 2 C(qn) (the 2-power tor*
*sion
in CSL2(qn)(A)), and choose Y 2 C(q2n) such that Y 2= X. Then the following hol*
*d.
(a)E* has type I.
(b)EU4= Eijk, E0ijk| i, j, k 2 Z (a finite set), where
Eijk= and E0ijk= .
(c)xC(Eijk) = [[(-I)i, (-I)j, (-I)k]] and xC(E0ijk) = [[(-I)i, (-I)j, (-I)k]].b*
*A.
(d)All of the subgroups E0ijkhave type II. The subgroup Eijkhas type I if and o*
*nly if
i j (mod 2), and lies in C (is conjugate to E*) if and only if i j k (*
*mod
2). The subgroups E000, E001, and E100thus represent the three conjugacy cla*
*sses
of rank four elementary abelian subgroups of Spin7(qn) (and E* = E000).
(e)For any ' 2 n Aut(H(qn)) (see Definition 2.4), if E0, E002 EU4are such th*
*at
'(E0) = E00, then '(xC(E0)) = xC(E00).
Proof.(a) The set
(I, 0) , (A, 0) , (B, 0) , (AB, 0) , (0, A) , (0, B) , (0, AB)
is a basis of eigenvectors for the action of E* on Vn = M2(Fqn) M02(Fqn). (Si*
*nce the
matrices A, B, and AB all have order 4 and determinant one, each has as eigenva*
*lues
the two distinct fourth roots of unity, and hence they all have trace zero.) Si*
*nce all of
these have determinant one, E* has type I by definition.
(b) Consider the subgroups
i j k i j k fi
R0 = !(C(q1 )3) \ S(qn) = [[X , X , X ]], [[X Y, X Y, X Y ]] fii, j, k 2*
* Z
and
R1 = CS(qn)(__) = R0..
Clearly, each subgroup E 2 EU4is contained in
CS(qn)(U) = S0(qn) = R0.<[[Bi, Bj, Bk]]>.
All involutions in this subgroup are contained in R1 = R0.<[[B, B, B]]>, and th*
*us E
R1. Hence E \ R0 has rank 3, which implies that E (the 2-torsion i*
*n R0).
Since all elements of order two in the coset R0.bBhave the form
[[XiB, XjB, XkB]] or [[XiY B, XjY B, XkY B]]
for some i, j, k, this shows that E must be one of the groups Eijkor E0ijk. (N*
*ote in
particular that E* = E000.)
14 Construction of 2-local finite groups
(c) By Proposition A.9(a),nthe element xC(E) 2 E is characterized uniquely by *
*the
property that xC(E) = g-1_q (g) for some g 2 Spin7(q1 ) such that gEg-1 2 C. We
now apply this explicitly to the subgroups Eijkand E0ijk.
For each i, Y -i(XiB)Y i= Y -2iXiB = B. Hence for each i, j, k,
[[Y i, Y j, Y k]]-1.Eijk.[[Y i, Y j, Y k]] = E*
and n
_q ([[Y i, Y j, Y k]]) = [[Y i, Y j, Y k]].[[(-I)i, (-I)j, (-I)k]*
*].
Hence
xC(Eijk) = [[(-I)i, (-I)j, (-I)k]].
Similarly, if we choose Z 2 CSL2(q1()A) such that Z2 = Y , then for each i,
(Y iZ)-1(XiY B)(Y iZ) = B.
Hence for each i, j, k,
[[Y iZ, Y jZ, Y kZ]]-1.E0ijk.[[Y iZ, Y jZ, Y kZ]] = E*.
n
Since _q (Z) = ZA,
n i j k i j k i j k
_q ([[Y Z, Y Z, Y Z]]) = [[Y Z, Y Z, Y Z]].[[(-I) A, (-I) A, (-I) A]],
and hence
xC(E0ijk) = [[(-I)iA, (-I)jA, (-I)kA]].
(d) This now follows immediately from point (c) and Proposition A.9(b,c).
(e) By Definition 2.4, n is generated by Inn(H(qn)) and the permutations of t*
*he
three factors in H(q1 ) ~= (SL2(q1 )3)={ (I, I, I)}. If ' 2 n is a permutatio*
*n au-
tomorphism, then it permutes the elements of EU4, and preserves the elements xC*
*(-)
by the formulas in (c). If ' 2 Inn(H(qn)) and '(E0) = E00for E0, E002 EU4, then
'(xC(E0)) = xC(E00) by definition of xC(-); and so the same property holds for *
*all
elements of n.
Following the notation introduced in Section 1, for P, Q S(qn), Hom Spin7(q*
*n)(P, Q)
denotes the set of homomorphisms from P to Q induced by conjugation by some
element of Spin7(qn). Also, if P, Q S(qn) \ H(qn), Hom n(P, Q) denotes the *
*set
of homomorphisms induced by restriction of an element of n. Let Fn = FSol(qn) *
*be
the fusion system over S(qn) generated by Spin7(qn) and n. In other words, for*
* each
P, Q S(qn), Hom Fn(P, Q) is the set of all composites
'1 '2 'k
P = P0 ---! P1 ---! P2 ---! . . .---!Pk-1 ---! Pk = Q,
where Pi S(qn) for all i, and each 'i lies in Hom Spin7(qn)(Pi-1, Pi) or (if *
*Pi-1, Pi
H(qn)) Hom n(Pi-1, Pi). This clearly defines a fusion system over S(qn).
Proposition 2.9. Fix n 1. Let E S(qn) be an elementary abelian subgroup of
rank 3 which contains U, and such that CS(qn)(E) 2 Syl2(CSpin7(qn)(E)). Then
{' 2 AutFn(CS(qn)(E)) | '(z) = z} = AutSpin7(qn)(CS(qn)(E)). (1)
Proof.Set
Spin = Spin7(qn), S = S(qn), = n, and F = Fn
for short. Consider the subgroups
R0 = R0(qn) def=!(C(q1 )3) \ S and R1 = R1(qn) def=CS(____) = .
Ran Levi and Bob Oliver *
* 15
Here, R0 is generated by elements ofnthe form [[X1, X2, X3]], where either Xi2 *
*C(qn),
or X1 = X2 = X3 = X 2 C(q2n) and _q (X) = -X. Also, C(qn) 2 Syl2(CSL2(qn)(A))
is cyclic of order 2k 4, where 2k is the largest power which divides qn 1; *
*and C(q2n)
is cyclic of order 2k+1. So
R0 ~=(C2k)3 and R1 = R0 o ,
where bB= [[B, B, B]] has order 2 and acts on R0 via (g 7! g-1). Note that
____= <[[ I, I, I]], [[A, A, A]]> ~=C32
is the 2-torsion subgroup of R0.
We claim that
R0 is the only subgroup of S isomorphic to (C2k)3. (2)
To see this, let R0 S be any subgroup isomorphic to (C2k)3, and let E0 ~=C32be*
* its
2-torsion subgroup. Recall that for any 2-group P , the Frattini subgroup Fr(P *
*) is the
subgroup generated by commutators and squares in P . Thus
E0 Fr(R0) Fr(S)
(note that [[B, B, I]] = (ø.[[B, I, I]])2). Any elementary abelian subgroup of *
*rank 4 in
Fr(S) would have to contain ____(the 2-torsion in R0 ~=C32k), and this is im*
*possi-
ble since no element of the coset R0.[[B, B, I]] commutes with Ab. Thus, rk(Fr*
*(S)) =
3. Hence U E0, since otherwise ____ would be an elementary abelian sub-
group of Fr(S) of rank 4. This in turn implies that R0 CS(U), and hence
that E0 Fr(CS(U)) R0. Thus E0 = ____(the 2-torsion in R0 again). Hence
R0 CS(____) = , and it follows that R0= R0. This finishes the proof*
* of (2).
Choose generators x1, x2, x3 2 R0 as follows. Fix X 2 CSL2(q1()A) of order 2k*
*, and
Y 2 CSL2(q2n)(A) of order 2k+1 such that Y 2= X. Set x1 = [[I, I, X]], x2 = [[X*
*, I, I]],
k-1 2k-1 2k-1
and x3 = [[Y, Y, Y ]]. Thus, x21 = z, x2 = z1, and (x3) = bA.
Now let E S(qn) be an elementary abelian subgroup of rank 3 which contains *
*U,
and such that CS(qn)(E) 2 Syl2(CSpin(E)). In particular, E R1 = CS(qn)(U). Th*
*ere
are two cases to consider: that where E R0 and that where E R0.
Case 1: Assume E R0. Since R0 is abelian of rank 3, we must have E = ____,
the 2-torsion subgroup of R0, and CS(E) = R1. Also, by (2), neither R0 nor R1 *
*is
isomorphic to any other subgroup of S; and hence
ff
AutF (Ri) = AutSpin(Ri), Aut (Ri) for i = 0, 1. (4)
By Proposition A.8, AutSpin(E) is the group of all automorphisms of E which s*
*end
z to itself. In particular, since H(qn) = CSpin(U), AutH(qn)(E) is the group of*
* all auto-
morphisms of E which are the identity on U. Also, = Inn(H(qn)).b 3, where b 3*
*sends
bA= [[A, A, A]] to itself and permutes the nontrivial elements of U = {[[ I, I*
*, I]]}.
Hence Aut (E) is the group of all automorphisms which send U to itself. So if*
* we
identify Aut(E) ~=GL3(Z=2) via the basis {z, z1, bA}, then
Aut Spin(E) = T1 def=GL12(Z=2) = (aij) 2 GL3(Z=2) | a21= a31= 0
and
Aut (E) = T2 def=GL21(Z=2) = (aij) 2 GL3(Z=2) | a31= a32= 0 .
16 Construction of 2-local finite groups
By (2) (and since E is the 2-torsion in R0),
NSpin(E) = NSpin(R0) and {fl 2 | fl(E) = E} = {fl 2 | fl(R0) = R0*
*}.
Since CSpin(E) = CSpin(R0)., the only nonidentity element of Aut Spin(R0) o*
*r of
Aut (R0) which is the identity on E is conjugation by bB, which is -I. Hence re*
*striction
from R0 to E induces isomorphisms
Aut Spin(R0)={ I} ~=AutSpin(E) and Aut (R0)={ I} ~=Aut (E).
Upon identifying Aut(R0) ~=GL3(Z=2k) via the basis {x1, x2, x3}, these can be r*
*egarded
as sections
~i:Ti-----! GL3(Z=2k)={ I} = SL3(Z=2k) x {~I | ~ 2 (Z=2k)*}={ I}
of the natural projection from GL3(Z=2k)={ I} to GL3(Z=2), which agree on the g*
*roup
T0 = T1 \ T2 of upper triangular matrices.
We claim that ~1 and ~2 both map trivially to the second factor. Since this f*
*actor
is abelian, it suffices to show that T0 is generated by [T1, T1] \ T0 and [T2, *
*T2] \ T0, and
that each Ti is generated by [Ti, Ti] and T0 _ and this is easily checked. (Not*
*e that
T1 ~=T2 ~= 4.)
By carrying out the above procedure over the field Fq2n, we see that both of *
*these
sections ~i can be lifted further to SL3(Z=2k+1) (still agreeing on T0). So by *
*Lemma
A.10, there is a section
~: GL3(Z=2) -----! SL3(Z=2k)
which extends both ~1 and ~2. By (4), AutF (R0) = Im(~).<- I>.
We next identify Aut F(R1). By Lemma 2.8(a), E* def= Spin7(q*
*n)
is a subgroup of rank 4 and type I. So by Proposition A.8, Aut Spin(E*) contain*
*s all
automorphisms of E* ~= C42which send z 2 Z(Spin) to itself. Hence for any x 2
NSpin(R1), since cx(z) = z, there is x1 2 NSpin(E*) such that cx1|E = cx|E (i.e*
*., xx-112
CSpin(E)) and cx1(Bb) = bB(i.e., [x1, bB] = 1). Set x2 = xx-11.
Since CSpin(U) = H(qn) Im(!), we see that CSpin(E) = K0., where
K0 = !(CSL2(q1()A)3) \ Spin
is abelian, R0 2 Syl2(K0), and bBacts on K0 by inversion. Upon replacing x1 by *
*bBx1
and x2 by x2bB-1if necessary, we can assume that x2 2 K0. Then
[x2, bB] = x2.(Bbx2bB-1)-1 = x22,
while by the original choice of x, x1 we have
[x2, bB] = [xx-11, bB] = [x, bB] 2 R0.
Thus x222 R0 2 Syl2(K0), and hence x2 2 R0 R1. Since x = x2x1 was an arbitrary
element of NSpin(R1), this shows that NSpin(R1) R1.CSpin(Bb), and hence that
Aut Spin(R1) = Inn(R1).{' 2 AutSpin(R1) | '(Bb) = bB}. (5)
Since Aut (R1) is generated by its intersection with AutSpin(R1) and the grou*
*p b 3
which permutes the three factors in H(q1 ) (and since the elements of b 3all fi*
*x bB), we
also have
Aut (R1) = Inn(R1).{' 2 Aut (R1) | '(Bb) = bB}.
Ran Levi and Bob Oliver *
* 17
Together with (4) and (5), this shows that AutF (R1) is generated by Inn(R1) to*
*gether
with certain automorphisms of R1 = R0.which send bBto itself. In other word*
*s,
fi
Aut F(R1) = Inn(R1). ' 2 Aut(R1) fi'(Bb) = bB, '|R0 2 AutF (R0)
fi
= Inn(R1). ' 2 Aut(R1) fi'(Bb) = bB, '|R0 2 ~(GL3(Z=2)) .
Thus
fi
' 2 AutF (R1) fi'(z) = z
fi
= Inn(R1). ' 2 Aut(R1) fi'(Bb) = bB, '|R0 2 ~(T1) = AutSpin(R0) = AutSpin(R1),
the last equality by (5); and (1) now follows.
Case 2: Now assume that E R0. By assumption, U E (hence E CS(E)
CS(U)), and CS(E) is a Sylow subgroup of CSpin(E). Since CS(E) is not isomorphic
to R1 = CS() (by (2)), this shows that E is not Spin-conjugate to .
By Proposition A.8, Spin contains exactly two conjugacy classes of rank 3 subgr*
*oups
containing z, and thus E must have type II. Hence by Proposition A.8(d), CS(E) *
*is
elementary abelian of rank 4, and also has type II.
Let C be the Spin7(qn)-conjugacy class of the subgroup E* = ____~=C42,*
* which
by Lemma 2.8(a) has type I. Let E0 be the set of all subgroups of S which are e*
*le-
mentary abelian of rank 4, contain U, and are not in C. By Lemma 2.8(e), for a*
*ny
' 2 Iso (E0, E00) and any E0 2 E0, E00def='(E0) 2 E0, and ' sends xC(E0) to xC(*
*E00).
The same holds for ' 2 IsoSpin(E0, E00) by definition of the elements xC(-) (Pr*
*oposi-
tion A.9). Since CS(E) 2 E0, this shows that all elements of Aut F(CS(E)) send*
* the
element xC(CS(E)) to itself. By Proposition A.9(c), Aut Spin(CS(E)) is the gro*
*up of
automorphisms which are the identity on the rank two subgroup ; a*
*nd
(1) now follows.
One more technical result is needed.
Lemma 2.10. Fix n 1, and let E, E0 S(qn) be two elementary abelian subgroups*
* of
rank three which contain U, and which are n-conjugate. Then E and E0are Spin7(*
*qn)-
conjugate.
Proof.By [Sz, 3.6.3(ii)], -I is the only element of order 2 in SL2(q1 ). Consid*
*er the
sets fi
J1 = X 2 SL2(qn) fiX2 = -I
and fi n
J2 = X 2 SL2(q2n) fi_q (X) = -X, X2 = -I .
n qn
Here, as usual, _q is induced by the field automorphism (x 7! x ). All elemen*
*ts in
J1 are SL2(q)-conjugate (this follows, for example, from [Sz, 3.6.23]), and we *
*claim the
same is true for elements of J2.
n
Let SL*2(qn) be the group of all elements X 2 SL2(q2n) such that _q (X) = X.
This is a group which contains SL2(qn) with index 2. Let k be such that the Sy*
*low
2-subgroups of SL2(qn) have order 2k; then k 3 since |SL2(qn)| = qn(q2n - 1).*
* Any
S 2 Syl2(SL*2(qn)) is quaternion of order 2k+1 16 (see [Go , Theorem 2.8.3]) *
*and
its intersection with SL2(qn) is quaternion of order 2k, so all elements in S \*
* J2 are
S-conjugate. It follows that all elements of J2 are SL*2(qn)-conjugate. If X, X*
*0 2 J2
and X0 = gXg-1 for g 2 SL*2(qn), then either g 2 SL2(qn) or gX 2 SL2(qn), and in
either case X and X0 are conjugate by an element of SL2(qn).
18 Construction of 2-local finite groups
By Proposition 2.5(a),
E, E0 CSpin7(qn)(U) = H(qn) def=!(SL2(q1 )3) \ Spin7(qn).
Thus E = and E0 = , where the*
* Xi are all
in J1 or all in J2, and similarly for the X0i. Also, since E and E0 are n-con*
*jugate
(and each element of n leaves U = invariant), the Xi and X0imust all b*
*e in
the same set J1 or J2. Hence they are all SL2(qn)-conjugate, and so E and E0 a*
*re
Spin7(qn)-conjugate.
We are now ready to show that the fusion systems Fn are saturated, and satisf*
*y the
conditions listed in Theorem 2.1.
Proposition 2.11. For a fixed odd prime power q, let S(qn) S(q1 ) Spin7(q1 )
be as defined above. Let z 2 Z(Spin7(q1 )) be the central element of order 2. *
* Then
for each n, Fn = FSol(qn) is saturated as a fusion system over S(qn), and satis*
*fies the
following conditions:
(a)For all P, Q S(qn) which contain z, if ff 2 Hom (P, Q) is such that ff(z) *
*= z,
then ff 2 Hom Fn(P, Q) if and only if ff 2 Hom Spin7(qn)(P, Q).
