SATURATED FUSION SYSTEMS OVER 2-GROUPS
BOB OLIVER AND JOANA VENTURA
Abstract. We develop methods for listing, for a given 2-group S, all nonc*
*onstrained
centerfree saturated fusion systems over S. These are the saturated fusio*
*n systems
which could, potentially, include minimal examples of exotic fusion syste*
*ms: fusion
systems not arising from any finite group. To test our methods, we carry *
*out this
program over four concrete examples: two of order 27 and two of order 210*
*. Our long
term goal is to make a wider, more systematic search for exotic fusion sy*
*stems over
2-groups of small order.
For any prime p and any finite p-group S, a saturated fusion system over S is*
* a
category F whose objects are the subgroups of S, whose morphisms are injective *
*group
homomorphisms between the objects, and which satisfy certain axioms due to Puig
and described here in Section 2. Among the motivating examples are the categor*
*ies
F = FS(G) where G is a finite group with Sylow p-subgroup S: the morphisms in
FS(G) are the group homomorphisms between subgroups of S which are induced by
conjugation by elements of G. A saturated fusion system F which does not arise*
* in
this fashion from a group is called "exotic".
When p is odd, it seems to be fairly easy to construct exotic fusion systems *
*over
p-groups (see, e.g., [BLO2 , x9], [RV ], and [Rz ]), although we are still ver*
*y far from
having any systematic understanding of how they arise. But when p = 2, the only
examples we know are those constructed by Levi and Oliver [LO ], based on earli*
*er
work by Solomon [S2] and Benson [Be ]. The smallest such example known is over*
* a
group of order 210, and it is possible that there are no exotic examples over s*
*maller
groups. Our goal in this paper is to take a first step towards developing techn*
*iques for
systematically searching for exotic fusion systems, a search which eventually c*
*an be
carried out in part using a computer.
A fusion system F is constrained (Definition 2.3) if it contains a normal p-s*
*ubgroup
which contains its centralizer; any constrained fusion system is the fusion sys*
*tem of
a unique finite group with analogous properties ([BCGLO1 , Proposition C]). A*
* fusion
system F over S is centerfree (Definition 2.3) if there is no element 1 6= z 2 *
*Z(S) such
that each morphism in F extends to a morphism between subgroups containing z wh*
*ich
sends z to itself. By [BCGLO2 , Corollary 6.14], if there is such a z, and if*
* F is exotic,
then there is a smaller exotic fusion system F= over S=. Thus all minimal*
* exotic
fusion systems must be nonconstrained and centerfree, and these conditions prov*
*ide a
convenient class of fusion systems to search for and list.
If F is a saturated fusion system over any p-group S, then the F-essential su*
*b-
groups of S are the proper subgroups P S which "contribute new morphisms" to
the category F: it is the smallest set of objects such that each morphism in S *
*is a
composite of restrictions of automorphisms of essential subgroups and of S itse*
*lf. We
___________
2000 Mathematics Subject Classification. Primary 20D20. Secondary 20D45, 20D0*
*8.
Key words and phrases. finite groups, 2-groups, fusion, simple groups.
B. Oliver is partially supported by UMR 7539 of the CNRS.
J. Ventura is partially supported by FCT/POCTI/FEDER and grant PDCT/MAT/58497*
*/2004.
1
2 BOB OLIVER AND JOANA VENTURA
refer to Definition 2.3, Proposition 2.5, and Corollary 2.6 for more details. W*
*e define
a critical subgroup of S to be one which could, potentially, be essential in so*
*me fusion
system over F. The precise definition (Definition 3.1) is somewhat complicated *
*(and
stated without reference to fusion systems), and involves the existence of subg*
*roups of
Out(P ) which contain strongly embedded subgroups. Thus Bender's classification*
* of
groups with strongly embedded subgroups (at the prime 2) plays a central role i*
*n our
work. In addition, one important thing about critical subgroups is that the 2-g*
*roups
we have studied contain very few of them (even those 2-groups which support many
"interesting" saturated fusion systems), and we have developed some fairly effi*
*cient
techniques for listing them.
Thus, the first step when trying to find all saturated fusion systems over a *
*2-group S
is to list its critical subgroups. Afterwards, for each critical P (and for P =*
* S), one com-
putes Out(P ), and determines which subgroups of Out(P ) can occur as AutF (P )*
* if P
is F-essential. The last step is then to put this all together: to see which co*
*mbinations
of essential subgroups and their automorphism groups can generate a nonconstrai*
*ned
centerfree saturated fusion system F.