(b)CFn(z) = FS(qn)(Spin7(qn)) as fusion systems over S(qn).
(c)All involutions of S(qn) are Fn-conjugate.
Furthermore, Fm Fn for m|n. The union of the Fn is thus a category FSol(q1 )
whose objects are the finite subgroups of S(q1 ).
Proof.We apply Proposition 1.2, where p = 2, G = Spin7(qn), S = S(qn), Z = =
Z(G); and U and CG(U) = H(qn) are as defined above. Also, = n Aut(H(qn)).
Condition (a) in Proposition 1.2 (all noncentral involutions in G are conjugate*
*) holds
since all subgroups in E2 are conjugate (Proposition A.8), and condition (b) ho*
*lds by
definition of . Condition (c) holds since
{fl 2 | fl(z) = z} = Inn(H(qn)).= AutNG(U)(H(qn))
by definition, since H(qn) = CG(U), and by Proposition 2.5(b). Condition (d) w*
*as
shown in Proposition 2.9, and condition (e) in Lemma 2.10. So by Proposition 1.*
*2, Fn
is a saturated fusion system, and CFn(Z) = FS(qn)(Spin7(qn)).
The last statement is clear.
3.Linking systems and their automorphisms
We next show the existence and uniqueness of centric linking systems associat*
*ed to
the FSol(q), and also construct certain automorphisms of these categories analo*
*gous to
the automorphisms _q of the group Spin7(qn). One more technical lemma about ele-
mentary abelian subgroups, this time about their F-conjugacy classes, is first *
*needed.
Lemma 3.1. Set F = FSol(q). For each r 3, there is a unique F-conjugacy class*
* of
elementary abelian subgroups E S(q) of rank r. There are two F-conjugacy clas*
*ses
of rank four elementary abelian subgroups E S(q): one is the set C of subgro*
*ups
Spin7(q)-conjugate to E* = , while the other contains the other *
*conjugacy
class of type I subgroups as well as all type II subgroups. Furthermore, Aut F*
*(E) =
Ran Levi and Bob Oliver *
* 19
Aut(E) for all elementary abelian subgroups E S(q) except when E has rank four
and is not F-conjugate to E*, in which case
Aut F(E) = {ff 2 Aut(E) | ff(xC(E)) = xC(E)}.
Proof.By Lemma 2.8(d), the three subgroups
E* = , E001= , E100=
(where X is a generator of C(q)) represent the three Spin7(q)-conjugacy classes*
* of rank
four subgroups. Clearly, E100and E001are 1-conjugate, hence F-conjugate; and by
Lemma 2.8(e), neither is 1-conjugate to E*. This proves that there are exactly*
* two
F-conjugacy classes of such subgroups.
Since E* and E001both are of type I in Spin7(q), their Spin7(q)-automorphism *
*groups
contain all automorphisms which fix z (see Proposition A.8). By Lemma 2.8(e), z*
* is
fixed by all -automorphisms of E001, and so AutF (E001) is the group of all au*
*tomor-
phisms of E001which send z = xC(E001) to itself. On the other hand, E* contains
automorphisms (induced by permuting the three coordinates of H) which permute t*
*he
three elements z, z1, zz1; and these together with AutSpin(E*) generate Aut(E*).
It remains to deal with the subgroups of smaller rank. By Proposition A.8 ag*
*ain,
there is just one Spin7(q)-conjugacy class of elementary abelian subgroups of r*
*ank one
or two. There are two conjugacy classes of rank three subgroups, those of type *
*I and
those of type II. Since E100is of type II and E001of type I, all rank three sub*
*groups
of E001have type I, while some of the rank three subgroups of E100have type II.*
* Since
E001is F-conjugate to E100, this shows that some subgroup of rank three and typ*
*e II is
F-conjugate to a subgroup of type I, and hence all rank three subgroups are con*
*jugate
to each other. Finally, AutF (E) = Aut(E) whenver rk(E) 3 since any such grou*
*p is
F-conjugate to a subgroup of E* (and we have just seen that AutF(E*) = Aut(E*)).
To simplify the notation, we now define
FSpin(qn) def=FS(qn)(Spin7(qn))
for all 1 n 1: the fusion system of the group Spin7(qn) at the Sylow subgro*
*up
S(qn). By construction, this is a subcategory of FSol(qn). We write
OSol(qn) = O(FSol(qn)) and OSpin(qn) = O(FSpin(qn))
for the corresponding orbit categories: both of these have as objects the subgr*
*oups of
S(qn), and have as morphism sets
Mor OSol(qn)(P, Q) = Hom FSol(qn)(P, Q)= Inn(Q) Rep(P, Q)
and
MorOSpin(qn)(P, Q) = Hom FSpin(qn)(P, Q)= Inn(Q) .
Let OcSol(qn) OSol(qn) and OcSpin(qn) OSpin(qn) be the centric orbit catego*
*ries; i.e.,
the full subcategories whose objects are the FSol(qn)- or FSpin(qn)-centric sub*
*groups of
S(qn). (We will see shortly that these in fact have the same objects.)
The obstructions to the existence and uniqueness of linking systems associate*
*d to the
fusion systems FSol(qn), and to the existence and uniqueness of certain automor*
*phisms
of those linking systems, lie in certain groups which were identified in [BLO2 *
* ] and
[BLO1 ]. It is these groups which are shown to vanish in the next lemma.
20 Construction of 2-local finite groups
Lemma 3.2. Fix a prime power q, and let
ZSol(q): OcSol(q) ----! Ab and ZSpin(q): OcSpin(q) ----! Ab
be the functors Z(P ) = Z(P ). Then for all i 0,
lim-i(ZSol(q)) = 0 = lim-i(ZSpin(q)).
OcSol(q) OcSpin(q)
Proof.Set F = FSol(q) for short. Let P1, . .,.Pk be F-conjugacy class represent*
*atives
for all F-centric subgroups Pi S(q), arranged such that |Pi| |Pj| for i j*
*. For
each i, let Zi ZSol(q) be the subfunctor defined by setting Zi(P ) = ZSol(q)(P*
* ) if P
is conjugate to Pj for some j i and Zi(P ) = 0 otherwise. We thus have a filt*
*ration
0 = Z0 Z1 . . .Zk = ZSol(q)
of ZSol(q) by subfunctors, with the property that for each i, the quotient func*
*tor
Zi=Zi-1vanishes except on the conjugacy class of Pi (and such that (Zi=Zi-1)(Pi*
*) =
ZSol(q)(Pi)). By [BLO2 , Proposition 3.2],
lim-*(Zi=Zi-1) ~= *(Out F(Pi); Z(Pi))
for each i. Here, *( ; M) are certain graded groups, define in [JMO , x5] for*
* all finite
groups and all finite Z(p)[ ]-modules M. We claim that *(Out F(Pi); Z(Pi)) *
*= 0
except when Pi= S(q) or S0(q) (see Definition 2.6).
Fix an F-centric subgroup P S(q). For each j 1, let j(Z(P )) = {g 2
j
Z(P ) | g2 = 1}, and set E = 1(Z(P )) _ the 2-torsion in the center of P . Fo*
*r each
j
j 1, let j(Z(P )) = {g 2 Z(P ) | g2 = 1}, and set E = 1(Z(P )) _ the 2-tors*
*ion in
the center of P . We can assume E is fully centralized in F (otherwise replace *
*P and
E by appropriate subgroups in the same F-conjugacy classes).
Assume first that Q def=CS(q)(E) P , and hence that NQ(P ) P . Then any
x 2 NQ(P )r P centralizes E = 1(Z(P )). Hence for each j, x acts trivially *
*on
j(Z(P ))= j-1(Z(P )), since multiplication by pj-1sends this group NQ(P )=P -l*
*inearly
and monomorphically to E. Since cx is a nontrivial element of Out F(P ) of p-p*
*ower
order,
*(Out F(P ); j(Z(P ))= j-1(Z(P ))) = 0
for all j 1 by [JMO , Proposition 5.5], and thus *(Out F(P ); Z(P )) = 0.
Now assume that P = CS(q)(E) = P , the centralizer in S(q) of a fully F-centr*
*alized
elementary abelian subgroup. Since there is a unique conjugacy class of elemen*
*tary
abelian subgroup of any rank 3, CS(q)(E) always contains a subgroup C42, and *
*hence
P contains a subgroup C42which is self centralizing by Proposition A.8(a). This*
* shows
that Z(P ) is elementary abelian, and hence that Z(P ) = E.
We can assume P is fully normalized in F, so
Aut S(q)(P ) 2 Syl2(Aut F(P ))
by condition (I) in the definition of a saturated fusion system. Since P = CS(*
*q)(E)
(and E = Z(P )), this shows that
Ker OutF(P ) ---! AutF(E)
has odd order. Also, since E is fully centralized, any F-automorphism of E exte*
*nds to
an F-automorphism of P = CS(q)(E), and thus this restriction map between automo*
*r-
phism groups is onto. By [JMO , Proposition 6.1(i,iii)], it now follows that
i(Out F(P ); Z(P )) ~= i(Aut F(E); E). (1)
Ran Levi and Bob Oliver *
* 21
By Lemma 3.1, AutF (E) = Aut(E), except when E lies in one certain F-conjugacy
class of subgroups E ~=C42; and in this case P = E and AutF (E) is the group of*
* auto-
morphisms fixing the element xC(E). In this last (exceptional) case, O2(Aut F(E*
*)) 6= 1
(the subgroup of elements which are the identity on E=), so
*(Out F(P ); Z(P )) = *(Aut F(E); E) = 0 (2)
by [JMO , Proposition 6.1(ii)]. Otherwise, when AutF (E) = Aut(E), by [JMO , *
*Propo-
sition 6.3] we have
8
>:
0 otherwise.
By points (1), (2), and (3), the groups *(Out F(P ); Z(P )) vanish except in t*
*he two
cases E = or E = U, and these correspond to P = S(q) or P = NS(q)(U) = S0(q*
*).
We can assume that Pk = S(q) and Pk-1 = S0(q). We have now shown that
lim-*(Zk-2) = 0, and thus that ZSol(q) has the same higher limits as Zk=Zk-2. H*
*ence
lim-j(ZSol(q)) = 0 for all j 2, and there is an exact sequence
0 ---! lim-0(ZSol(q)) ----! lim-0(Zk=Zk-1)----! lim1(Zk-1=Zk-2)
~=Z=2 - ~=Z=2
----! lim-1(ZSol(q)) ---! 0.
One easily calculates that lim-0(ZSol(q)) = 0, and hence we also get lim-1(ZSol*
*(q)) = 0.
The proof that lim-i(ZSpin(q)) = 0 for all i 1 is similar, but simpler. If *
*F = FSpin(q),
then for any F-centric subgroup P S(q), there is an element x 2 NS(P )r P such
that [x, P ] = , and cx is a nontrivial element of O2(Out F(P )). Thus
*(Out F(P ); Z(P )) = 0
for all such P by [JMO , Proposition 6.1(ii)] again.
We are now ready to construct classifying spaces BSol(q) for these fusion sys*
*tems
FSol(q). The following proposition finishes the proof of Theorem 2.1, and also *
*contains
additional information about the spaces BSol(q).
To simplify notation, we write LcSpin(qn) = LcS(qn)(Spin7(qn)) (n 1) to den*
*ote the
centric linking system for the group Spin7(qn). The field automorphism (x 7! x*
*q)
induces an automorphism of Spin7(qn) which sends S(qn) to itself; and this in t*
*urn
induces automorphisms _qF= _qF(Sol), _qF(Spin), and _qL(Spin) of the fusion sys*
*tems
FSol(qn) FSpin(qn) and of the linking system LcSpin(qn).
Proposition 3.3. Fix an odd prime q, and n 1. Let S = S(qn) 2 Syl2(Spin7(qn))*
* be
as defined above. Let z 2 Z(Spin7(qn)) be the central element of order 2. Then *
*there
is a centric linking system
L = LcSol(qn) ---i--!FSol(qn)
associated to the saturated fusion system F def=FSol(qn) over S, which has the *
*following
additional properties.
(a)A subgroup P S is F-centric if and only if it is FSpin(qn)-centric.
22 Construction of 2-local finite groups
(b)LcSol(qn) contains LcSpin(qn) as a subcategory, in such a way that ß|LcSpin(*
*qn)is the
usual projection to FcSpin(qn), and that the distinguished monomorphisms
ffiP
P ---! AutL(P )
for L = LcSol(qn) are the same as those for LcSpin(qn).
(c)Each automorphism of LcSpin(qn) which covers the identity on FcSpin(qn) exte*
*nds to
an automorphism of LcSol(qn) which covers the identity on FcSol(qn). Further*
*more,
such an extension is unique up to composition with the functor
Cz: LcSol(qn) -----! LcSol(qn)
which is the identity on objects and sends ff 2 Mor LcSol(qn)(P, Q) to bzOff*
* Obz-1
(öc njugation by z").
(d)There is a unique automorphism _qL2 Aut (LcSol(qn)) which covers the automor-
phism of FSol(qn) induced by the field automorphism (x 7! xq), which extends*
* the
automorphism of LcSpin(qn) induced by the field automorphism, and which is t*
*he
identity on ß-1(FSol(q)).
Proof.By Proposition 2.11, F = FSol(qn) is a saturated fusion system over S = S*
*(qn) 2
Syl2(Spin7(qn)), with the property that CF (z) = FSpin(qn). Point (a) follows *
*as a
special case of [BLO2 , Proposition 2.5(a)].
Since lim-i(ZSol(qn))= 0 for i = 2, 3 by Lemma 3.2, there is by [BLO2 , Pr*
*opo-
OcSol(qn)
sition 3.1] a centric linking system L = LcSol(qn) associated to F, which is un*
*ique up
to isomorphism (an isomorphism which commutes with the projection to FSol(qn) a*
*nd
with the distinguished monomorphisms). Furthermore, ß-1(FSpin(qn)) is a linking*
* sys-
tem associated to FSpin(qn), such a linking system is unique up to isomorphism *
*since
lim-2(ZSpin(qn)) = 0 (Lemma 3.2 again), and this proves (b).
(c) By [BLO1 , Theorem 6.2] (more precisely, by the same proof as that used in
[BLO1 ]), the vanishing of lim-i(ZSol(qn)) for i = 1, 2 (Lemma 3.2) shows that*
* each
automorphism of F = FSol(qn) lifts to an automorphism of L, which is unique up
to a natural isomorphism of functors; and any such natural isomorphism sends ea*
*ch
object P S to a isomorphism bgfor some g 2 Z(P ). Similarly, the vanishing *
*of
lim-i(ZSpin(qn)) for i = 1, 2 shows that each automorphism of FSpin(qn) lifts t*
*o an
automorphism of LcSpin(qn), also unique up to a natural isomorphism of functors*
*. Since
LcSol(qn) and LcSpin(qn) have the same objects by (a), this shows that each aut*
*omorphism
of LcSpin(qn) which covers the identity on FcSpin(qn) extends to a unique autom*
*orphism
of LcSol(qn) which covers the identity on FSol(qn).
It remains to show, for any 2 Aut(LcSol(qn)) which covers the identity on F*
*cSol(qn)
and such that |LcSpin(qn)= Id, that is the identity or conjugation by z. We*
* have
already noted that must be naturally isomorphic to the identity; i.e., that t*
*here are
elements fl(P ) 2 Z(P ), for all P in LcSol(qn), such that
(ff) = fl(Q) Off Ofl(P )-1 for all ff 2 Mor LcSol(qn)(P, Q), all P,*
* Q.
Since is the identity on LcSpin(qn), the only possibilities are fl(P ) = 1 fo*
*r all P (hence
= Id), or fl(P ) = z for all P (hence is conjugation by z).
(d) Now consider the automorphism _qF2 Aut(FSol(qn)) induced by the field auto-
morphism (x 7! xq) of Fqn. We have just seen that this lifts to an automorphism*
* _qL
of LcSol(qn), which is unique up to natural isomorphism of functors. The restri*
*ction of
Ran Levi and Bob Oliver *
* 23
q q
_L to LcSpin(qn), and the automorphism _L(Spin) of LcSpin(qn) induced directly *
*by the
field automorphism, are two liftings of _qF|FSpin(qn), and hence differ by a na*
*tural iso-
morphism of functors which extends to a natural isomorphism of functors on LcSo*
*l(qn).
Upon composing with this natural isomorphism, we can thus assume that _qLdoes
restrict to the automorphism of LcSpin(qn) induced by the field automorphism.
Now consider the action of _qLon AutL(S0(q)), which by assumption is the iden*
*tity
on Aut LcSpin(q)(S0(q)), and in particular on ffi(S0(q)) itself. Thus, with re*
*spect to the
extension
1 ---! S0(q) ----! AutL(S0(q)) ----! 3 ---! 1,
_qLis the identity on the kernel and on the quotient, and hence is described by*
* a cocycle
j 2 Z1( 3; Z(S0(q))) ~=Z1( 3; (Z=2)2).
Since H1( 3; (Z=2)2) = 0, j must be a coboundary, and thus the action of _qLon
AutL(S0(q)) is conjugation by an element of Z(S0(q)). Since it is the identity*
* on
AutLcSpin(q)(S0(q)), it must be conjugation by 1 or z. If it is conjugation by *
*z, then we
can replace _qL(on the whole category L) by its composite with z; i.e., by its *
*composite
with the functor which is the identity on objects and sends ff 2 Mor L(P, Q) to*
* bzOff Obz.