To illustrate how this procedure works in practice, we finish by listing all *
*noncon-
strained centerfree saturated fusion systems over four 2-groups: two of order 2*
*7 and
two of order 210. We chose them because each is the Sylow subgroup of several *
*"in-
teresting" simple or almost simple groups; in fact, each is the Sylow 2-subgrou*
*p of
at least one sporadic simple group. The groups we chose are the Sylow 2-subgro*
*ups
of M22, M23, and McL ; J2 and J3; He , M24, and GL5(2); and Co3. The last case*
* is
particularly interesting because it is also the Sylow subgroup of the only know*
*n exotic
fusion system over a 2-group of order 210.
Not surprisingly, we found no new exotic fusion systems over any of these four
groups, and a much wider and more systematic search will be needed to have much
hope of finding new exotic examples. For example, over the group S = UT5(2) of *
*upper
triangular 5 x 5 matrices over F2, we show (Theorem 6.8) that the only nonconst*
*rained
centerfree saturated fusion systems are those of the simple groups He, M24, and*
* GL5(2).
Likewise, over the Sylow 2-subgroup of Co3, we show (Theorem 7.8) that each such
fusion system is either the fusion system of Co3, or that of the almost simple *
*group
Aut(P Sp6(3)), or the exotic fusion system Sol(3) constructed in [LO ]. Thus in*
* these
cases, we repeat in part the well known results of Held [He ] and Solomon [S2],*
* except
that we only classify fusion systems over these 2-groups, and do not try to lis*
*t all groups
which contain them as Sylow subgroups. But the techniques we use are somewhat
different, and we hope that they can eventually make possible a more systematic*
* search
for exotic fusion systems.
This approach also makes it easy to determine all automorphisms of the fusion
systems we classify. We don't state it here explicitly, but using the informati*
*on given
about the fusion systems over the four 2-groups we study, one can easily determ*
*ine
their automorphisms, and check they all extend to automorphisms of the associat*
*ed
groups.
The paper is organized as follows. The first section contains background resu*
*lts on
finite groups, their automorphism groups, and strongly embedded subgroups, while
Section 2 contains background results on fusion systems. Then, in Section 3, cr*
*itical
subgroups are defined, and techniques developed for determining the critical su*
*bgroups
of a given 2-group. Afterwards, in Sections 4-7, we present our examples, descr*
*ibing
the nonconstrained centerfree saturated fusion systems over four different 2-gr*
*oups.
SATURATED FUSION SYSTEMS OVER 2-GROUPS 3
We would like in particular to thank Kasper Andersen, who helped revive our i*
*nterest
in this program by doing a computer search for some critical subgroups; and Andy
Chermak, for (among other things) suggesting we look at the Sylow subgroup of t*
*he
Janko groups J2 and J3.
1. Background results
We collect here some results about groups and their automorphisms which will *
*be
needed later. Almost all of them are either well known, or follow from well kn*
*own
constructions.
We first recall some standard notation. For any group G and any prime p, Op(G)
denotes the largest normal p-subgroup (the intersection of the Sylow p-subgroup*
*s of
G), and Op(G) denotes the smallest normal subgroup of p-power index. Also, Op0(*
*G)
0
denotes the largest normal subgroup of order prime to p, and Op (G) denotes the
smallest normal subgroup of index prime to p.
1.1___Automorphisms_of_p-groups___
We first consider conditions which can be used to show that certain automorph*
*isms
of a p-group P lie in O2(Aut (P )). Recall that the Frattini subgroup Fr(P ) o*
*f a p-
group P is the subgroup generated by commutators and p-th powers; i.e., the sma*
*llest
normal subgroup whose quotient is elementary abelian. It has the property that*
* if
g1, . .,.gk 2 P are elements whose classes generate P=Fr(P ), then they also ge*
*nerate
P .
Lemma 1.1. Fix a prime p, a p-group P , a subgroup P0 Fr(P ), and a sequence *
*of
subgroups
P0 P1 . . .Pk = P.
Set
fi -1
A = ff 2 Aut(P ) fix ff(x) 2 Pi-1, all x 2 Pi, all i = 1, . .,.k Aut(P *
*) :
the group of automorphisms which leave each Piinvariant, and which induce the i*
*dentity
on each quotient group Pi=Pi-1. Then A is a p-group. If the Pi are all characte*
*ristic
in P , then A C Aut(P ), and hence A Op(Aut (P )).
Proof.To prove that A is a p-group, it suffices to show that each element ff 2 *
*A has
p-power order. This follows, for example, from [G , Theorems 5.1.4 & 5.3.2]. Th*
*e last
statement is then clear.
We next turn to the problem of determining Out (P ) for a p-group P . In the *
*next
lemma, for any group G and any normal subgroup H C G, we let Aut(G, H) Aut(G)
denote the group of automorphisms ff of G such that ff(H) = H, and set Out(G, H*
*) =
Aut(G, H)=Inn(G).
Lemma 1.2. Fix a group G and a normal subgroup H C G such that CG(H) H
(i.e., H is centric in G). Then there is an exact sequence
1 ---! H1(G=H; Z(H)) -----! Out (G, H) --Res---!
O 2
NOut(H)(Out G(H))=Out G(H) -----! H (G=H; Z(H)), (1)
where all maps except (possibly) O are homomorphisms. If, furthermore, H is abe*
*lian
and the extension of H by G=H is split, then the restriction map is onto.
4 BOB OLIVER AND JOANA VENTURA
Proof.We first prove that there is an exact sequence of the following form:
j Res
1 ---! Z1(G=H; Z(H)) -----! Aut (G, H) -----!
eO 2
NAut(H)(Aut G(H)) -----! H (G=H; Z(H)). (2)
The restriction map is well defined, since for all ff 2 Aut (G, H) and all g 2*
* G,
(ff|H )cg(ff|H )-1 = cff(g)2 Aut G(H). Also, Z1(G=H; Z(H)) denotes the group o*
*f 1-
cocycles: those maps ! :G=H -! Z(H) such that
!(g1g2H) = !(g1H).g1!(g2H)g-11.
Let j(!) 2 Aut(G) be the automorphism j(!)(g) = !(gH).g for all g 2 G. This cle*
*arly
defines an injective homomorphism from Z1(G=H; Z(H)) into Ker(Res).
Now assume ff 2 Ker(Res); thus ff 2 Aut(G) is such that ff|H = IdH. Then for *
*all
g 2 G, cg = cff(g)2 Aut(H), and so ff(g) = !(gH).g for some ! :G=H ! Z(H). Also,
for all g, h 2 G,
!(ghH) = ff(gh).(gh)-1 = ff(g)(ff(h)h-1)g-1 = !(gH).g(!(hH)),
and hence ! 2 Z1(G=H; Z(H)). This proves the exactness of (2)at Aut(G, H).
We next prove that Im (Res) = eO-1(0). Fix some ' 2 NAut(H)(Aut G(H)), and l*
*et
_ 2 Aut (G=Z(H)) be defined by _(gZ(H)) = g0Z(H) if 'cg'-1 = cg0in Aut (H).
This defines _ uniquely since H is centric. The obstruction to extending ' and *
*_ to an
automorphism of G is the same as the obstruction to two extensions of H by G=H *
*(with
the same outer action of G=H on H) being isomorphic, and thus lies in H2(G=H; Z*
*(H))
(cf. [Mc , Theorem IV.8.8]). More explicitly, choose any map of sets b':G ---! *
*G such
that for all g 2 G and h 2 H, b'(g) 2 _(gZ(H)) and b'(hg) = '(h)'b(g). Then
b'(gh) = b'(ghg-1.g) = '(ghg-1)'b(g) = c_(gZ(H))'(h) .b'(g) = b'(g)'(h);
for all g 2 G and h 2 H, where the third equality holds since 'cgZ(H)= c_(gZ(H)*
*)',
and the fourth since b'(g) 2 _(gZ(H)). Using this, one shows there is a map
O: G=H x G=H ---! Z(H) defined by 'b(g1g2) = O(g1H, g2H).b'(g1)'b(g2).
This is a 2-cocycle by the associativity of G, and a different choice of b'chan*
*ges O by a
coboundary. Thus ' extends to b'2 Aut(G) if and only if [O] = 0 in H2(G=H; Z(H)*
*).
This proves the exactness of (2). The exactness of (1)now follows by dividing*
* out
by the short exact sequence
1 ---! Aut Z(H)(G) -----! Inn(G) -----! Aut G(H) ---! 1.
Note in particular that the group of 1-coboundaries with coefficients in Z(H) c*
*orre-
sponds exactly to AutZ(H)(G).