In this way, we can assume that _qLis the identity on AutL(S0(q)). By constru*
*ction,
every morphism in FSol(q) is a composite of morphisms in FSpin(q) and restricti*
*ons
of automorphisms in FSol(q) of S0(q). Since _qLis the identity on ß-1(FSpin(q))*
*, this
shows that it is the identity on ß-1(FSol(q)).
It remains to check the uniqueness of _qL. If _0 is another functor with the*
* same
properties, then by (e), (_0)-1 O_qLis either the identity or conjugation by z;*
* and the
latter is not possible since conjugation by z is not the identity on ß-1(FSol(q*
*)).
This finishes the construction of the classifying spaces BSol(q) = |LSol(q)|^*
*2for the
fusion systems constructed in Section 2. We end the section with an explanatio*
*n of
why these are not the fusion systems of finite groups.
Proposition 3.4. For any odd prime power q, there is no finite group G whose fu*
*sion
system is isomorphic to that of FSol(q).
Proof.Let G be a finite group, fix S 2 Syl2(G), and assume that S ~= S(q) 2
Syl2(Spin7(q)), and that the fusion system FS(G) satisfies conditions (a) and (*
*b) in
Theorem 2.1. In particular, all involutions in G are conjugate, and the centra*
*lizer
of any involution z 2 G has the fusion system of Spin7(q). When q 3 (mod 8),
Solomon showed [So, Theorem 3.2] that there is no finite group whose fusion sys*
*tem
has these properties. When q 1 (mod 8), he showed (in the same theorem) that
there is no such G such that bH def=CG(z)=O20(CG(z)) is isomorphic to a subgrou*
*p of
Aut(Spin7(q)) which contains Spin7(q) with odd index. (Here, O20(-) means larg*
*est
odd order normal subgroup.)
Let G be a finite group whose fusion system is isomorphic to FSol(q), and aga*
*in
0
set Hb def=CG(z)=O20(CG(z)) for some involution z 2 G. Set H = O2 (Hb=): t*
*he
smallest normal subgroup of Hb= of odd index. Then H has the fusion system *
*of
7(q) ~=Spin7(q)=Z(Spin7(q)). We will show that H ~= 7(q0) for some odd prime p*
*ower
0 0
q0. It then follows that O2 (Hb) ~=Spin7(q ), thus contradicting Solomon's theo*
*rem and
proving our claim.
The following "classification freeä rgument for proving that H ~= 7(q0) for *
*some q0
was explained to us by Solomon. We refer to the appendix for general results ab*
*out
24 Construction of 2-local finite groups
the groups Spinn(q) and n(q). Fix S 2 Syl2(H). Thus S is isomorphic to a Sylow
2-subgroup of 7(q), and has the same fusion.
0
We first claim that H must be simple. By definition (H = O2 (Hb=)), H has
no proper normal subgroup of odd index, and H has no proper normal subgroup of
odd order since any such subgroup would lift to an odd order normal subgroup of
bH= CG(z)=O20(CG(z)). Hence for any proper normal subgroup N C H, Q def=N \ S
is a proper normal subgroup of S, which is strongly closed in S with respect to*
* H
in the sense that no element of Q can be H-conjugate to an element of Sr Q. Usi*
*ng
Lemma A.4(a), one checks that the group 7(q) contains three conjugacy classes *
*of
involutions, classified by the dimension of their (-1)-eigenspace. It is not ha*
*rd to see
(by taking products) that any subgroup of S which contains all involutions in o*
*ne
of these conjugacy classes contains all involutions in the other two classes as*
* well.
Furthermore, S is generated by the set of all of its involutions, and this show*
*s that
there are no proper subgroups which are strongly closed in S with respect to H.*
* Since
we have already seen that the intersection with S of any proper normal subgroup*
* of H
would have to be such a subgroup, this shows that H is simple.
Fix an isomorphism
' 0
S -------!~S 2 Syl2( 7(q))
=
which preserves fusion. Choose x0 2 S0 whose (-1)-eigenspace is 4-dimensional, *
*and
such that is fully centralized in FS0( 7(q)). Then
CO7(q)(x0) ~=O+4(q) x O3(q)
by Lemma A.4(c). Since +4(q) O+4(q) and 3(q) O3(q) both have index 4,
C 7(q)(x0) is isomorphic to a subgroup of O+4(q) x O3(q) of index 4, and contai*
*ns a
normal subgroup K0~= +4(q) x 3(q) of index 4. Since is fully centralized,*
* CS0(x0)
is a Sylow 2-subgroup of C 7(q)(x0), and hence S00def=S0\ K0 is a Sylow 2-subgr*
*oup of
K0.
Set x = '-1(x0) 2 S. Since S ~=S0 have the same fusion in H and 7(q), CS(x) *
*~=
CS0(x0) have the same fusion in CH (x) and C 7(q)(x0). Hence H1(CH (x); Z(2)) *
* ~=
H1(C 7(q)(x0); Z(2)) (homology is determined by fusion), both have order 4, and*
* thus
CH (x) also has a unique normal subgroup K C H of index 4. Set S0 = K \ S. Thus
'(S0) = S00, and using Alperin's fusion theorem one can show that this isomorph*
*ism
is fusion preserving with respect to the inclusions of Sylow subgroups S0 K a*
*nd
S00 K0.
Using the isomorphisms of Proposition A.5:
+4(q) ~=SL2(q) xSL2(q) and 3(q) ~=P SL2(q),
we can write K0= K01xK02, where K01~=SL2(q) and K02~=SL2(q) x P SL2(q). Set
S0i= S0\K0i2 Syl2(K0i); thus S00= S01xS02. Set Si= '-1(S0i), so that S0 = S*
*1xS2
is normal of index 4 in CS(x). The fusion system of K thus splits as a central *
*product
of fusion systems, one of which is isomorphic to the fusion system of SL2(q).
We now apply a theorem of Goldschmidt, which says very roughly that under the*
*se
conditions, the group K also splits as a central product. To make this more pre*
*cise,
let Kibe the normal closure of Siin K C CH (x). By [Gd , Corollary A2], since S*
*1 and
S2 are strongly closed in S0 with respect to K,
[K1, K2] .O20(K).
Ran Levi and Bob Oliver *
* 25
Using this, it is not hard to check that Si 2 Syl2(Ki). Thus K1 has same fusio*
*n as
SL2(q) and is subnormal in CH (x) (K1 C K C CH (x)), and an argument similar to
that used above to prove the simplicity of H shows that K1=(.O20(K1)) is sim*
*ple.
Hence K1 is a 2-component of CH (x) in the sense described by Aschbacher in [As*
*1 ].
By [As1 , Corollary III], this implies that H must be isomorphic to a Chevalley*
* group
of odd characteristic, or to M11. It is now straightforward to check that among*
* these
groups, the only possibility is that H ~= 7(q0) for some odd prime power q0.
4.Relation with the Dwyer-Wilkerson space
We now want to examine the relation between the spaces BSol(q) which we have *
*just
constructed, and the space BDI(4) constructed by Dwyer and Wilkerson in [DW1 ].
Recall that this is a 2-complete space characterized by the property that its c*
*oho-
mology is the Dickson algebra in four variables over F2; i.e., the ring of inva*
*riants
F2[x1, x2, x3, x4]GL4(2). We show, for any odd prime power q, that BDI(4) is ho*
*motopy
equivalent to the 2-completion of the union of the spaces BSol(qn), and that BS*
*ol(q) is
homotopy equivalent to the homotopy fixed point set of an Adams map from BDI(4)
to itself.
We would like to define an infinite "linking system" LcSol(q1 ) as the union *
*of the
finite categories LcSol(qn), and then set BSol(q1 ) = |LcSol(q1 )|^2. The diffi*
*culty with this
approach is that a subgroup which is centric in the fusion system FSol(qm ) nee*
*d not be
centric in a larger fusion system FSol(qn) (for m|n). To get around this proble*
*m, we
define LccSol(qn) LcSol(qn) to be the full subcategory whose objects are thos*
*e subgroups
of S(qn) which are FSol(q1 )-centric; or equivalently FSol(qk)-centric for all *
*k 2 nZ.
Similarly, we define LccSpin(qn) to be the full subcategory of LcSpin(qn) whose*
* objects are
those subgroups of S(qn) which are FSpin(q1 )-centric. We can then define LcSo*
*l(q1 )
and LcSpin(q1 ) to be the unions of these categories.
For these definitions to be useful, we must first show that |LccSol(qn)|^2has*
* the same
homotopy type as |LcSol(qn)|^2. This is done in the following lemma.
Lemma 4.1. For any odd prime power q and any n 1, the inclusions
|LccSol(qn)|^2 |LcSol(qn)|^2 and |LccSpin(qn)|^2 |LcSpin(qn)|*
*^2
are homotopy equivalences.
Proof.It clearly suffices to show this when n = 1.
Recall, for a fusion system F over a p-group S, that a subgroup P S is F-ra*
*dical
if OutF (P ) is p-reduced; i.e., if Op(Out F(P )) = 1. We will show that
all FSol(q)-centric FSol(q)-radical subgroups of S(q) are also FSol(q1 )-cen*
*tric(1)
and similarly
all FSpin(q)-centric FSpin(q)-radical subgroups of S(q) are also FSpin(q1()-ce*
*ntric.2)
In other words, (1) says that for each P S(q) which is an object of LcSol(q) *
*but not
of LccSol(q), O2 Out FSol(q)(P ) 6= 1. By [JMO , Proposition 6.1(ii)], this i*
*mplies that
*(Out FSol(q)(P ); H*(BP ; F2)) = 0.
26 Construction of 2-local finite groups
Hence by [BLO2 , Propositions 3.2 & 2.2] (and the spectral sequence for a homo*
*topy
colimit), the inclusion LccSol(q) LcSol(q) induces an isomorphism
c ~= * cc
H* |LSol(q)|; F2 ------! H |LSol(q)|; F2 ,
and thus |LccSol(q)|^2' |LcSol(q)|^2. The proof that |LccSpin(q)|^2' |LcSpin(q*
*)|^2is similar,
using (2).
Point (2) is shown in Proposition A.12, so it remains only to prove (1). Set*
* F =
FSol(q), and set Fk = FSol(qk) for all 1 k 1. Let E Z(P ) be the 2-torsio*
*n in
the center of P , so that P CS(q)(E). Set
8
>> if rk(E) = 1
><
if rk(E) = 2
E0=
>>if rk(E) = 3
>:
E if rk(E) = 4
in the notation of Definition 2.6. In all cases, E is F-conjugate to E0 by Lem*
*ma
3.1. We claim that E0 is fully centralized in Fk for all k < 1. This is clear*
* when
rk(E0) = 1 (E0 = Z(S(qk))), follows from Proposition 2.5(a) when rk(E0) = 2, and
from Proposition A.8(a) (all rank 4 subgroups are self centralizing) when rk(E0*
*) = 4.
If rk(E0) = 3, then by Proposition A.8(d), the centralizer in Spin7(qk) (hence *
*in S(qk))
of any rank 3 subgroup has an abelian subgroup of index 2; and using this (toge*
*ther
with the construction of S(qk) in Definition 2.6), one sees that E0 is fully ce*
*ntralized
in Fk.
_ If E0 6= E, choose ' 2 Hom F (E, S(q)) such that '(E) = E0; then ' extends to
' 2 Hom F(CS(q)(E), S(q)) by condition_(II) in the definition of a saturated f*
*usion
system, and we can replace P by '(P ) and E by '(E). We can thus assume that
E is fully centralized in Fk for each k < 1. So by [BLO2 , Proposition 2.5(a)*
*], P
is Fk-centric if and only if it is CFk(E)-centric; and this also holds when k =*
* 1.
Furthermore, since OutCF(E)(P ) C OutF (P ), O2(Out CF(E)(P )) is a normal 2-su*
*bgroup
of OutF (P ), and thus
O2 Out CF(E)(P ) O2(Out F(P )).
Hence P is CF (E)-radical if it is F-radical. So it remains to show that
all CF (E)-centric CF (E)-radical subgroups of S(q) are also CF1 (E)-centric*
*.(3)
If rk(E) = 1, then CF (E) = FSpin(q) and CF1 (E) = FSpin(q1 ), and (3) follow*
*s from
(2). If rk(E) = 4, then P = E = CS(q1 )(E) by Proposition A.8(a), so P is F1 -c*
*entric,
and the result is clear.
If rk(E) = 3, then by Proposition A.8(d), CF (E) CF1 (E) are the fusion sys*
*tems
of a pair of semidirect products Ao C2 A1 oC2, where A A1 are abelian and *
*C2
acts on A1 by inversion. Also, E is the full 2-torsion subgroup of A1 , since o*
*therwise
rk(A1 ) > 3 would imply A1 oC2 Spin7(q1 ) contains a subgroup C52(contradicti*
*ng
Proposition A.8). If P A, then either Out CF(E)(P ) has order 2, which contra*
*dicts
the assumption that P is radical; or P is elementary abelian and Out CF(E)(P ) *
*= 1,
in which case P Z(Ao C2) is not centric. Thus P A; P \ A E contains all
2-torsion in A1 , and hence P is centric in A1 oC2.
If rk(E) = 2, then by Proposition 2.5(a), CF1 (E) and CF (E) are the fusion s*
*ystems
of the groups
H(q1 ) ~=SL2(q1 )3={ (I, I, I)} (4)
Ran Levi and Bob Oliver *
* 27
and
H(q) = H(q1 ) \ Spin7(q) H0(q) def=SL2(q)3={ (I, I, I)}.
If P S(q) is centric and radical in the fusion system of H(q), then by Lemma *
*A.11(c),
its intersection with H0(q) ~=SL2(q)3={ (I, I, I)} is centric and radical in th*
*e fusion
system of that group. So by Lemma A.11(a,f),
P \ H0(q) ~=(P1 x P2 x P3)={ (I, I, I)} (5)
for some Piwhich are centric and radical in the fusion system of SL2(q). Since *
*the Sylow
2-subgroups of SL2(q) are quaternion [Go , Theorem 2.8.3], the Pimust be nonabe*
*lian
and quaternion, so each Pi={ I} is centric in P SL2(q1 ). Hence P \ H0(q) is ce*
*ntric
in SL2(q)3={ (I, I, I)} by (5), and so P is centric in H(q1 ) by (4).
We would like to be able to regard BSpin7(q) as a subcomplex of BSol(q), but *
*there
is no simple natural way to do so. Instead, we set
BSpin07(q) = |LccSpin(q)|^2 |LccSol(q)|^2 BSol(q);
then BSpin07(q) ' BSpin7(q)^2by [BLO1 , Proposition 1.1] and Lemma 4.1. Also, *
*we
write
BSol0(q) = |LccSol(q)|^2 BSol(q) def=|LcSol(q)|^2
to denote the subcomplex shown in Lemma 4.1 to be equivalent to BSol(q); and
set BSpin07(q1 ) = |LcSpin(q1 )|^2. From now on, when we talk about the inclus*
*ion of
BSpin7(q) into BSol(q), as long as it need only be well defined up to homotopy,*
* we
mean the composite
BSpin7(q) ' BSpin07(q) BSol0(q)
(for some choice of homotopy equivalence). Similarly, if we talk about the inc*
*lu-
sion of BSol(qm ) into BSol(qn) (for m|n) where it need only be defined up to h*
*omo-
topy, we mean these spaces identified with their equivalent subcomplexes BSol0(*
*qm )
BSol0(qn).
Lemma 4.2. Let q be any odd prime. Then for all n 1,
H*(BSol(qn); F2)___! H*(BH(qn); F2)C3
| |
| | (1)
# #
H*(BSpin7(qn); F2)__! H*(BH(qn); F2)
(with all maps induced by inclusions of groups or spaces) is a pullback square.
Proof.By [BLO2 , Theorem B], H*(BSol(qn); F2) is the ring of elements in the c*
*oho-
mology of S(qn) which are stable relative to the fusion. By the construction in*
* Section
2, the fusion in Sol(qn) is generated by that in Spin7(qn), together with the p*
*ermu-
tation action of C3 on the subgroup H(qn) Spin7(qn), and hence (1) is a pullb*
*ack
square.
Proposition 4.3. For each odd prime q, there is a category LcSol(q1 ), together*
* with a
functor
ß :LcSol(q1 ) ------! FSol(q1 ),
such that the following hold:
(a)For each n 1, ß-1(FSol(qn)) ~=LccSol(qn).
28 Construction of 2-local finite groups
(b)There is a homotopy equivalence
''
BSol(q1 ) def=|LcSol(q1 )|^2-------! BDI(4)
'
such that the following square commutes up to homotopy
ffi(q1 ) 1
BSpin07(q1 )^2_____! BSol(q )
| |
''0'| '''| (1)
# b #
ffi
BSpin(7)^2________! BDI(4) .
Here, j0 is the homotopy equivalence_of [FM ], induced by some fixed choice *
*of em-
bedding of the Witt vectors for Fq into C, while ffi(q1 ) is the union of th*
*e inclusions
|LccSpin(qn)|^2 |LccSol(qn)|^2, and bffiis the inclusion arising from the c*
*onstruction of
BDI(4) in [DW1 ].
Furthermore, there is an automorphism _qL2 Aut(LcSol(q1 )) of categories which *
*satis-
fies the conditions:
(c)the restriction of _qLto each subcategory LccSol(qn) is equal to the restric*
*tion of _qL2
Aut (LcSol(qn)) as defined in Proposition 3.3(d);
(d)_qLcovers the automorphism _qFof FSol(q1 ) induced by the field automorphism
(x 7! xq); and
(e)for each n, (_qL)n fixes LccSol(qn).