It remains to prove the last statement. Assume that H is abelian, and that G*
* =
HK where H \ K = 1. Thus K projects isomorphically onto G=H ~= Out G(H),
and so we can identify these groups. Fix fi 2 Aut (H) = Out (H), and assume fi*
* 2
NAut(H)(Aut G(H)). Let fl 2 Aut(K) be the automorphism of K ~=Aut G(H) induced
by conjugation by fi. Then there is an automorphism ff 2 Aut(G) such that ff|H *
*= fi
and ff|K = fl, and this shows that the restriction maps are surjective in this *
*case.
1.2___Strongly_embedded_subgroups____
Strongly embedded subgroups of a finite group play a central role in this pap*
*er. We
begin with the definition.
SATURATED FUSION SYSTEMS OVER 2-GROUPS 5
Definition 1.3. Fix a primefp.iFor any finite group G, a subgroup G0 G is cal*
*led
strongly embedded at p if pfi|G0|, and for all g 2 Gr G0, G0\ gG0g-1 has order *
*prime
to p. A subgroup G0 G is strongly embedded if it is strongly embedded at 2.
The following proposition describes one way to characterize strongly embedded*
* sub-
groups.
Lemma 1.4. Fix a prime p, a finite group G, and a Sylow p-subgroup S G. Set
G0 = 1.
Then G contains a strongly embedded subgroup at p if and only if G0 G. More
precisely, if G0 G, then G0 is strongly embedded at p. Conversely, if H G *
*is a
strongly embedded subgroup at p, then G0 gHg-1 for some g 2 G, and thus G0 *
*G.
Proof.Assume that G0 G; we show thatfG0iis strongly embedded in G at p. By
construction, G0 S 6= 1, and so pfi|G0|. Let g 2 G be such that gG0g-1 \ G0
has order a multiple of p; we must show that g 2 G0. By assumption, there is an
element x 2 gG0g-1 \ G0 of order p. Since S 2 Sylp(G0) (S 2 Sylp(G) since P is *
*fully
normalized), there are elements h, k 2 G0 such that hxh-1, k(g-1xg)k-1 2 S. Thus
1 6= hxh-1 2 S \ (hgk-1)S(hgk-1)-1 ;
so hgk-1 2 G0 by (1), and thus g 2 G0.
Conversely, assume that H G is a strongly embedded subgroup at p. Then H
contains a Sylow p-subgroup of G, so gHg-1 S for some g 2 G, and H0def=gHg-1f*
*isi
also strongly embedded. Hence for g 2 G0, gSg-1 \ S 6= 1 by (1), so pfi|gH0g-1 *
*\ H0|,
and g 2 H0 since it is strongly embedded. Thus G0 H0.
The classification of all finite groups with strongly embedded subgroups is d*
*ue to
Bender.
Theorem 1.5 (Bender). Let G be a finite group with strongly embedded subgroup (*
*at
the prime 2). Fix a Sylow subgroup S 2 Syl2(G). Then either S is cyclic or quat*
*ernion,
0 n n
or O2 (G=O20(G)) is isomorphic to one of the simple groups P SL2(2 ), P SU3(2 )*
*, or
Sz(2n) (where n 2, and n is odd in the last case).
Proof.See [Be ].
The following lemma about F2-representations of groups with strongly embedded
subgroups (at the prime 2) will play a key role in the next section, and in lat*
*er appli-
cations. Much stronger results are, in fact, true; we limit it here to results *
*which are
fairly easy to prove or which will be needed for applications later in this pap*
*er.
When ff is an automorphism of a vector space V , we write [ff, V ] = Im[V -ff*
*-Id---!V ].
Lemma 1.6. Let G be a finite group with strongly embedded subgroup, and let V b*
*e an
F2-vector space on which G acts faithfully. Fix some S 2 Syl2(G), and let 1 6= *
*s 2 S
be any nonidentity element. Then the following hold.
(a) If |S| = 2k, then dimF2(V ) 2k.
(b) If Z(S) ~=Cn2, then rk([s, V ]) n.
(c) If S ~=C2k or Q2k, and k > 1, then rk([s, V ]) 2.
6 BOB OLIVER AND JOANA VENTURA
Proof.The result is clear when |S| = 2 (rk(V ) 2.rk([s, V ]) 2), so we assu*
*me
|S| 4. If s 2 S has order 4, then [s2, V ] [s, V ] ((s2- Id)(v) = (s - Id*
*)(v + s(v))),
so it suffices to prove (b) and (c) when s is an involution. Also, since all in*
*volutions in
G are conjugate by [Sz2, Lemma 6.4.4], it suffices to prove (b) for just one in*
*volution
s in S. We handle the case where Z(S) is noncyclic in Case 1, and the case wher*
*e S
is cyclic or quaternion in Case 2.
0
Case 1: Assume first that O2 (G=O20(G)) ~=L where L is simple. We first note t*
*he
following:
(1) For any n > 1, there is a prime power pa such that pa|(2n - 1), but pa-(2l-*
* 1) for
l < n. Furthermore, if n 6= 2, 6, we can choose p 6= 3.
To see this, let n be the n-th cyclotomic polynomial, and let p be any p*
*rime
dividing n(2). Thus p|(2n - 1)=(2m - 1) for all m < n dividing n. So if we*
* let pa
be the largest power of p which divides 2n - 1, then pa does not divide pm *
*- 1 for
m < n dividing n, and thus n is the order of 2 mod pa. If p = 3, then n = 2*
*.3r
r-1 3r-1
(since |(Z=3a)x| = 2.3a-1), so n(X) = X2.3 - X + 1, n(2) 3 (mod 9),
and so we can choose p > 3 dividing n(2) if n > 6 ( n(2) > 3).
(2) Fix an odd prime power q, and let n be the order of 2 (mod q). If V is a fa*
*ithful
Cq-representation over F2, then rk(V ) n. If V is a faithful D2q-represen*
*tation
over F2, and g 2 D2qis an involution, then rk([g, V ]) n=2. *
* _
To see this, fix a generator_a of Cq, and consider the set of eigenvalues*
* in F2 of
the action of a on V (or on F2 F2V ). This set includes primitive pa-th ro*
*ots of
unity, and is invariant under the action of the Frobenius automorphism u 7!*
* u2.
Hence there are at least n eigenvalues, and so rk(V ) n. If this extends*
* to an
action of the dihedral group, then g switches eigenspaces for eigenvalues u*
* and
u-1 (and the eigenvalue -1 doesn't occur), so rk([g, V ]) n=2.
(3) If L contains a dihedral subgroup of order 2pa for some odd prime p, then G
contains a dihedral subgroup of order 2pa.
_ _ _ a _ _ _
To see this, fix g, h2 L such that |g| = p , |h| = 2, and is dihed*
*ral. Let
_ *
* _
C G be the coset sent to g, and let h 2 G be any element of order 2 sent *
*to h.
There is an involution oe on C which sends g to (hgh-1)-1, and since C has *
*odd
order, there is some bg2 C which is fixed by oe. Then his dihedral of *
*order
2m where pa|m, and contains a dihedral subgroup of order 2pa.
There are three subcases to consider.
Case 1A: Assume L ~=P SL2(q), where q = 2k. Then S ~=Ck2. Choose pa|(q2 - 1) as
in (1) above. Then pa|(q + 1), since q + 1 and q - 1 are relatively prime and p*
*a-(q - 1)
by assumption. Since L contains a dihedral subgroup of order 2(q + 1), G contai*
*ns a
dihedral subgroup D of order 2pa by (3). Hence by (2), rk(V ) 2n, and rk([g, *
*V ]) n
for any involution g 2 D.
Case 1B: Next assume L ~=Sz(q) for odd q = 2n 8. Then |S| = q2 and Z(S) ~=Cn*
*2.
This time, choose pa|(q4 - 1) as in (1). Since
p __ p __
q4 - 1 = 24n- 1 = (q + 2q+ 1)(q + 2q+ 1)(q2 - 1),
where the factors are relatively prime (andpsince_pa-(q2- 1) by assumption), th*
*is shows
that pa divides one of the factorspq__ 2q + 1. By [H3 , TheorempXI.3.10],_L co*
*ntains
dihedral subgroups of order 2(q + 2q+ 1) and of order 2(q - 2q+ 1). Hence G
SATURATED FUSION SYSTEMS OVER 2-GROUPS 7
contains a dihedral subgroup D of order 2pa by (3). By (2) again, this implies*
* that
rk(V ) 4n, and rk([g, V ]) 2n for all involutions g 2 D.