Proof.By Proposition 2.11, the inclusions Spin7(qm ) Spin7(qn) for all m|n in*
*duce
inclusions of fusion systems FSol(qm ) FSol(qn). Since the restriction of a *
*linking
system over FccSol(qn) is a linking system over FccSol(qm ), the uniqueness of *
*linking systems
(Proposition 3.3) implies that we get inclusions LccSol(qm ) LccSol(qn). We d*
*efine LcSol(q1 )
to be the union of the finite categories LccSol(qn). (More precisely, fix a se*
*quence of
positive integers n1|n2|n3| . .s.uch that every positive integer divides some n*
*i, and set
1[
LcSol(q1 ) = LccSol(qni).
i=1
Then by uniqueness again, we can identify LccSol(qn) for each n with the approp*
*riate
subcategory.)
Let ß :LcSol(q1 ) --! FSol(q1 ) be the union of the projections from LccSol(q*
*ni) to
FSol(qni) FSol(q1 ). Condition (a) is clearly satisfied. Also, using Proposit*
*ion 3.3(d)
and Lemma 4.1, we see that there is an automorphism _qLof LcSol(q1 ) which sati*
*sfies
conditions (c,d,e) above. (Note that by the fusion theorem as shown in [BLO2 ,*
* The-
orem A.10], morphisms in LcSol(qn) are generated by those between radical subgr*
*oups,
and hence by those in LccSol(qn).)
It remains only to show that |LcSol(q1 )|^2' BDI(4), and to show that square *
*(1)
commutes. The space BDI(4) is 2-complete by its construction in [DW1 ]. By Lem*
*ma
4.1,
c n * n
H*(BSol(q1 ); F2) ~=lim-H* |LSol(q )|; F2 = lim-H BSol(q ); F2 .
n n
Ran Levi and Bob Oliver *
* 29
Hence by Lemma 4.2 (and since the inclusions BSpin7(qn) --! BSol(qn) commute wi*
*th
the maps induced by inclusions of fields Fqm Fqn), there is a pullback square
H*(BSol(q1 ); F2)_! H*(BH(q1 ); F2)C3
| |
| | (2)
# #
H*(BSpin7(q1 ); F2)_! H*(BH(q1 ); F2) .
Also, by [FM , Theorem 1.4], there are maps
3
BSpin7(q1 ) ----! BSpin(7) and BH(q1 ) ----! B SU(2) ={ (I, I, I)}
which induce isomorphisms of F2-cohomology, and hence homotopy equivalences aft*
*er
2-completion. So by Propositions 4.7 and 4.9 (or more directly by the computati*
*ons
in [DW1 , x3]), the pullback of the above square is the ring of Dickson invari*
*ants in the
polynomial algebra H*(BC42; F2), and thus isomorphic to H*(BDI(4); F2).
Point (b), including the commutativity of (1), now follows from the following*
* lemma.
Lemma 4.4. Let X be a 2-complete space such that H*(X; F2) is the Dickson alge-
f
bra in 4 variables. Assume further that there is a map BSpin(7) --! X such that
H*(f|BC42; F2) is the inclusion of the Dickson invariants in the polynomial alg*
*ebra
H*(BC42; F2). Then X ' BDI(4). More precisely, there is a homotopy equivalence
between these spaces such that the composite
f
BSpin(7) ------! X ' BDI(4)
is the inclusion arising from the construction in [DW1 ].
Proof.In fact, Notbohm [Nb , Theorem 1.2] has proven that the lemma holds even
without the assumption about BSpin(7) (but with the more precise assumption that
H*(X; F2) is isomorphic as an algebra over the Steenrod algebra to the Dickson *
*alge-
bra). The result as stated above is much more elementary (and also implicit in *
*[DW1 ]),
so we sketch the proof here.
Since H*(X; F2) is a polynomial algebra, H*( X; F2) is isomorphic as a graded
vector space to an exterior algebra on the same number of variables, and in par*
*ticular
is finite. Hence X is a 2-compact group. By [DW3 , Theorem 8.1] (the central*
*izer
decomposition for a p-compact group), there is an F2-homology equivalence
hocolim-----!(ff) ---'---!X.
A
Here, A is the category of pairs (V, '), where V is a nontrivial elementary ab*
*elian
2-group, and ' : BV --! X makes H*(BV ; F2) into a finitely generated module o*
*ver
H*(X; F2) (see [DW2 , Proposition 9.11]). Morphisms in A are defined by letti*
*ng
Mor A((V, '), (V 0, '0)) be the set of monomorphisms V --! V 0of groups which *
*make
the obvious triangle commute up to homotopy. Also,
ff: Aop--! Top is the functorff(V, ') = Map (BV, X)'.
By [DW1 , Lemma 1.6(1)] and [La , Th'eor`eme 0.4], A is equivalent to the cate*
*gory of
elementary abelian 2-groups E with 1 rk(E) 4, whose morphisms consist of all
'
group monomorphisms. Also, if BC2 --! X is the restriction of f to any subgroup
C2 Spin(7), then in the notation of Lannes,
TC2(H*(X; F2); '*) ~=H*(BSpin(7); F2)
30 Construction of 2-local finite groups
by [DW1 , Lemmas 16.(3), 3.10 & 3.11], and hence
H*(Map (BC2, X)'; F2) ~=H*(BSpin(7); F2)
by Lannes [La , Th'eor`eme 3.2.1]. This shows that
i j
Map (BC2, X)' ^2' BSpin(7)^2,
and thus that the centralizers of other elementary abelian 2-groups are the sam*
*e as their
centralizers in BSpin(7)^2. In other words, ff is equivalent in the homotopy c*
*ategory
to the diagram used in [DW1 ] to define BDI(4). By [DW1 , Proposition 7.7] (a*
*nd the
remarks in its proof), this homotopy functor has a unique homotopy lifting to s*
*paces.
So by definition of BDI(4),
^
X ' hocolim-----!(ff) 2 ' BDI(4).
A
Set B_q def=|_qL|, a self homotopy equivalence of BSol(q1 ) ' BDI(4). By cons*
*truc-
tion, the restriction of B_q to the maximal torus of BSol(q1 ) is the map induc*
*ed by
x 7! xq, and hence this is an Ä dams mapä s defined by Notbohm [Nb ]. In fact,
by [Nb , Theorem 3.5], there is an Adams map from BDI(4) to itself, unique up to
homotopy, of degree any 2-adic unit.
Following Benson [Be ], we define BDI4(q) for any odd prime power q to be the
homotopy fixed point set of the Z-action on BSol(q1 ) ' BDI(4) induced by the
Adams map B_q. By öh motopy fixed point set" in this situation, we mean that the
following square is a homotopy pullback:
BDI4(q) ____________! BSol(q1 )
| |
| |
# #
(Id,B_q) 1 1
BSol(q1 )______!BSol(q ) x BSol(q ).
The actual pullback of this square is the subspace BSol(q) of elements fixed by*
* B_q,
and we thus have a natural map BSol(q) -ffi0-!BDI4(q).
Theorem 4.5. For any odd prime power q, the natural map
BSol(q) ----ffi0---!BDI4(q)
'
is a homotopy equivalence.
Proof.Since BDI(4) is simply connected, the square used to define BDI4(q) remai*
*ns
a homotopy pullback square after 2-completion by [BK , II.5.3]. Thus BDI4(q) i*
*s 2-
complete. Also, BSol(q) def=|LcSol(q)|^2is 2-complete since |LcSol(q)| is 2-go*
*od [BLO2 ,
Proposition 1.11], and hence it suffices to prove that the map between these sp*
*aces is
an F2-cohomology equivalence. By Lemma 4.2, this means showing that the followi*
*ng
commutative square is a pullback square:
H*(BDI4(q); F2) ___! H*(BH(q); F2)C3
| |
| | (1)
# #
H*(BSpin7(q); F2)___! H*(BH(q); F2) .
Here, the maps are induced by the composite
BSpin7(q) ' BSpin07(q)^2 BSol(q) ------! BDI4(q)
Ran Levi and Bob Oliver *
* 31
and its restriction to BH(q). Also, by Proposition 4.3(b), the following diagra*
*m com-
mutes up to homotopy:
incl 1 ''0
BSpin7(q) ___! BSpin7(q )____! BSpin(7)
| | |
ffi(q)| ffi(q1|) bffi| (2)
# # #
incl 1 ''
BSol(q) _____! BSol(q )_____! BDI(4)
By [Fr, Theorem 12.2], together with [FM_, x1], for any connected reductive L*
*ie group
G and any algebraic epimorphism _ on G(F q) with finite fixed subgroup, there i*
*s a
homotopy pullback square
_ incl _
B(G(F q)_)^2_________!BG(F q)^2
incl#| #| (3)
_ (Id,B_) _ _
BG(F q)^2______!BG(F q)^2x BG(F q)^2.
We need to apply this when G = Spin7or G = H = (SL2)3={ (I, I, I)}. In particul*
*ar,
if _ =__q is the automorphism induced by the_field automorphism_(x 7! xq), then
Spin7(F q)_ = Spin7(q) by Lemma A.3, and H(F q)_ = H(q) def=H(F q) \ Spin7(q). *
*We
thus get a description of BSpin7(q) and BH(q) as homotopy pullbacks.
_
By [FM , Theorem 1.4], BG(F q)^2' BG(C)^2. Also, we can replace the complex
__ def
Lie groups Spin7(C) and H(C) by maximal compact subgroups Spin(7) and H =
SU(2)3={ (I, I, I)}, since these have the same homotopy type.
_
If we set R = H*(BG(F q); F2) ~=H*(BG(C); F2), then there are Eilenberg-Moore
spectral sequences
_
E2 = Tor*R Rop(R, R) =) H*(B(G(F q)_); F2);
where the (R Rop)-module structure on R is defined by setting (a b).x = a.x.B_(*
*b).
When G = Spin7or H, then R is a polynomial algebra by Proposition 4.7 and the a*
*bove
remarks, and B_ acts on R via the identity. The above spectral sequence thus sa*
*tisfies
the hypotheses of [Sm , Theorem II.3.1], and hence collapses. (Alternatively, n*
*ote that
in this case, E2 is generated multiplicatively by E0,*2and E-1,*2by (5) below.)*
* Similarly,
when R = H*(BDI(4); F2), there is an analogous spectral sequence which converge*
*s to
H*(BDI4(q); F2), and which collapses for the same reason. By the above remarks,*
* these
spectral sequences are natural with respect to the inclusions BH(-) BSpin7(-)*
*, and
(using the naturality of _q shown in Proposition 3.3(d)) of BSpin7(-) into BSol*
*(-) or
BDI(4).
To simplify the notation, we now write
__
A def=H*(BDI(4); F2), B def=H*(BSpin(7); F2), and C def=H*(H ; F2)
to denote these cohomology rings. The Frobenius automorphism _q acts via the id*
*entity
on each of them. We claim that the square
Tor*A Aop(A, A)__!Tor*C Cop(C, C)C3
| |
| | (4)
# #
Tor*B Bop(B, B)___! Tor*C Cop(C, C)
32 Construction of 2-local finite groups
is a pullback square. Once this has been shown, it then follows that in each d*
*egree,
square (1) has a finite filtration under which each quotient is a pullback squa*
*re. Hence
(1) itself is a pullback.
For any commutative F2-algebra R, let R=F2 denote the R-module generated by
elements dr for r 2 R with the relations dr = 0 if r 2 F2,
d(r + s) = dr + ds and d(rs) = r.ds + s.dr.
Let *R=F2denote the ring of Kähler differentials: the exterior algebra (over *
*R) of
R=F2= 1R=F2. When R is a polynomial algebra, there are natural identifications
Tor*R Rop(R, R) ~=HH*(R; R) ~= *R=F2. (5)
The first isomorphism holds for arbitrary algebras, and is shown, e.g., in [Wb *
*, Lemma
9.1.3]. The second holds for smooth algebras over a field [Wb , Theorem 9.4.7]*
* (and
polynomial algebras are smooth as shown in [Wb , x9.3.1]). In particular, the *
*iso-
morphisms (5) hold for R = A, B, C, which are shown to be polynomial algebras in
Proposition 4.7 below. Thus, square (4) is isomorphic to the square
* C
*A=F2__! C=F2 3
| |
| | (6)
# #
*B=F2___! *C=F2,
which is shown to be a pullback square in Propositions 4.7 and 4.9 below.
It remains to prove that square (6) in the above proof is a pullback square. *
*In what
follows, we let Di(x1, . .,.xn) denote the i-th Dickson invariant in variables *
*x1, . .,.xn.
This is the (2n- 2n-i)-th symmetric polynomial in the elements (equivalently in*
* the
nonzero elements) of the F2-vector space F2. We refer to [Wi ] fo*
*r more
detail. Note that what he denotes cn,iis what we call Dn-i(x1, . .,.xn).
Lemma 4.6. For any n,
Y
D1(x1, . .,.xn+1)= (xn+1 + x) + D1(x1, . .,.xn)2
x2F2
Xn
n 2n-i 2
= x2n+1+ xn+1Di(x1, . .,.xn) + D1(x1, . .,.xn) .
i=1
Proof.The first equality is shown in [Wi , Proposition 1.3(b)]; here we prove t*
*hem both
simultaneously. Set Vn = F2. Since oei(Vn) = 0 whenever 2n - i i*
*s not a
power of 2 (cf. [Wi , Proposition 1.1]),
X2n
D1(x1, . .,.xn+1)= oei(Vn).oe2n-i(xn+1 + Vn)
i=0
Y Xn
= (xn+1 + x) + Di(x1, . .,.xn).oe2n-i(xn+1 + Vn).
x2Vn i=1
Also, since oei(Vn) = 0 for 0 < i < 2n-1 as noted above,
Xk ( n-1
n-i 0 if 0 < k < 2
oek(xn+1 + Vn) = xk-in+1.2k-ioei(Vn) = n-1
i=0 D1(x1, . .,.xn)if k = 2 .
Ran Levi and Bob Oliver *
* 33
This proves the first equality, and the second follows since
Y n 2nX n n Xn n-i
(xn+1 + x) = x2n+1+ x2n-i+1oei(Vn) = x2n+1+ x2n+1Di(x1, . .,.xn).
x2Vn i=1 i=1
In the following proposition (and throughout the rest of the section), we wor*
*k with
the polynomial ring F2[x, y, z, w], with the natural action of GL4(F2). Let
GL22(F2), GL31(F2) GL4(F2)
be the subgroups of automorphisms of V def=F2which leave invariant *
*the sub-
spaces and , respectively. Also, let GL220(F2) GL22(F2) be th*
*e subgroup
of automorphisms which are the identity modulo . Thus, when described in *
*terms
of block matrices (with respect to the given basis {x, y, z, w}),
2 2
GL31(F2) = A0X1 , GL2(F2) = B0YC , and GL20(F2) = B0YI ,
for A 2 GL3(F2), X a column vector, B, C 2 GL2(F2), and Y 2 M2(F2).
We need to make more precise the relation between V (or the polynomial ring
F2[x, y, z, w]) and the cohomology of Spin(7). To do this, let W Spin(7) be*
* the
inverse image of the elementary abelian subgroup
*
* ff
diag(-1, -1, -1, -1, 1, 1, 1), diag(-1, -1, 1, 1, -1, -1, 1), diag(-1, 1, -1*
*, 1, -1, 1, -1)
SO(7).
Thus, W ~=C42. Fix a basis {j, j0, ,, i} for W , where i 2 Z(Spin(7)) is the no*
*ntrivial
element. Identify V = W *in such a way that {x, y, z, w} V is the dual bas*
*is to
{j, j0, ,, i}. This gives an identification
H*(BW ; F2) = F2[x, y, z, w],
arranged such that the action of NSpin(7)(W )=W on V = consists o*
*f all
automorphisms which leave invariant, and thus can be identified with *
*the
action of GL31(F2). Finally, set
__
H = CSpin(7)(,) ~=Spin(4) xC2 Spin(3) ~=SU(2)3={ (I, I, I)}
(the central product). Then in the same way, the action of N_H(W )=W on H*(BW ;*
* F2)
can be identified with that of GL220(F2).
Proposition 4.7. The inclusions
__
BW -----! BH -----! BSpin(7) -----! BDI(4)
as defined above, together with the identification H*(BW ; F2) = F2[x, y, z, w]*
*, induce
isomorphisms
A def=H*(BDI(4); F2) = F2[x, y, z, w]GL4(F2)= F2[a8, a12, a14, a15]
3(F )
B def=H*(BSpin(7); F2) = F2[x, y, z, w]GL1 2= F2[b4, b6, b7, b8](*)
__ 2
C def=H*(BH ; F2) = F2[x, y, z, w]GL20(F2)= F2[c2, c3, c04, c004] ;
where
a8 = D1(x, y, z, w), a12= D2(x, y, z, w), a14= D3(x, y, z, w), a15= D4(x, y, *
*z, w);
Y
b4 = D1(x, y, z), b6 = D2(x, y, z), b7 = D3(x, y, z), b8 = (w +*
* ff);
ff2
34 Construction of 2-local finite groups
and
Y Y
c2 = D1(x, y), c3 = D2(x, y), c04= (z + ff), c004= (w + ff*
*).
ff2 ff2
Furthermore,
__
(a)the natural action of 3 on H ~=SU(2)3={ (I, I, I)} induces the action on C *
*which
fixes c2, c3 and permutes {c04, c004, c04+ c004}; and
(b)the above variables satisfy the relations
a8= b8 + b24 a12= b8b4 + b26 a14= b8b6 + b27 a15= b8b7
b4= c04+ c22 b6= c2c04+ c23 b7= c3c04 b8= c004(c04+ c004) .
Proof.The formulas for A = H*(BDI(4); F2) are shown in [DW1 ]. From [DW1 , Le*
*m-
mas 3.10 & 3.11], we see there are (some) identifications
3(F ) * __ GL2*
*(F )
H*(BSpin(7); F2) ~=F2[x, y, z, w]GL1 2 and H (BH ; F2) ~=F2[x, y, z, w] 2*
*0 2.