Case 1C: Now assume L ~=P SU3(q) for q = 2n with n 2. Then |S| = q3 = 23n and
Z(S) ~=Cn2. Let pa|(q6-1) be as in (1), with p > 3. Then pa|(q3+1) = (q2-q+1)(q*
*+1),
pa - (q + 1), and so pa|(q2 - q + 1) since p > 3 cannot divide both factors (si*
*nce
q2- q + 1 = (q + 1)(q - 2) + 3). Also, L contains a cyclic subgroup of order (q*
*2- q + 1)
or (q2 - q + 1)=3 (which comes from regarding Fq6as a 3-dimensional Fq2-vector *
*space
3 a
with hermitian product (x, y) = xyq ). So there is a cyclic subgroup of order p*
* in L
and hence also in G, and rk(V ) 6n by (2).
Next let pa|(q2- 1) be as in (1). In particular, pa|(q + 1). Let D L be the*
* dihedral
subgroup of order 2(q + 1) generated by diagonalimatricesjdiag(u, u-1, 1) for u*
* 2 Fq2
0 10
with uq+1 = 1 and by the permutation matrix -100001. By (3), L and G both cont*
*ain
dihedral subgroups of order 2pa; and rk([s, V ]) n by (2).
Case 2: Now assume S is cyclic or quaternion of order 2k with k 2, andfseti
H = O20(G). Recall that s 2 S is the (unique) involution. For each prime pfi|H|*
*, the
number of Sylow p-subgroups of H is odd, and hence there is at least one subgro*
*up
Hp 2 Sylp(H) which is normalized by S. Since H is generated by these Hp, at lea*
*st
one of them is not centralized by s. So upon replacing H by some appropriate Sy*
*low
subgroup Hp and replacing G by HpS, we can assume that H = O20(G) is a p-group.
__ _
Set V = F2 F2V , regarded as a representation_of G = HS. Set H0 = Z(H): a
nontrivial abelian p-group._Set bH0= Hom (H0, F*2),_regarded as the set of irre*
*ducible
L __ __
characters of irreducible F 2[H0]-modules. Write V = O2Hb0VO, where V O is*
* the
subspace generated by all 1-dimensional H0-subrepresentations having character *
*O.
Assume first that [s, H0] 6= 1. Since H0 acts faithfully on V , and since th*
*e only
s-invariant characters_of H0 are those which factor through H0=[s, H0], there i*
*s some
O 2 Hb0 such that V O 6= 0 and is not s-invariant._ Then O is in a free S-orbit*
* of
characters, and hence rk(V ) = dim_F2(V ) |S| and rk([s, V ]) 1_2|S|.
Now assume [s, H0] = 1. Then [s, V ] is a nontrivial H0-subrepresentation of*
* V ,
and so rk([s, V ]) 2. Also, H isfnonabeliani(s acts faithfully on H and trivi*
*ally on
H0 = Z(H)), so |H| p3. Now, p3fi|GLm (2)| only if m 3r where r is the small*
*est
integer such that p|(2r - 1). This proves that rk(V ) 6 if p = 3 and rk(V ) *
* 9 if
p > 3. So we are done if |S| 8, or if |S| 16 and p > 3.
Let g 2 S be a generator if S ismcyclic, or a generator of an index 2 subgrou*
*p if S is
quaternion. Let m be such that g2 = s. Then
m
V % (g - Id).V % (g - Id)2.V % (g - Id)3.V % . .%.(g - Id)2 .V = [s, V ].
Thus (
1_|S| + 2 if S is cyclic
rk(V ) 2m + rk([s, V ]) 21
_4|S| + 2 if S is quaternion.
Using these inequalities and the other information given above, one now checks *
*that
for |S| = 2k, dim(V ) 2k in all cases except possibly when p = 3 and S ~=Q16.
In this case, S must act faithfully on H=Fr(H) (Lemma 1.1), which must have r*
*ank
at least 4 since Q16 is not a subgroup of GL2(3) (and the irreducible represent*
*ations
of a 2-group have dimension a power of 2). Since H is nonabelian, |H| 35. But*
* 35
does not divide |GL7(2)|, and so rk(V ) 8 in this case.
8 BOB OLIVER AND JOANA VENTURA
1.3___General_results_on_groups_
The following result is useful when listing subgroups of Out (P ), for a p-gr*
*oup P ,
which have a given Sylow p-subgroup. The most important case is that where Q =
Op(G), G = QH, and H0 2 Sylp(H); but we will also have other applications which
require the more general setting.
Proposition 1.7. Fix a prime p, a finite group G, and a normal abelian p-subgro*
*up
Q C G. Let H G be such that Q \ H = 1, and let H0 H be of index prime to p.