__
From the explicit choices of subgroups W H Spin(7) as described above (and
by the descriptions in Proposition_A.8 of the automorphism groups), the images *
*of
H*(BSpin(7); F2) and H*(BH ; F2) in F2[x, y, z, w] are seen to be contained in *
*the
rings of invariants, and hence these isomorphisms actually are equalities as cl*
*aimed.
We next prove the equalities in (*) between the given rings of invariants and*
* poly-
nomial algebras. The following argument was shown to us by Larry Smith. If k *
*is
a field and V is an n-dimensional vector space over k, then a system of parame*
*ters
in the polynomial algebra k[V ] is a set of n homogeneous elements f1, . .,.fn *
*such
that k[V ]=(f1, . .,.fn) is finite dimensional over k. By [Sm2 , Proposition 5.*
*5.5], if V is
an n-dimensional k[G]-representation, and f1, . .,.fn 2 k[V ]G is a system of p*
*arame-
ters the product of whose degrees is equal to |G|, then k[V ]G is a polynomial *
*algebra
with f1, . .,.fn as generators. By [Sm2 , Proposition 8.1.7], F2[x, y, z, w] is*
* a free finitely
generated module over the ring generated by its Dickson invariants (this holds *
*for poly-
nomial algebras over any Fp), and thus F2[x, y, z, w]=(a8, a12, a14, a15) is fi*
*nite. (This
can also be shown directly using the relation in Lemma 4.6.) So assuming the re*
*lations
in point (b), the quotients F2[x, y, z, w]=(b4, b6, b7, b8) and F2[x, y, z, w]=*
*(c2, c3, c04, c004) are
also finite. In each case, the product of the degrees of the generators is clea*
*rly equal
to the order of the group in question, and this finishes the proof of the last *
*equality in
the second and third lines of (*).
It remains to prove points (a) and (b). Using Lemma 4.6, the ci are expresse*
*d as
polynomials in x, y, z, w as follows:
c2 = D1(x, y) = x2 + xy + y2
c3 = D2(x, y) = xy(x + y)
(1)
c04= D1(x, y, z) + D1(x, y)2 = z4 + z2D1(x, y) + zD2(x, y) = z4 + z2c2 + zc3
c004= D1(x, y, w) + D1(x, y)2 = w4 + w2D1(x, y) + wD2(x, y) = w4 + w2c2 + wc3.
In particular,
Y
c04+ c004= (z + w)4 + (z + w)2D1(x, y) + (z + w)D2(x, y) = (z + w +(ff).*
*2)
ff2
Ran Levi and Bob Oliver *
* 35
Furthermore, by (1), we get
Sq1(c2)= c3
Sq1(c3)= Sq1(c04) = Sq1(c004) = 0
Sq2(c3)= x2y2(x + y) + xy(x + y)3 = c2c3
Sq2(c04)= z4c2 + z2c22+ zc2c3 = c2c04 (3)
Sq3(c04)= Sq1(c2c04) = c3c04
Sq2(c004)= c2c004
Sq3(c004)= c3c004.
__
The permutation action_of_ 3 on H ~=SU(2)3={ (I, I, I)} permutes the three e*
*l-
ements i, ,, i + , of Z(H ) W , and thus (via the identification V = W *descr*
*ibed
above) induces the identity on x, y 2 V and permutes the elements {z, w, z + w}*
* mod-
2(F )
ulo . Hence the induced action of 3 on C = F2[V ]GL20 2 is the restricti*
*on of the
action on F2[V ] = F2[x, y, z, w] which fixes x, y and permutes {z, w, z + w}. *
*So by (1)
and (2), we see that this action fixes c2, c3 and permutes the set {c04, c004, *
*c04+ c004}. This
proves (a).
It remains to prove the formulas in (b). From (1) and (3) we get
b4= D1(x, y, z) = c04+ c22,
b6= D2(x, y, z) = Sq2(b4) = c2c04+ c23,
b7= D3(x, y, z) = Sq1(b6) = c3c04.
Also, by (1) and (2),
Y i Y j i Y j
b8 = (w + ff) = (w + ff) . (w + z + ff) = c004(c04+ c004*
*).
ff2 ff2 ff2
This proves the formulas for the bi in terms of ci. Finally, we have
a8 = D1(x, y, z, w) = b8 + b24,
a12= D2(x, y, z, w) = Sq4(b8 + b24) = Sq4(c004(c04+ c004) + (c04+ c22)2)
= c04c004(c04+ c004) + c22c004(c04+ c004) + c22c042+ c43= b8b4 + b26
a14= D3(x, y, z, w) = Sq2(a12) = c2c04c004(c04+ c004) + c23c004(c04+ c00*
*4) + c23c042
= b8b6 + b27
a15= D4(x, y, z, w) = Sq1(a14) = c3c04c004(c04+ c004) = b8b7;
and this finishes the proof of the proposition.
Lemma 4.8. Let ~ 2 Aut(C) be the algebra involution which exchanges c04and c004*
*and
leaves c2 and c3 fixed. An element of C will be called "~-invariant" if it is f*
*ixed by this
involution. Then the following hold:
(a)If fi 2 B is ~-invariant, then fi 2 A.
(b)If fi 2 B is such that fi.c04iis ~-invariant, then fi = fi0.bi8for some fi02*
* A.
Proof.Point (a) follows from Proposition 4.7 upon regarding A, B, and C as the *
*fixed
subrings of the groups GL4(F2), GL31(F2) and GL220(F2) acting on F2[x, y, z, w]*
*, but also
follows from the following direct argument. Let m be the degree of fi as a poly*
*nomial
36 Construction of 2-local finite groups
in b8; we argue by induction on m. Write fi = fi0 + bm8.fi1, where fi1 2 F2[b4*
*, b6, b7],
and where fi0 has degree < m (as a polynomial in b8). If m = 0, then fi = fi1 2
F2[b4, b6, b7] F2[c2, c3, c04], and hence fi 2 F2[c2, c3] since it is ~-invar*
*iant. But from
the formulas in Proposition 4.7(b), we see that F2[b4, b6, b7] \ F2[c2, c3] con*
*tains only
constant polynomials (hence it is contained in A).
Now assume m 1. Then, expressed as a polynomial in c2, c3, c04, c004, the *
*largest
power of c004which occurs in fi is c0042m. Since fi is ~-invariant, the highest*
* power of c04
which occurs is c042m; and hence by Proposition 4.7(b), the total degree of eac*
*h term in
fi1 (its degree as a polynomial in b4, b6, b7) is at most m. So for each term b*
*r4bs6bt7in fi1,
br4bs6bt7bm8- am-r-s-t8ar12as14at15
is a sum of terms which have degree < m in b8, and thus lies in A by the induct*
*ion
hypothesis.
To prove (b), note first that since fi.c04iis ~-invariant and divisible by c0*
*4i, it must also
be divisible by c004i, and hence c004i|fi. Furthermore, by Proposition 4.7, all*
* elements of B
as well as c04are invariant under the involution which fixes c04and sends c0047*
*! c04+ c004.
Thus (c04+ c004)i|fi. Since b8 = c004(c04+ c004), we can now write fi = fi0.bi8*
*for some fi 2 B.
Finally, since
fi.c04i= fi0.c04i.c004i.(c04+ c004)i
is ~-invariant, fi0 is also ~-invariant, and hence fi02 A by (a).
2(F )
Note that C3 3 = GL2(F2) act on C = F2[x, y, z, w]GL20 2: via the action *
*of
GL22(F2)=GL220(F2), or equivalently by permuting c04, c004, and c04+ c004(and f*
*ixing c2, c3).
Thus A = B\CC3, since GL4(F2) is generated by the subgroups GL31(F2) and GL22(F*
*2).
This is also shown directly in the following lemma.
Proposition 4.9. The following square is a pullback square, where all maps are *
*induced
by inclusions between the subrings of F2[x, y, z, w]:
* C3
*A=F2__! C=F2
| |
| |
# #
*B=F2___! *C=F2.
Proof.Let ~ be the involution of Lemma 4.8: the algebra involution of C which e*
*x-
changes c04and c004and leaves c2 and c3 fixed. By construction, all elements in*
* the image
of *B=F2are invariant under the involution which fixes c04(and c2, c3), and se*
*nds c004to
c04+ c004. Hence elements in the image of *B=F2are fixed by C3 if and only if *
*they are
fixed by 3, if and only if they are ~-invariant. So it will suffice to show th*
*at all of the
above maps are injective, and that all ~-invariant elements in the image of *B*
*=F2lie in
the image of *A=F2. The injectivity is clear, and the square is a pullback for*
* 0-=F2by
Lemma 4.8.
Fix a ~-invariant element
! = P1db4 + P2db6 + P3db7 + P4db8
(1)
= P2c04dc2 + P3c04dc3 + P4c04dc004+ (P1 + P2c2 + P3c3 + P4c004) dc042 1*
*B=F2,
where Pi2 B for each i. By applying ~ to (1) and comparing the coefficients of *
*dc2 and
dc3, we see that P2c04and P3c04are ~-invariant. Also, upon comparing the coeffi*
*cients
of dc04, we get the equation
P1 + P2c2 + P3c3 + P4c004= ~(P4)c004. (2)
Ran Levi and Bob Oliver *
* 37
So by Lemma 4.8, P2 = P20b8 and P3 = P30b8 for some P20, P302 A. Upon subtracti*
*ng
P20da14+ P30da15= P2db6 + P3db7 + (P20b6 + P30b7) db8
from ! and introducing an appropriate modification to P4, we can now assume that
P2 = P3 = 0. With this assumption and (2), we have
P1 + P4c004= ~(P4c04) = ~(P4).c004,
so that
P1c04= (P4 + ~(P4))c04c004 (3)
is ~-invariant. This now shows that P1 = P10b8 for some P102 A, and upon subtra*
*cting
P10da12 from ! we can assume that P1 = 0. This leaves ! = P4db8 = P4da8. By (3)
again, P4 is ~-invariant, so P4 2 A by Lemma 4.8 again, and thus ! 2 1A=F2.
The remaining cases are proved using the same techniques, and so we sketch th*
*em
more briefly. To prove the result in degree two, fix a ~-invariant element
! = P1db4db6 + P2db4db7 + P3db4db8 + P4db6db7 + P5db6db8 + P6db7db8
= P4c042dc2dc3 + (P1c04+ P4c3c04+ P5c04c004) dc2dc04+ P5c042dc2dc004
+ (P2c04+ P4c2c04+ P6c04c004) dc3dc04+ P6c042dc3dc004
+ (P3c04+ P5c2c04+ P6c3c04) dc04dc0042 2B=F2.
Using Lemma 4.8, we see that P4 = P40b28, and hence can assume that P4 = 0. One
then eliminates P1 and P2, then P5 and P6, and finally P3.
If
! = P1db4db6db7 + P2db4db6db8 + P3db4db7db8 + P4db6db7db8
= (P1c042+ P4c042c004) dc2dc3dc04+ (P2c042+ P4c3c042) dc2dc04dc004
+ (P3c042+ P4c2c042) dc3dc04dc004+ P4c043dc2dc3dc0042 3B=F2
is ~-invariant, then we eliminate successively P1, then P4, then P2 and P3.
Finally, if
! = P db4db6db7db8 = P c043dc2dc3dc04dc0042 4B=F2
is ~-invariant, then P = P 0b38for some P 02 A by Lemma 4.8 again, and so ! =
P 0da8da12da14da152 4A=F2.
Appendix
Appendix A. Spinor groups over finite fields
Let F be any field of characteristic 6= 2. Let V be a vector space over F ,*
* and let
b: V --! F be a nonsingular quadratic form. As usual, O(V, b) denotes the grou*
*p of
isometries of (V, b), and SO(V, b) the subgroup of isometries of determinant 1.*
* We
will be particularly interested in elementary abelian 2-subgroups of such ortho*
*gonal
groups.
Lemma A.1. Fix an elementary abelian 2-subgroup E O(V, b). For each irreducib*
*le
character Ø 2 Hom (E, { 1}), let Vffl V denote the corresponding eigenspace:*
* the
subspace of elements v 2 V such that g(v) = Ø(g).v for all g 2 E. Then the rest*
*riction
of b to each subspace Vfflis nonsingular, and V is the orthogonal direct sum of*
* the Vffl.
38 Construction of 2-local finite groups
Proof.Elementary.
We give a very brief sketch of the definition of spinor groups via Clifford a*
*lgebras;
for more details we refer to [Di, xII.7] or [As2 , x22]. Let T (V ) denote the *
*tensor algebra
of V , and set
C(V, b) = T (V )=<(v v) - b(v)> :
the Clifford algebra of (V, b). To simplify the notation, we regard F as a sub*
*ring of
C(V, b), and V as a subgroup of its additive group; thus the class of v1 . .*
* .vk
will be written v1. .v.k. Note that vw + wv = 0 if v, w 2 V and v ? w. Hence
if dim F(V ) = n, and {v1, . .,.vn} is an orthogonal basis, then the set of 1 a*
*nd all
vi1. .v.ikfor i1 < . .<.ik (1 k n) is an F -basis for C(V, b).
Write C(V, b) = C0 C1, where C0 and C1 consist of classes of elements of ev*
*en or
odd degree, respectively. Let G C(V, b)* denote the group of invertible eleme*
*nts u
such that uV u-1 = V , and let ß :G --! O(V, b) be the homomorphism
(
(v 7! -uvu-1) if u 2 C1
ß(u) =
(v 7! uvu-1) if u 2 C0.
In particular, for any nonisotropic element v 2 V (i.e., b(v) 6= 0), v 2 G and *
*ß(v) is
the reflection in the hyperplane v? . By [Di, xII.7], ß is surjective and Ker(ß*
*) = F *.
Let J be the antiautomorphism of C(V, b) induced by the antiautomorphism v1
. . .vk 7! vk . . .v1 of T (V ). Since O(V, b) is generated by hyperplane ref*
*lections,
G is generated by F *and nonisotropic elements v 2 V . In particular, for any *
*u =
~.v1. .v.k2 G,
J(u).u = ~2 . vk. .v.1. v1. .v.k= ~2 . b(v1) . .b.(vk) 2 F *= Ker(ß);
implying that ß(J(u)) = ß(u)-1 for all u 2 G. There is thus a homomorphism
e`:G -------! F * defined by e`(u) = u.J(u).
In particular, e`(~) = ~2 for ~ 2 F * G, while for any set of nonisotropic ele*
*ments
v1, . .,.vk of V ,
e`(v1. .v.k) = (v1. .v.k)(vk. .v.1) = b(v1) . .b.(vk).
Hence e`factors through a homomorphism
`V,b:O(V, b) -------! F *=F *2= F *={u2| u 2 F *},
called the spinor norm.
Set G+ = ß-1(SO(V, b)) = G \ C0, and define
Spin(V, b) = Ker(e`|G+ ) and (V, b) = Ker(`V,b|SO(V,b)).
In particular, (V, b) has index 2 in SO(V, b) if F is a finite field, and (V,*
* b) =
SO(V, b) if F is algebraically closed (all units are squares). We thus get a co*
*mmutative
diagram
Ran Levi and Bob Oliver *
* 39
1 1 1
fflffl| fflffl||~7!~2 fflffl||
1 ____//_{ 1}__________//F_*_________//F *2__//1
| || ||
| | |
| | |
fflffl| fflffl|e` fflffl|
(A.2) 1 __//_Spin(V,_b)______//G+__________//_F_*__//_1
| |
| | |
| i | |
| | |
fflffl| fflffl|`V,b fflffl|
1 ____// (V, b)_____//SO(V, b)_____//F *=F *2//_1
fflffl| fflffl| fflffl|
1 1 1
where all rows and columns are short exact, and where all columns are central e*
*xten-
sions of groups. If dim(V ) 3 (or if dim(V ) = 2 and the form b is hyperbolic*
*), then
(V, b) is the commutator subgroup of SO(V, b) [Di, xII.8].
The following lemma follows immediately from this description of Spin(V, b), *
*together
with the analogous description of the corresponding spinor group over the algeb*
*raic
closure of F .
__ __ __ _
Lemma A.3. Let F be the algebraic closure of F , and set V = F F V and b= Id_F*
* b.
__ _
Then Spin(V, b) is the subgroup_of Spin(V , b) consisting of those elements fix*
*ed by all
Galois automorphisms _ 2 Gal(F =F ).
For any nonsingular quadratic form b on a vector space V , the discriminant o*
*f b (or
of V ) is the determinant of the corresponding symmetric bilinear form B, relat*
*ed to b
by the formulas
b(v) = B(v, v) and B(v, w) = 1_2b(v + w) - b(v) - b(w) .
Note that the discriminant is well defined only modulo squares in F *. When W *
* V
is a subspace, then we define the discriminant of W to mean the discriminant of*
* b|W .
In what follows, we say that the discriminant of a quadratic form is a square o*
*r a
nonsquare to mean that it is the identity or not in the quotient group F *=F *2.
Lemma A.4. Fix an involution x 2 SO(V, b), and let V = V+ V- be its eigenspace
decomposition. Then the following hold.
(a)x 2 (V, b) if and only if the discriminant of V- is a square.
(b)If x 2 (V, b), then it lifts to an element of order 2 in Spin(V, b) if and *
*only if
dim (V-) 2 4Z.
(c)If x 2 (V, b), and if ff 2 (V, b) is such that [x, ff] = 1, then ff = ff+ *
* ff-, where
ff 2 O(V , b). Also, the liftings of x and ff commute in Spin(V, b) if and *
*only if
det(ff-) = 1.
Proof.Let {v1, . .,.vk} be an orthogonal basis for V- (k is even). Then x = ß(v*
*1. .v.k)
in the above notation, since ß(vi) is the reflection in the hyperplane vi?. Hen*
*ce by the
commutativity of Diagram (A.2),
`V,b(x) b(v1). .b.(vk) = det(b|V-) (mod F *2).
Thus x 2 (V, b) = Ker(`V,b) if and only if V- has square discriminant.
40 Construction of 2-local finite groups
In particular, if x 2 (V, b), then the product of the b(vi) is a square, and*
* hence
(upon replacing v1 by a scalar multiple) we can assume that b(v1). .b.(vk) = 1.*
* Then
exdef=v1. .v.k2 Spin(V, b) = Ker(e`). Since vw = -wv in the Clifford algebra wh*
*enever
v ? w, and since (vi)2 = b(vi) for each i,
(
1 if k 0 (mod 4)
ex2= (-1)k(k-1)=2.(v1)2. .(.vk)2 = (-1)k(k-1)=2=
-1 if k 2 (mod 4) .
This proves (b).
It remains to prove (c). The first statement (ff = ff+ ff-) is clear. Fix*
* liftings
eff2 C(V , b)*. Rather than defining the direct sum of an element of C(V+, b) w*
*ith
an element of C(V-, b), we regard the groups C(V , b)* as (commuting) subgroups*
* of
C(V, b)*, and set
eff= eff+Oeff-= eff-Oeff+2 Spin(V, b).
Let ex= v1. .v.kbe as above. Clearly, excommutes with all elements of C(V+, b).*
* Since
(v1. .v.k).vi= (-1)k-1.vi.(v1. .v.k) = -vi.(v1. .v.k)
for i = 1, . .,.k, we have ex.fi = (-1)i.fi.exfor all fi 2 Ci(V-, b) (i = 0, 1)*
*. In particular,
since [eff+, eff-] = 1, [ex, eff] = [ex, eff-] = det(ff-), and this finishes th*
*e proof.
We will need explicit isomorphisms which describe Spin3(F ) and Spin4(F ) in *
*terms
of SL2(F ). These are constructed in the following proposition, where M02(F ) d*
*enotes
the vector space of matrices of trace zero. Note that the determinant is a nons*
*ingular
quadratic form on M2(F ) and on M02(F ), in both cases with square discriminant.
Proposition A.5. Define
æ3: SL2(F ) ------! (M02(F ), det)
and
æ4: SL2(F ) x SL2(F ) ------! (M2(F ), det)
by setting
æ3(A)(X) = AXA-1 and æ4(A, B)(X) = AXB-1.
Then æ3 and æ4 are both epimorphisms, and lift to unique isomorphisms
je3 0
SL2(F ) ------!~ Spin(M2(F ), det)
=
and
je4
SL2(F ) x SL2(F ) ------!~ Spin(M2(F ), det).
=
Proof.See [Ta , pp. 142, 199] for other ways of defining these isomorphisms. By*
* Lemma
A.3, it suffices to prove this (except for the uniqueness of the lifting) when *
*F is alge-
braically closed. In particular, (M02(F ), det) = SO(M02(F ), det) and (M2(F *
*), det) =
SO(M2(F ), det) in this case.
For general V and b, the group SO(V, b) is generated by reflections fixing n*
*on-
isotropic subspaces (i.e., of nonvanishing discriminant) of codimension 2 (cf. *
* [Di,
xII.6(1)]). Hence to see that æ3 and æ4 are surjective, it suffices to show th*
*at such
elements lie in their images. A codimension 2 reflection in SO(M02(F ), det) is*
* of the
form RX (the reflection fixing the line generated by X) for some X 2 M02(F ) wh*
*ich
is nonisotropic (i.e., det(X) 6= 0). Since F is algebraically closed, we can a*
*ssume
X 2 SL2(F ). Then X2 = -I (since Tr(X) = 0 and det(X) = 1), and RX = æ3(X)
since it has order 2 and fixes X. Thus æ3 is onto.
Ran Levi and Bob Oliver *
* 41
Similarly, any 2-dimensional nonisotropic subspace W V has an orthonormal b*
*asis
{Y, Z}, and ZY -1and Y -1Z have trace zero (since they are orthogonal to the id*
*entity
matrix) and determinant one. Hence their square is -I, and one repeats the abo*
*ve
argument to show that RW = æ4(ZY -1, Y -1Z). So æ4 is onto.
The liftings eæmexist and are unique since SL2(F ) is the universal central e*
*xtension
of P SL2(F ) (or universal among central extensions by 2-groups if F = F3).
We now restrict to the case F = Fq where q is an odd prime power. We refer to
[As2 , x21] for a description of quadratic forms in this situation, and the not*
*ation for
the associated orthogonal groups. If n is odd and b is any nonsingular quadrati*
*c form
on Fnq, then every nonsingular quadratic form is isomorphic to ub for some u 2 *
*F*q,
and hence one can write SOn(q) = SO(Fnq, b) = SO(Fnq, ub) without ambiguity. If
n is even, then there are exactly two isomorphism classes of quadratic forms on*
* Fnq;
and one writes SO+n(q) = SO(Fnq, b) when b is the hyperbolic form (equivalently,
has discriminant (-1)n=2 modulo squares), and SO-n(q) = SO(Fnq, b) when b is not
hyperbolic (equivalently, has discriminant (-1)n=2.u for u 2 F*qnot a square). *
* This
notation extends in the obvious way to n(q) and Spinn(q).
The following lemma does, in fact, hold for for orthogonal representations ov*
*er arbi-
trary fields of characteristic 6= 2. But to simplify the proof (and since we we*
*re unable
to find a reference), we state it only in the case of finite fields.
Lemma A.6. Assume F = Fq, where q is a power of an odd prime. Let V be an
F -vector space, and let b be a nonsingular quadratic form on V . Let P O(V, *
*b) be
a 2-subgroup which is orthogonally irreducible; i.e., such that V has no splitt*
*ing as an
orthogonal direct sum of nonzero P -invariant subspaces. Then dimF (V ) is a po*
*wer of
2; and if dim(V ) > 1 then b has square discriminant.
Proof.This means showing that each orthogonal group O(Fqn, b), such that either*
* n is
not a power of 2, or n = 2k 2 and the quadratic form b has nonsquare discrimi*
*nant,
contains some subgroup Om (q) x On-m (q) (for 0 < m < n) of odd index. We refer*
* to
the standard formulas for the orders of these groups (see [Ta , p.165]): if ffl*
* = 1 then
n-1Y Yn
2 *
*2i
|Offl2n(q)| = 2qn(n-1)(qn - ffl) (q2i- 1) and |O2n+1(q)| = 2qn (q*
* - 1).
i=1 i=1
We will also use repeatedly the fact that for all 0 < i < 2k (k 1), the large*
*st powers
k+i i 2k+i *
* i
of 2 dividing (q2 - 1) and (q - 1) are the same. In other words, (q - 1)=(*
*q - 1)
is invertible in Z(2).
For any n 1,
__|O2n+1(q)|__ n qn + ffl
= q .______
|Offl2n(q)|.|O1(q)| 2
is odd for an appropriate choice of ffl. Thus, there are no irreducibles of odd*
* dimension.
Assume n is not a power of 2, and write n = 2k + m where 0 < m < 2k and k 1.
Then
_m-1 k ! _ k ! _ k !
_____|Offl2n(q)|_ m2k+1 Y q2(2 +i)- 1 q2 +m - ffl q2 + 1
= q . __________ . _________ . _______ ,
|O+2k+1(q)|.|Offl2m(q)| i=1 q2i- 1 qm - ffl 2
42 Construction of 2-local finite groups
and each factor is invertible in Z(2). When n = 2m = 2k and k 1, then O-2n(q)*
* is the
orthogonal group for the quadratic form with nonsquare discriminant, and
_ m-1 !
____|O-2n(q)|___ 2m2 Y q2(m+i)- 1 q2m + 1
= q . __________ ._______,
|O+2m(q)|.|O-2m(q)| i=1 q2i- 1 2
and again each factor is invertible in Z(2). Finally,
___|Offl2(q)|_ q - ffl
= _____
|O1(q)|.|O1(q)| 2
is odd whenever q 1 (mod 4) and ffl = -1, or q 3 (mod 4) and ffl = +1; and *
*these are
precisely the cases where the quadratic form on Fq2 has nonsquare discriminant.
We must classify the conjugacy classes of those elementary abelian 2-subgroup*
*s of
Spin7(q) which contain its center. The following definition will be useful when*
* doing
this.
Definition A.7. Fix an odd prime power q, and identify SO7(q) = SO(F7q, b) and
Spin7(q) = Spin(F7q, b), where b is a nonsingular quadratic form with square di*
*scrimi-
nant. An elementary abelian 2-subgroup of SO7(q) or of Spin7(q) will be called *
*of type
I if its eigenspaces all have square discriminant (with respect to b), and of t*
*ype II oth-
erwise. Let En be the set of elementary abelian 2-subgroups in Spin7(q) which c*
*ontain
Z(Spin7(q)) ~=C2 and have rank n. Let EInand EIInbe the subsets of En consistin*
*g of
those subgroups of types I and II, respectively.
In the following two propositions, we collect together the information which *
*will
be needed about elementary abelian 2-subgroups of Spin7(q). We fix Spin7(q) =
Spin(V, b), where V ~=F7q, and b is a nonsingular quadratic form with square d*
*is-
criminant. Let z 2 Z(Spin7(q))_be_the generator. For any subgroup H Spin7(q) *
*or
_
any element g 2 Spin7(q), let H and gdenote their images in 7(q) SO7(q)._For*
* each
elementary abelian 2-subgroup E Spin7(q), and each character Ø 2 Hom (E , { 1*
*}),
Vffl V denotes the eigenspace of Ø (and V1 denotes the eigenspace of the trivi*
*al char-
acter). Also (when z 2 E), Aut (E, z) denotes the group of all automorphisms o*
*f E
which send z to itself.
Proposition A.8. For any odd prime power q, the following table describes the n*
*um-
bers of Spin7(q)-conjugacy classes in each of the sets EInand EIIn, the dimensi*
*ons and
discriminants of the eigenspaces of subgroups in these sets, and indicates in w*
*hich cases
AutSpin7(q)(E) contains all automorphisms which fix z.
______________________________________________________________
| Set of subgroups |E|I | EI | EII | EI |EII |
|___________________________|_|2__|__3___|__3___|___4___|_4__|__
| Nr. conj. classes |1| | 1 | 1 | 2 | 1 |
|____________________________|_|__|______|______|______|______|
| dim(V ) | |3 | 1 | 0 |
|______1___________________|_|____|_____________|____________|_
| dim(V ), Ø 6= 1 |4| | 2 | 1 |
|______ffl__________________|_|__|______________|____________|_
| discr(V , b) sq|u|ares|quaren|onsq. |_ | _ |
|________1___________________|_|____|______|______|____|_____|_
| discr(V , b), Ø 6= 1 squ|a|res|quaren|onsq.s|quare |both |
|________ffl__________________|_|___|______|______|______|____ |
| Aut (E) = Aut(E, z) |y|es |yes |yes | yes |no |
|_____Spin7(q)_____________|_|_____|______|______|_______|____|
There are no subgroups in E2 of type II, and no subgroups of rank 5. Furtherm*
*ore,
we have:
(a)For all E 2 E4, CSpin7(q)(E) = E.
Ran Levi and Bob Oliver *
* 43
__ __
(b)If E, E02 EI4, then E0 = gE g-1 for some g 2 SO7(q), and E and E0 are Spin7(*
*q)-
conjugate if and only if g 2 7(q).
_ _ __
(c)If E 2 EII4, then there is_a unique element 1 6= x = x(E) 2 E with the prop*
*erty
_
that for 1 6= Ø 2 Hom (E , { 1}), Vfflhas square discriminant if Ø(x) = 1 a*
*nd
_ *
* __
nonsquare discriminant if Ø(x) = -1. Also, the_image of AutSpin7(q)(E) in Au*
*t(E )
is the group of all automorphisms which send x to itself; and if X E denot*
*es
_ __ *
* __
the inverse image of E, then AutSpin7(q)(E) contains all automorphisms *
*of E
which are the identity on X and the identity modulo .
(d)If E 2 E3, then CSpin7(q)(E) = Ao C2, where A is abelian and C2 acts on A by
inversion. If E 2 EII3, then the Sylow 2-subgroups of CSpin7(q)(E) are elem*
*entary
abelian of rank 4 (and type II).
Proof.Write Spin= Spin7(q) for short. Fix an elementary abelian subgroup E Sp*
*in
such that z 2 E.
Step 1: We first_show that rk(E) 4, and that the dimensions of the eigenspa*
*ces
Vfflfor Ø 2 Hom (E , { 1}) are as described in the table.
__
By Lemma A.4, every involution_in E has a 4-dimensional (-1)-eigenspace.__In
particular, if rk(E) = 2, (E ~=C2), then dim (Vffl) = 4 for 1 6= Ø 2 Hom (E , *
*{ 1}),
while dim(V1) = 3.
Now assume rk(E) = n for some n > 2. Assume we_have_shown, for all E0 2 En-1,
that the_eigenspace of the trivial character of E 0is r-dimensional._For each 1*
* 6= Ø 2
Hom (E , { 1}), let Effl2 En-1 be the subgroup_such that E ffl= Ker(Ø); then V1*
* Vfflis
the eigenspace of the trivial character of Effl= Ker(Ø), and thus dim(V1)+dim (*
*Vffl) = r.
Hence all nontrivial characters of E have_eigenspaces of the same dimension. Si*
*nce there
are 2n-1- 1 nontrivial characters of E, we have dim(V1) + (2n-1- 1) dim(Vffl) =*
* 7, and
these two equations completely determine dim(V1) and dim(Vffl). Using this proc*
*edure,
the dimensions of the eigenspaces are shown inductively to be equal_to those gi*
*ven by
the table. Also, this shows that rk(E) 4, since otherwise rk(E ) 4, so the *
*Vfflfor
Ø 6= 1 must be trivial (they cannot all have dimension 1), so E acts on V via*
* the
identity, which contradicts the assumption that E Spin7(q).
Step 2: We next_show_that EII2= ;, describe the discriminants of the eigenspa*
*ces
of characters of E for E 2 En (for all n), and show that subgroups of rank 4 ar*
*e self
centralizing. In particular, this proves (a) together with the first statement *
*of (c).
If E 2 E2, then_E = for some noncentral involution g 2 Spin7(q), and t*
*he
_
eigenspaces of E = have square discriminant by Lemma A.4(a) (and since the
ambient space V has square discriminant by assumption). Thus EII2= ;.
__ _
If E 2 E3, then the sum of any two eigenspaces of E_is an eigenspace of g for*
* some
g 2 Er . Hence the sum of any two eigenspaces of E has square discriminant, *
*so
either all of the eigenspaces have square discriminant (E 2 EI3), or all of the*
* eigenspaces
have nonsquare discriminant (E 2 EII3).
__
Assume rk(E) = 4. We have seen that all eigenspaces_of E are 1-dimensional.
By Lemma A.4(c), for each a 2 CSpin7(q)(E), a(Vffl) =_Vfflfor each Ø 6= 1, and *
*since
dim(Vffl) = 1 it must act on each Vfflvia Id._Thus a 2 7(q) has order 2; let*
* V be its
eigenspaces. Then dim(V-) is even since det(a) = 1, and V- has square discrimin*
*ant
44 Construction of 2-local finite groups
by Lemma A.4(a). If dim(V-) = 4, then |a| = 2 (Lemma A.4(b)), and hence a 2 E
since otherwise would have rank 5. If dim (V-)_= 2, then V- is the sum *
*of
the_eigenspaces_of two distinct_characters_Ø1,_Ø2 of E , there is some g 2 E su*
*ch that
Ø1(g) 6= Ø2(g), hence det(g|V-) = Ø1(g)Ø2(g) = -1, so [g, a] = z by Lemma A.4(c*
*),
and this contradicts the assumption that [a, E] = 1. If dim(V-) = 6, then_V- is*
* the
sum of the eigenspaces of all but one of the nontrivial characters of E , and t*
*his gives
a similar contradiction to the assumption [a, E] = 1. Thus, CSpin7(q)(E) = E.
_
Now assume that E 2 EII4, and let x 2 O7(q) be the element which acts via - Id
on eigenspaces with nonsquare discriminant, and via the identity on those with *
*square_
discriminant. Since b has square discriminant on_V , the number of eigenspaces *
*of E on
which the discriminant is nonsquare is even, so x 2 7(q) by Lemma A.4(a),_and *
*lifts
to an element x 2 Spin7(q). Also, for each g 2 E, the (-1)-eigenspace of g has *
*square
discriminant_(Lemma A.4(a) again), hence contains an even number of eigenspaces*
* of
E of nonsquare discriminant, and by Lemma A.4(c) this shows that [g, x] = 1. Th*
*us
x 2 CSpin7(q)(E) = E, and this proves the first statement in (c).
Step 3: We next check the numbers of conjugacy classes of subgroups in each of*
* the
sets EInand EIIn, and describe AutSpin(E) in each case. This finishes the proof*
* of (b)
and (c), and of all points in the above table.
From the above description, we see immediately_that_if E and E0 have the same
_ _ 0
rank and type, then any isomorphism ff 2 Iso(E , E0),_such_that ff(x(E)) = x(E *
*) if
E, E0 2 EII4, has the property that for all Ø 2 Hom (E 0, { 1}), Vffland VfflOf*
*fhave the
same dimension and the same discriminant (modulo squares). Hence for any such f*
*f, __ __
there is an element g 2 O7(q) such that g(VfflOff) = Vfflfor all Ø; and ff = cg*
* 2 Iso(E , E0)
for such g. Upon replacing g by -g if necessary, we can assume that g 2 SO7(q).*
* This
shows that
__ __
E, E0 have the same rank and type =) E and E 0are SO7(q)-conjugate (1)
and also that (
__
__ Aut(E ) if E =2EII
Aut SO7(q)(E ) = __ _ 4 (2)
Aut(E , x(E))if E 2 EII4.