Consider the set
0 fi 0 0 0
H = H G fiH \ Q = 1, QH = QH, H0 H .
Then for each H02 H, there is g 2 CQ(H0) such that H0= gHg-1.
Proof.Fix H0 2 H, and define O: H ! Q by setting O(h) = h0h-1, where h0 is the
unique element in H0\hQ. Then O 2 Z1(H; Q) is a 1-cocycle (by direct computatio*
*n),
and O|H0 = 1. Since H1(H; Q) injects into H1(H0; Q) (Q is abelian and [H : H0]*
* is
prime to p), this means that O is a coboundary, and hence H0= gHg-1 for some g *
*2 Q.
Also, [g, H0] = 1 since [g, h] = (ghg-1)h-1 2 Q \ H0= 1 for each h 2 H0.
As an example of why Q must be assumed abelian in the above proposition, cons*
*ider
the group G = GL2(3) (and p = 2). Set
ff
Q = O2(G) = (-0110), (111-1)~=Q8.
Consider the subgroups
ff 0 ff ff
H = (1101), (100-1)~= 3, H = (-1011), (100-1), and H0 = (100-1).
Then H and H0 are both splittings of the surjection G ! G=Q ~= 3 which contain *
*H0
as Sylow 2-subgroup, but theyfarefnot conjugate in G. Instead, H0is G-conjugate*
* to the
subgroup H00= (1101), (-1001). The 1-cocycle H ! Q which sends the subgroup of
order three to I and its complement to -I is nontrivial in H1(H; Q8), but its r*
*estriction
is trivial in H1(H0; Q8).
The following very elementary lemma will be used later to list subgroups of a*
* given
2-group which are not normal, and have index two in their normalizers.
Lemma 1.8. Assume S is a 2-group with a normal subgroup S0 C S, such that S0 and
S=S0 are both elementary abelian. Set k = rk(S=S0), and assume k 2. Let P be *
*the
set of all subgroups P S such that P is not normal and |NS(P )=P | = 2. Fix P*
* S,
and set P0 = P \ S0, ` = rk(P S0=S0) = rk(P=P0), and m = rk(S0=P0). Then P 2 P
if and only if one of the following holds:
(a) m = 1, 0 ` < k: P0 [x, S0] for x 2 P S0; either P0 [xi, S0] for each
x 2 Sr P S0, or ` = 1, k = 2, P0 [S0, S], and P0 [S, S]. (2` classes if
P0 [S0, S] and 1 class if P0 [S0, S].)
(b) m = 2, ` 1: [x, P0] P0 for x 2 P S0r S0, [x, P0] P0 if x 2 Sr P S0; a*
*nd
P0 [x, S0] for some x 2 P S0r S0. (1 class if ` = 1 and 2 classes if ` = *
*2.)
(c) m = 3, 4, ` = k = 2: the action of S=S0 on S0=P0 ~=Cm2 has fixed subspace *
*of
rank one. Equivalently, S0=P0 is a free F2[S=S0]-module if m = 4, and has i*
*ndex
two in a free module if m = 3. (2 classes if m = 3 and 1 class if m = 4.)
In each case, the number of classes given is the number of conjugacy classes of*
* P
inducing fixed subgroups P0 S0 and P S0=S0 S=S0.
SATURATED FUSION SYSTEMS OVER 2-GROUPS 9
Proof.Since P 6C S, P0 S0, and ` < k if rk(S0=P0) = 1. Since NS0(P )=P0 is
the fixed subspace of the action of P=P0 on S0=P0, CS0=P0(P=P0) has rank one. S*
*ince
F2[P=P0] is injective as a module over itself, there is a P=P0-equivariant homo*
*morphism
from S0=P0 to F2[P=P0] which is an isomorphism on fixed sets, and which must be
injective since otherwise the kernel would have nontrivial fixed subspace. In p*
*articular,
m = rk(S0=P0) |P=P0| = 2`.
An element x 2 S normalizes P0 if and only if [x, P0] P0; and when m = 1, t*
*his
is the case if and only if [x, S0] P0. This explains most of the conditions l*
*isted for
x 2 Sr S0; it remains only to check the conditions which imply NS(P ) P S0 wh*
*en
` = 1, k = 2, and P0 C S. In this situation, P S0=P0 ~=C22(m = 1) or D8 (m = 2),
and has index 2 in S=P0. Also, P=P0 is a subgroup of order 2 and index 2 in its
S=P0-normalizer, and this is possible only when m = 1 and S=P0 ~=D8; thus when
[S, S] P0.