We next claim that
__
E =2EI4=) 9fl 2 SO7(q)r 7(q) such that[fl, E] = 1 . (3)
To prove this, choose 1-dimensional nonisotropic_summands W Vffland W 0 V_,
where Ø, _ are two distinct characters of E, and where W has square discriminan*
*t and
W 0has nonsquare discriminant._Let fl 2 SO7(q) be the involution_with (-1)-eige*
*nspace
W W 0. Then [fl, E] = 1, since fl sends each eigenspace of E to itself, and f*
*l =2 7(q)
since its (-1)-eigenspace has nonsquare discriminant (Lemma A.4(a)).
__
If E has rank 4 and type I, then Aut (E ) ~= GL3(F2) is simple,_and in partic*
*ular
has no subgroup of index 2. Hence by (2), each element of Aut (E ) is_induced *
*by
conjugation by an_element_of 7(q). Also, if g 2 SO7(q) centralizes E, then g(V*
*ffl) = Vffl
for all Ø 2 Hom (E , { 1}), g acts via - Idon an even number of eigenspaces (si*
*nce it
has determinant +1), and hence g 2 7(q) by Lemma A.4(a). Thus
__
E 2 EI4=) NSO7(q)(E ) 7(q) (4)
Ran Levi and Bob Oliver *
* 45
If E =2EI4, then by (3), for any g 2 SO7(q), there is fl 2 SO7(q)r__7(q)_such*
* that__ __
cg|E = cgfl|E, and either g or gfl lies in 7(q). Thus IsoSO7(q)(E , E0) = Iso *
*7(q)(E , E0) for
any E0. Together with (1), this shows that E is Spin-conjugate to all other sub*
*groups
of the same rank and type, and together with (2) it shows that
( __
__ Aut(E ) if E =2EII4
Im Aut Spin(E) --! Aut(E ) = __ _ (5)
Aut(E , x(E))if E 2 EII4.
__ __ __
If E 2 EI4, then by (4) and (2), Aut 7(q)(E ) = Aut SO7(q)(E ) = Aut_(E ),_and*
*_so (5)
also holds in this case. Furthermore, for any g 2 SO7(q)r 7(q), E and gE g-1*
* are
representatives for two distinct_ 7(q)-conjugacy classes _ since by (4), no ele*
*ment of
the coset g. 7(q) normalizes E .
We have now determined in all cases the number of conjugacy_classes_of subgro*
*ups
of a given rank and type, and the image of AutSpin(E) in Aut(E ). We next claim*
* that
if rk(E) < 4 or E 2 EI4, then
fi
E =2EII4=) Aut Spin(E) ff 2 Aut(E) fiff(z) = z, ff Id(mod ) .(6)
Together with (5), this will finish the proof that AutSpin(E) is the group of a*
*ll auto-
morphisms of E which send z to itself. We also claim that
fi
E 2 EII4=) Aut Spin(E) ff 2 Aut(E) fiff|X = IdX, ff Id(mod ) ,(7)
_ __
where X E denotes the inverse image of E , and this will finish the *
*proof
of (c).
We prove (6) and (7) together._ Fix ff 2 Aut(E) (ff 6= Id) which sends z to i*
*tself,
induces the identity_on E , and such that ff|X = IdX if E 2 EII4. Then ther*
*e is
_
1 6=_Ø 2 Hom (E , { 1}) such that ff(g) = g when Ø(g) = 1 and ff(g) = zg when
Ø(g) = -1. Choose any character _ such that V_ 6= 0 and V_ffl6= 0, and let W *
*V_
and W 0 V_fflbe 1-dimensional nonisotropic_subspaces with_the same discriminant
(this is possible when E 2 EII4since x(E)_2 Ker(Ø)). Let g 2 O7(q) be the_invol*
*ution
whose (-1)-eigenspace is W W 0. Then g 2 7(q) by Lemma A.4(a), so g lifts to
g 2 Spin7(q), and using Lemma A.4(c) one sees that cg = ff.
Step 4: It remains_to prove (d). Assume E 2 E3. Let 1 = Ø1, Ø2, Ø3, Ø4 be the *
*four
characters of E , and set Vi = Vffli. Then dim (V1) = 1, dim (Vi) = 2 for i = *
*2, 3, 4,
and the Vi either all have square_discriminantLor all have nonsquare discrimina*
*nt. For
each g 2 CSpin(E), we can write g = 4i=1gi, where gi 2 O(Vi, bi). For each pa*
*ir of
distinct indices i, j 2 {2, 3, 4}, there is some g 2 E whose (-1)-eigenspace is*
* Vi Vj,
and hence det(gi gj) = 1 by Lemma A.4(c). This shows that the giall have the s*
*ame
determinant. Let A CSpin(E) be the subgroup of index 2 consisting of those g *
*such
that det(gi) = 1 for all i.
_ _ _
Now, SO1(F q) = 1, while SO2(F q) ~=F*qis the group of diagonal matrices of t*
*he form
_
diag(u, u-1) with respect_to a hyperbolic basis of Fq2. Thus A is contained_in *
*a central
extension of C2 by (F *q)3, and any such extension is abelian since H2((F *q)3)*
* = 0. Hence
A is abelian. The groups_O2 (q) are all dihedral (see [Ta , Theorem 11.4]). Hen*
*ce for
any g 2 CSpin(E)r A, ghas order 2 and (-1)-eigenspace of dimension 4 (its inter*
*section
with each Viis 1-dimensional), and hence |g| = 2 by Lemma A.4(b). Thus all elem*
*ents
of CSpin(E)r A have order 2, so the centralizer must be a semidirect product of*
* A with
a group of order 2 which acts on it by inversion.
46 Construction of 2-local finite groups
Now assume that E 2 EII3; i.e., that the Vi all have nonsquare discriminant. *
*Then
for i = 2, 3, 4, SO(Vi, bi) has order q 1, whichever is not a multiple of 4 (*
*see [Ta ,
Theorem 11.4] again). Thus if g 2 A CSpin(E) has 2-power order, then gi = I*
* _
for each i, the number of i for which gi = Id is even (since the (-1)-eigenspac*
*e of g
has square discriminant), and hence g 2 E. In other words, E 2 Syl2(A). A Syl*
*ow
2-subgroup of CSpin(E) is thus generated by E together with an element of order*
* 2
which acts on E by inversion; this is an elementary abelian subgroup of rank 4,*
* and is
necessarily of type II.
We also need some more precise information_about the subgroups of Spin7(q) of
rank 4 and type II. Let _q 2 Aut(Spin7(F q)) denote the automorphism induced by*
* the
field automorphism (x 7! xq). By Lemma A.3, Spin7(q) is precisely the subgroup*
* of
elements fixed by _q.
Proposition A.9. Fix an odd prime power q, and let z 2 Z(Spin7(q)) be the centr*
*al
involution. Let C and C0 denote the two conjugacy classes of subgroups E Spin*
*7(q)
of rank 4 and type I. Then the following hold.
_
(a)For each E 2 E4, there is an element a 2 Spin7(F q) such that aEa-1 2 C. For*
* any
such a, if we set
xC(E) def=a-1_q(a),
then xC(E) 2 E and is independent of the choice of a.
(b)E 2 C if and only if xC(E) = 1, and E 2 C0 if and only if xC(E) = z.
(c)Assume E 2 EII4, and set ø(E) = . Then rk(ø(E)) = 2, and
fi
Aut Spin7(q)(E) = ff 2 Aut(E) fiff|fi(E)= Id .
__ _____
The four eigenspaces of E contained in the (-1)-eigenspace of xC(E) all have*
* non-
square discriminant, and the other three eigenspaces all have square discrim*
*inant.
Proof.(a) For all E 2 E4, E has type I as a subgroup of Spin7(q2) since all el*
*ements_
of Fq are squares in Fq2. Hence by Proposition A.8(b), for all E0 2 C, there i*
*s a 2
____-1 __0 _ 4
SO7(q2) 7(q4) such that_aE a = E . Upon lifting a to a 2 Spin7(q ), this pr*
*oves
that there is a 2 Spin7(F q) such that aEa-1 2 C.
Fix any such a, and set
x = xC(E) = a-1_q(a).
For all g 2 E, _q(g) = g and _q(aga-1) = aga-1 since E, aEa-1 Spin7(q), and h*
*ence
aga-1 = _q(a).g._q(a-1) = a(xgx-1)a-1.
Thus, x 2 CSpin7(_Fq)(E), and so x 2 E since it is self centralizing in each Sp*
*in7(qk)
(Proposition A.8(a)).
*
* _
We next check that xC(E) is independent of the choice of a. Assume a, b 2 Spi*
*n7(F q)
are such that aEa-1 2 C and bEb-1 2 C. Then by Proposition A.8(b), there is
g 2 Spin7(q) such that gbE(gb)-1 = aEa-1. Set E0 = aEa-1 2 C, then gba-1 2
NSpin7(_Fq)(E0). Furthermore, since Aut Spin7(q)(E0) contains all automorphism*
*s which
send z to itself, and since E0 is self centralizing in each of the groups Spin7*
*(qk) (both
by Proposition A.8 again), we see that NSpin7(_Fq)(E0) is contained in Spin7(q)*
*. Thus,
ba-1 2 Spin7(q), so _q(ba-1) = ba-1; and this proves that xC(E) = a-1_q(a) = b-*
*1_q(b)
is independent of the choice of a.
Ran Levi and Bob Oliver *
* 47
(b) If E 2 C, then we can choose a = 1, and so xC(E) = 1.
_
If E 2 C0, then by Proposition A.8(b), there is a 2 Spin7(q2) such that a*
* 2
SO7(q)r 7(q)_and aEa-1_2 C._ Then _q(a) 6= a since a 2= Spin7(q) (Proposition
A.3), but _q(a) = a since a 2 SO7(q). Thus, xC(E) = a-1_q(a) = z in this case.
We have now shown that_xC(E) 2 if E has type I, and it remains to prove t*
*he
converse._Fix_a 2 Spin7(F_q) such that aEa-1 2 C. If xC(E) 2 , then _q(a) 2 *
*{a, za},
so _q(a) = a, and hence a 2 SO7(q). Conjugation by an element of SO7(q) sends
eigenspaces with square discriminant to eigenspaces with square discriminant, s*
*o all
eigenspaces of E must have square discriminant since all eigenspaces of aEa-1 d*
*o.
Hence E has type I.
(c) Now write Spin = Spin7(q) for short. Assume E 2 EII4, and set x = xC(E) and
ø(E) = . Then x =2 by (b), and thus ø(E) has rank 2.
By (a) (the uniqueness of x having the given properties), each element of Aut*
*Spin(E)
restricts to the identity on ø(E). We have already seen (Proposition A.8(c)) th*
*at there
_ __ __
is an element x(E) 2 E such that the image in Aut (E ) of Aut Spin(E) is the g*
*roup
_ _ _ __
of automorphisms which fix x(E), and this shows that x(E) = x: the image in E of
x. Since we already showed (Proposition A.8(c) again) that Aut Spin(E) contain*
*s all
automorphisms which are the identity on ø(E) and the identity modulo , this *
*finishes
the proof that AutSpin(E) is the group of all automorphisms which are the ident*
*ity on
ø(E). The last statement (about the discriminants of the eigenspaces) follows d*
*irectly
from the first statement of Proposition A.8(c).
Throughout the rest of the section, we collect some more technical results wh*
*ich will
be needed in Sections 2 and 4.
Lemma A.10. Fix k 2. Let A = e13(2k-1) 2 GL3(Z=2k) be the elementary matrix
which has off diagonal entry 2k-1 in position (1, 3). Let T1 and T2 be the two *
*maximal
parabolic subgroups of GL3(2):
T1 = GL12(Z=2) = (aij) 2 GL3(2) | a21= a31= 0
and
T2 = GL21(Z=2) = (aij) 2 GL3(2) | a31= a32= 0 .
Set T0 = T1 \ T2: the group of upper triangular matrices in GL3(2). Assume that
~i:Ti-----! SL3(Z=2k)
are lifts of the inclusions (for i = 1, 2) such that ~1|T0 = ~2|T0. Then there*
* is a
homomorphism
~: GL3(2) ---! SL3(Z=2k)
such that ~|T1= ~1, and either ~|T2= ~2, or ~|T2= cA O~2.
Proof.We first claim that any two liftings oe, oe0:T2 --! SL3(Z=2k) are conjuga*
*te by
an element of SL3(Z=2k). This clearly holds when k = 1, and so we can assume
inductively that oe oe0(mod 2k-1). Let M03(F2) be the group of 3 x 3 matrices*
* of trace
zero, and define æ: T2 --! M03(F2) via the formula
oe0(B) = (I + 2k-1æ(B)).oe(B)
for B 2 T2. Then æ is a 1-cocycle. Also, H1(T2; M03(F2)) = 0 by [DW1 , Lemma 4*
*.3]
(the module is F2[T2]-projective), so æ is the coboundary of some X 2 M03(F2), *
*and oe
and oe0differ by conjugation by I + 2k-1X.
48 Construction of 2-local finite groups
By [DW1 , Theorem 4.1], there exists a section ~ defined on GL3(2) such that*
* ~|T1=
~1. Let B 2 SL3(Z=2k) be such that ~|T2 = cB O~2. Since ~|T0 = ~2|T0, B must
commute with all elements in ~(T0), and one easily checks that the only such el*
*ements
are A = e13(2k-1) and the identity.
Recall that a p-subgroup P of a finite group G is p-radical if NG(P )=P is p-*
*reduced;
i.e., if Op(NG(P )=P ) = 1. (Here, Op(-) denotes the largest normal p-subgroup.*
*) We
say here that P is Fp(G)-radical if OutG (P ) (= OutFp(G)(P )) is p-reduced. In*
* Section
4, some information will be needed involving the F2(Spin7(q))-radical subgroups*
* of
Spin7(q) which are also 2-centric. We first note the following general result.
Lemma A.11. Fix a finite group G and a prime p. Then the following hold for any
p-subgroup P G which is p-centric and Fp(G)-radical.
(a)If G = G1x G2, then P = P1x P2, where Pi is p-centric in Gi and Fp(Gi)-radic*
*al.
(b)If P H C G, then P is p-centric in H and Fp(H)-radical.
(c)If H C G has p-power index, then P \ H is p-centric in H and Fp(H)-radical.
__ __ __ __
(d)If G_C G has_p-power_index, then P = G \ P for some P G which is p-centric
in G and Fp(G )-radical.
(e)If Q C G is a central p-subgroup, then Q P , and P=Q is p-centric in G=Q a*
*nd
Fp(G=Q)-radical.
ff -1
(f)If eG-- i G is an epimorphism such that Ker(ff) Z(Ge), then ff (P ) is p-*
*centric
in eGand Fp(Ge)-radical.
Proof.Point (a) follows from [JMO , Proposition 1.6(ii)]: P = P1xP2 for Pi Gi*
*since
P is p-radical, and Pi must be p-centric in Gi and Fp(Gi)-radical since
CG(P ) = CG1(P1) x CG2(P2) and Out G(P ) ~=Out P1(G1) x OutP2(G2).
Point (b) holds since CH (P ) CG(P ) and Op(Out H(P )) Op(Out G(P )).
It remains to prove the other four points.
(e) Fix a central p-subgroup Q Z(G). Then P Q, since otherwise 1 6=
NQP (P )=P Op(NG(P )=P ). Also, P=Q is p-centric in G=Q, since otherwise the*
*re
would be x 2 Gr P of p-power order such that
1 6= [cx] 2 Ker OutG (P ) ---! OutG=Q(P=Q) x OutG(Q) Op(Out G(P )).
It remains only to prove that P=Q is Fp(G=Q)-radical, and to do this it suffice*
*s to
show that
Out G=Q(P=Q) ~=Out G(P ).
Equivalently, since P=Q and P are p-centric, we must show that
___NG=Q(P=Q)_____~ NG(P )
= ___________0;
C0G=Q(P=Q) x P=Q CG(P ) x P
and this is clear once we have shown that
C0G=Q(P=Q) ~=C0G(P ).
_ 0
Any x 2 CG=Q(P=Q) lifts to an element x 2 G of order prime to p, whose conjugat*
*ion
action on P induces the identity on Q and on P=Q. By [Go , Corollary 5.3.3], al*
*l such
automorphisms of P have p-power order, and thus x centralizes P . Since Q is a *
*p-group
Ran Levi and Bob Oliver *
* 49
and C0G=Q(P=Q) has order prime to p, this shows that the projection modulo Q se*
*nds
C0G=Q(P=Q) isomorphically to C0G(P ).
ff -1
(f) Let eG- -i G be an epimorphism whose kernel is central. Clearly, ff P is*
* p-
centric in eG. It remains only to prove that ff-1P is Fp(Ge)-radical, and to do*
* this it
suffices to show that
OutGe(ff-1P ) ~=Out G(P ).
Equivalently, since P and ff-1(P ) are p-centric, we must show that
____NGe(ff-1P_)__~ NG(P )
= ___________0;
C0eG(ff-1P ) x ff-1PCG(P ) x P
and this is clear once we have shown that
C0eG(ff-1P ) ~=C0G(P ).
This follows by exactly the same argument as in the proof of (e).
(c) Set P 0= P \ H for short. Let
i 0 0 0 0
NH (P 0) --- --i Out H(P ) ~=NH (P )=(CH (P ).P )
be the natural projection, and set
K = ß-1(Op(Out H(P 0))) NH (P 0).
Then K Op(NH (P 0)) is an extension of CH (P 0).P 0by the p-group Op(Out H(P *
*0)). It
suffices to show that p - [K:P 0], since this implies that Op(Out H(P 0)) = 1 (*
*i.e., P 0is
Fp(H)-radical), and that any Sylow p-subgroup of CH (P 0) is contained in P 0(h*
*ence P 0
is p-centric in H).
fi
Assume otherwise: that pfi[K:P 0]. Note first that P 0C NG(P ), and that NG(P*
* )
NG(K); i.e., NG(P ) normalizes P 0and K. The first statement is obvious, and t*
*he
second is verified by observing directly that NG(P ) normalizes NH (P 0) and CH*
* (P 0).
Thus the action of NG(P ) on K induces an action of NG(P ), and in particularfo*
*fiP ,
on K=P 0. Let K0=P 0denote the fixed subgroupfofithis action of P . Since pfi[K*
*:P 0] by
assumption, and since P is a p-group, pfi|K0=P 0|. A straightforward check also*
* shows
that K0 C NG(P ), and therefore that P K0 C NG(P ). Also, since P 0 K0 H,
P K0=P ~=K0=(P \ K0) = K0=P 0
is a normal subgroup of NG(P )=P of order a multiple of p. Since P is p-centric*
* in G
by assumption,
Out G(P ) = NG(P )=(CG(P ).P ) = NG(P )=(C0G(P ) x P ),
and hence the image of P K0=P in OutG (P ) is a normal subgroup which also has *
*order
a multiple of p.
By definition of K as an extension of CH (P 0).P 0by a p-group, if x 2 K has *
*order
prime to p, then x 2 CH (P 0). Hence if x 2 K0 has order prime to p, then for *
*every
z 2 P , [x, z] 2 P 0, so x acts trivially on P=P 0. Since x also centralizes P *
*0, it follows that
x centralizes P . This shows that the image of P K0=P in OutG (P ) is a p-group*
*, thus a
nontrivial normal p-subgroup of OutG (P ), and this contradicts the original as*
*sumption
that P is Fp(G)-radical.
50 Construction of 2-local finite groups
__
(d) Let G C G be a normal subgroup of p-power index and let P G be a p-centr*
*ic
and Fp(G)-radical subgroup. Let
i ffi
N_G(P ) --- --i Out _G(P ) ~=N_G(P ) C_G(P ).P
be the natural surjection, and set
K = ß-1 Op(Out _G(P )) N_G(P ).
__
Then K is an extension of C_G(P ).P by Op(Out _G(P )). Fix any P 2 Sylp(K). We *
*will
__ __ __ __
show that P \ G = P , and that P is p-centric in G and Fp(G )-radical.
For each x 2 K \ G NG(P ),
ß(x) 2 Op(Out _G(P )) \ OutG(P ) Op(Out G(P )) = 1.
Hence
0
x 2 Ker NG(P ) ----! Out_G(P ) = (C_G(P ).P ) \ G = CG(P ) . P ~=CG(P ) x *
*P,
where C0G(P ) CG(P ) is of order prime to p. Since the_opposite inclusion_is *
*obvious,
this shows that K \ G = C0G(P ) x P , and hence (since P 2 Sylp(K)) that P \ G *
*= P .
__
Next, note that (K \_G) C K and K=(K_\ G) G =G, and hence K=C0G(P ) has *
*__
p-power order. Since P 2 Sylp(K), P is an extension of P by K=(K \ G), and NK (*
*P )
is an extension of a subgroup of_(K \ G) = (C0G(P ) x_P )_by_K=(K \ G). Also, *
*an
element x 2 C0G(P ) normalizes P if and only if [x, P] 2 P \ C0G(P ) = 1. Hence
__ __ __ __ __
NK (P ) = CK (P ).P = C0G(P ) x P, (1)
__ __ __
where_C0G(P ) = C0G(P ) \ CG(P ) has order_prime_to p_and is_normal in NK (P_).*
*_Since
C_G(P ) C_G(P ) K, (1) shows that C_G(P ) C0G(P ) x P, and hence that P *
*is
__
p-centric in G .
__ __
It remains to show that P is Fp(G )-radical. Note first that K C N_G(P ) by c*
*onstruc-
__
tion, so for any x 2 N_G(P ), xP x-1 2 Sylp(K). Since K is an extension of C0G(*
*P ) x P
by the p-group_K=(K_\ G), and since C0G(P ) C K,_it follows_that K is a split e*
*xtension
of C0G(P ) by P . Hence for any x 2 N_G(P ), xP x-1 = yP y-1 for some y 2 C0G(*
*P ).
Consequently, the restriction map
__ __ __
N_G(P )=C_G(P ) ~=AutG_(P ) ------! Aut _G(P ) ~=N_G(P )=C_G(P )(2)
__ __ __ *
* __
is surjective. Also, if x 2 C_G(P ) K normalizes P , then x 2 NK (P ) ~=P x C*
*0G(P )
__ *
* __
by (1), and so cx 2 Inn(P ). Thus the kernel of the map in (2) is contained in *
*Inn(P ).
Consequently,
__ __ __
Out _G(P ) = Aut_G(P )= Inn(P ) ~=AutG_(P )= Aut_P(P ) ~=Out _G(P )=Op(Out _*
*G(P )),
__ __
and it follows that P is Fp(G )-radical.
This is now applied to show the following:
Proposition A.12. Fix an odd prime power q, and let P Spin7(q) be any sub- _
group which is 2-centric and F2(Spin7(q))-radical. Then P is centric in Spin7(*
*F q);
i.e., CSpin7(_Fq)(P ) = Z(P ).
Ran Levi and Bob Oliver *
* 51
Proof.Let z be the central involution in Spin7(q). By Lemma A.11(e), z 2 P , a*
*nd
__def
P = P= is 2-centric in 7(q) and is F2( 7(q))-radical._So by Lemma A.11(d),*
* there
is a 2-subgroup bP O7(q) such that bP\ 7(q) = P , and such that bPis 2-centri*
*c in
O7(q) and is F2(O7(q))-radical.
L m
Let V = i=1Vi be a maximal decomposition of V as an orthogonal direct sum of
bP-representations, and set bi = b|Vi. We assume these are arranged so that for*
* some
k, dim(Vi) > 1 when i k and dim(Vi) = 1 when i > k. Let V+ be the sum of those
1-dimensional components Viwith square discriminant, and let V- be the sum of t*
*hose
1-dimensional components Viwith nonsquare discriminant. We will be referring to*
* the
two decompositions
Mm Mk
(V, b) = (Vi, bi) = (Vi, bi) (V+, b+) (V-, b-),
i=1 i=1
both of which are orthogonal direct sums. We also write
_ (1) _
V (1)= Fq FqV and Vi = Fq FqVi,
and let b(1) and b(1)ibe the induced quadratic forms.
Step 1: For each i, set
Di= { IdVi} O(Vi, bi),
a subgroup of order 2; and write
Ym Y
D = Di O(V, b), and D = Di O(V , b ).
i=1 Vi V
Thus D and D are elementary abelian 2-groups of rank m and dim(V ), respective*
*ly.
We first claim that
bP D, (1)
and that
Ym
Pb = Pi where 8 i, Pi is 2-centric in O(Vi, bi) and F2(O(Vi, bi))-radica*
*l.(2)
i=1
Clearly, [D, bP] = 1 (and D is a 2-group), so D bPsince bPis 2-centric. This *
*proves
(1). The Viare thus distinct (pairwise nonisomorphic) as bP-representations, si*
*nce they
are pairwise nonisomorphic as D-representations. The decomposition as a sum of *
*Vi's
is thus unique (not only up to isomorphism), since Hom bP(Vi, Vj) = 0 for i 6= *
*j.
Let bCbe the group of elements of O(V, b) which send each Vito itself, and le*
*t bNbe
the group of elements which permute the Vi. By the uniqueness of the decomposit*
*ion
of V ,
Ym
bP.CO(V,b)(Pb) bC= O(Vi, bi) and NO(V,b)(Pb) bN.
i=1
Since bPis 2-centric in O(V, b) and F2(O(V, b))-radical, it is also 2-centric i*
*n Nb and
F2(Nb)-radical (this holds for any subgroup which contains NO(V,b)(Pb)). So by *
*Lemma
A.11(b) (and since Cb C Nb), Pb is 2-centric in Cb and F2(Cb)-radical. Point (*
*2) now
follows from Lemma A.11(a).
52 Construction of 2-local finite groups
Step 2: Whenever dim(Vi) > 1 (i.e., 1 i k), then by Lemma A.6, dim(Vi) is
even, and bihas square discriminant. So by Lemma A.4(a), - IdVi2 (Vi, bi) for *
*such
i. Together with (1), this shows that
__ Yk
P = bP\ 7(q) Dix (V+, b+) \ D+ x (V-, b-) \ D- . (3)
i=1
Also, by Lemma A.4(a) again,
(V , b ) \ D = SO(V ,b ) \ D
fi ff(4)
= - IdVi Vjfik+ 1 i < j m, Vi, Vj V .
__
Step 3: By (3) and (4), the Vi are distinct as P -representations (not only a*
*s Pb-
representations), except possibly when dim (V ) = 2. We first check that this *
*ex-
ceptional case_cannot_occur. If dim (V+) = 2 and its_two irreducible summands *
*are
isomorphic as P -representations, then the image of P under projection to O(V+,*
* b+)
is_just { IdV+}. Hence we can write V+ = W W 0, where W ?W 0are 1-dimensiona*
*l,
P-invariant, and have nonsquare discriminant. Also, dim(V-) is odd,_since V+ an*
*d_the_
Vifor i k are all even dimensional._So - IdV- W lies in C 7(q)(P ) but not in*
* P. But
this is impossible, since P is 2-centric in 7(q). The argument when dim(V-) = *
*2 is
similar.
__
The Vi are thus distinct as P -representations. So for all i 6= j, Hom P (Vi*
*, Vj) = 0,
and hence _
Hom _Fq[P](Vi(1), Vj(1)) ~=Fq FqHom Fq[P](Vi, Vj) = 0.
__ (1)
Thus any element of O(V (1), b(1)) which centralizes P sends each Vi to itsel*
*f. In
other words,
__ Ym (1) (1)
CSpin7(_Fq)(P )= C 7(_Fq)(P ) O(Vi , bi ).
i=1
__
If dim (V ) 2, then since P contains all involutions in O(V , b ) which ar*
*e Pb-
invariant and have even dimensional_(-1)-eigenspace (see (3)), Lemma A.4(c) sho*
*ws
that each element of Spin7(F q)_which commutes with P must act on V via Id. *
*Also,
for 1 i k, since - IdVi2 P by (3), each element in the centralizer of P act*
*s on Vi
with determinant 1 (Lemma A.4(c) again). We thus conclude that
Yk
CSpin7(_Fq)(P )= SO(Vi(1), b(1)i) x { IdV+} x { IdV-}.(5)
i=1
Step 4: We next show that
Yk
CSpin7(_Fq)(P )= { IdVi} x { IdV+} x { IdV-}. (6)
i=1
Using (5), this means showing, for each 1 i k, that
pri CSpin7(_Fq)(P )= { IdVi}; (7)
_ (1) (1)
where pri denotes the projection of O7(F q) = O(V (1), b(1)) to O(Vi , bi ). *
* By
Lemma A.6, dim(Vi) = 2 or 4. We consider these two cases separately.
Ran Levi and Bob Oliver *
* 53
Case 4A: If dim(Vi) = 4, then by (2) and Lemma A.11(c), Pi0def=Pi\ (Vi, bi) is
2-centric in (Vi, bi) and is F2( (Vi, bi))-radical. Also, by Proposition A.5,
(Vi, bi) ~= +4(q) ~=SL2(q) xC2 SL2(q).
By Lemma A.11(a,f), under this identification, we have Pi0= QxC2Q0, where Q and*
* Q0
are 2-centric in SL2(q) and F2(SL2(q))-radical. The Sylow 2-subgroups of SL2(q)*
* are
quaternion groups of order 8, all subgroups of a quaternion 2-group are quate*
*rnion
or cyclic, and cyclic 2-subgroups of SL2(q) cannot be both 2-centric and F2(SL2*
*(q))-
radical. So Q and_Q0 must be quaternion of order 8. By [Sz, 3.6.3], any cyc*
*lic
2-subgroup of SL2(F q) of order 4 is conjugate to a subgroup of_diagonal matr*
*ices,
whose centralizer is the group of all diagonal matrices in SL2(F q)._ Knowing t*
*his,
one easily_checks that all nonabelian quaternion 2-subgroups of SL2(F q) are ce*
*ntric in
SL2(F q). It follows that Pi0is centric in
_ _
SO(Vi(1), b(1)i) ~=SL2(F q) xC2 SL2(F q),
and hence that
0 0
priCSpin7(_Fq)(P )= CSO(V (1) (1)(Pi) = Z(Pi) = { IdVi}.
i ,bi )
Thus (7) holds in this case.
Case 4B: If dim(Vi) = 2, then O(Vi, bi) ~=O2 (q) is a dihedral group of order *
*2(q 1)
[Ta , Theorem 11.4]. Hence Pi 2 Syl2(O(Vi, bi)), since the Sylow subgroups are*
* the
only radical 2-subgroups of a dihedral group. Fix Vj for any k < j m, and cho*
*ose
ff 2 O(Vi, bi) of determinant (-1) whose (-1)-eigenspace has the same discrimin*
*ant
as Vj. Since Pi 2 Syl2(O(Vi, bi)), we_can assume (after conjugating if necessar*
*y) that
ff 2 Pi. Then (- IdVj) ff lies in P = bP\ 7(q). Hence for any g 2 CSpin7(_Fq*
*)(P )=,
pri(g) 2 O(Vi(1), b(1)i) leaves both eigenspaces of ff invariant, and has deter*
*minant 1
by (5). Thus pri(g) = IdVi; and so (7) holds in this case.
Step 5: Clearly, - IdV lies in SO(V , b ) if and only if dim(V ) is even (whi*
*ch is
the case for exactly one of the two spaces V ), and this holds if and only if -*
* IdV 2
(V , b ). Also, since each Vi for 1 i k has square discriminant (Lemma A.6
again), - IdVi2 (Vi, bi) for all such i. Thus (6) and (1) imply that
__
CSpin7(_Fq)(P )= bP\ 7(q) = P,
_
and hence that P is centric in Spin7(F q).
Proposition A.12 does not hold in general if Spin7(-) is replaced by an arbit*
*rary
algebraic group. For example, assume q is an odd prime power, and let P SL5(q)
be the group of diagonal matrices of 2-power order. Then P is_2-centric in SL5(*
*q) and
F2(SL5(q))-radical, but is definitely not 2-centric in SL5(F q).
References
[As1] M. Aschbacher, A characterization of Chevalley groups over fields of odd *
*order, Annals of
Math. 106 (1977), 353-398
[As2] M. Aschbacher, Finite group theory, Cambridge Univ. Press (1986)
[Be] D. Benson, Cohomology of sporadic groups, finite loop spaces, and the Dic*
*kson invariants,
Geometry and cohomology in group theory, London Math. Soc. Lecture notes *
*ser. 252, Cam-
bridge Univ. Press (1998), 10-23
54 Construction of 2-local finite groups
[BK] P. Bousfield & D. Kan, Homotopy limits, completions, and localizations, L*
*ecture notes in
math. 304, Springer-Verlag (1972)
[BLO1]C. Broto, R. Levi, & B. Oliver, Homotopy equivalences of p-completed clas*
*sifying spaces of
finite groups, preprint
[BLO2]C. Broto, R. Levi, & B. Oliver, The homotopy theory of fusion systems, pr*
*eprint
[BM] C. Broto & J. Møller, Homotopy finite Chevalley versions of p-compact gro*
*ups (in preparation)
[Di] J. Dieudonn'e, La g'eom'etrie des groupes classiques, Springer-Verlag (19*
*63)
[DW1] W. Dwyer & C. Wilkerson, A new finite loop space at the prime two, J. Ame*
*r. Math. Soc. 6
(1993), 37-64
[DW2] W. Dwyer & C. Wilkerson, Homotopy fixed-point methods for Lie groups and *
*finite loop
spaces, Annals of Math. 139 (1994), 395-442
[DW3] W. Dwyer & C. Wilkerson, The center of a p-compact group, The ~Cech cente*
*nnial, Contemp.
Math. 181 (1995), 119-157
[Fr] E. Friedlander, Etale homotopy of simplicial schemes, Princeton Univ. Pre*
*ss (1982)
[FM] E. Friedlander & G. Mislin, Cohomology of classifying spaces of complex L*
*ie groups and
related discrete groups, Comment. Math. Helv. 59 (1984), 347-361
[Gd] D. Goldschmidt, Strongly closed 2-subgroups of finite groups, Annals of M*
*ath. 102 (1975),
475-489
[Go] D. Gorenstein, Finite groups, Harper & Row (1968)
[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
[La] J. Lannes, Sur les espaces fonctionnels dont la source est le classifiant*
* d'un p-groupe ab'elien
'el'ementaire, Publ. I.H.E.S. 75 (1992)
[Nb] D. Notbohm, On the 2-compact group DI(4) (preprint)
[Pu] L. Puig, Unpublished notes
[Sm] L. Smith, Homological algebra and the Eilenberg-Moore spectral sequence, *
*Trans. Amer.
Math. Soc. 129 (1967), 58-93
[Sm2] L. Smith, Polynomial invariants of finite groups, A. K. Peters (1995)
[So] R. Solomon, Finite groups with Sylow 2-subgroups of type .3, J. Algebra 2*
*8 (1974), 182-198
[Sz] M. Suzuki, Group theory I, Springer-Verlag (1982)
[Ta] D. Taylor, The geometry of the classical groups, Heldermann Verlag (1992)
[Wb] C. Weibel, An introduction to homological algebra, Cambridge Univ. Press *
*(1994)
[Wi] C. Wilkerson, A primer on the Dickson invariants, Proc. Northwestern homo*
*topy theory
conference 1982, Contemp. Math. 19 (1983), 421-434
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
__