In all cases, the numbers of conjugacy classes of subgroups P with given P S0*
* and
P0 is equal to the order of H1(P=P0; S0=P0); except in case (a) when P0 [S0, *
*S], in
which case the two S0-conjugacy classes of such subgroups are conjugate in S.
2.Fusion systems
We first recall the definition of an (abstract) saturated fusion system. For *
*any group
G, and any x 2 G, cx denotes conjugation by x (cx(g) = xgx-1). For H, K G, we
write fi
Hom G (H, K) = ' 2 Hom (H, K) fi' = cx some x 2 G .
We also set AutG (H) = Hom G(H, H) ~=NG(H)=CG(H).
Definition 2.1 ([Pg ], [BLO2 , Definition 1.1]). A fusion system over a finite*
* p-group S
is a category F, with Ob (F) the set of all subgroups of S, which satisfies the*
* following
two properties for all P, Q S:
o Hom S(P, Q) Hom F(P, Q) Inj(P, Q); and
o each ' 2 Hom F(P, Q) is the composite of an isomorphism in F followed by an
inclusion.
When F is a fusion system over S, two subgroups P, Q S are said to be F-con*
*jugate
if they are isomorphic as objects of the category F. A subgroup P S is called*
* fully
centralized in F (fully normalized in F) if |CS(P )| |CS(P 0)| (|NS(P )| |N*
*S(P 0)|)
for all P 0 S which is F-conjugate to P .
Definition 2.2 ([Pg ], [BLO2 , Definition 1.2]). A fusion system F over a fini*
*te p-group
S is saturated if the following two conditions hold:
(I)(Sylow axiom) For all P S which is fully normalized in F, P is fully cen*
*tralized
in F and AutS(P ) 2 Sylp(Aut F(P )).
(II)(Extension axiom) If P S and ' 2 Hom F (P, S) are such that '(P ) is ful*
*ly
centralized, and if we set
N' = {g 2 NS(P ) | 'cg'-1 2 AutS('(P ))},
_ _
then there is ' 2 Hom F(N', S) such that '|P = '.
10 BOB OLIVER AND JOANA VENTURA
For any finite group G and any Sylow subgroup S 2 Sylp(G), the fusion system
of G (at p) is the category FS(G), whose objects are the subgroups of S, and wi*
*th
morphism sets Mor FS(G)(P, Q) = Hom G (P, Q). This is easily shown to be satur*
*ated
using the Sylow theorems (cf. [BLO2 , Proposition 1.3]). A saturated fusion sy*
*stem is
exotic if it is not the fusion system of any finite group.
The following definitions play a central role in this paper.
Definition 2.3. Fix a prime p, a p-group S, and a saturated fusion system F ove*
*r S.
Let P S be any subgroup.
o P is F-centric if CS(P 0) = Z(P 0) for all P 0which is F-conjugate to P .
o P is F-radical if Op(Out F(P )) = 1; i.e., if OutF (P ) contains no nontrivi*
*al normal
p-subgroup.
o P is F-essential if P is F-centric and fully normalized in F, and OutF (P ) *
*contains
a strongly embedded subgroup at p.
o P is_central in F if every morphism_' 2 Hom F(Q, R) in F extends to a morphi*
*sm
' 2 Hom F(P Q, P R) such that '|P = IdP.
o P is_normal in F if every morphism_' 2 Hom F(Q, R) in F extends to a morphism
' 2 Hom F(P Q, P R) such that '(P ) = P .
o The fusion system F is nonconstrained if there is no subgroup P S which is
F-centric and normal in F.
o For any ' 2 Aut(S), 'F'-1 denotes the fusion system over S defined by
Hom 'F'-1(P, Q) = '.Hom F('-1(P ), '-1(Q)).'-1
for all P, Q S.
When F = FS(G) for a finite group G with S 2 Sylp(G), then P S is F-centric
if and only if it is p-centric in G: that is, Z(P ) 2 Sylp(CG(P )), or equival*
*ently,
CG(P ) = Z(P ) x C0G(P ) for some (unique) subgroup C0G(P ) of order prime to p*
*. The
subgroup P is F-essential if and only if it is p-centric in G, NS(P ) 2 Sylp(NG*
*(P )), and
NG(P )=(P .CG(P )) has a strongly embedded subgroup.
We say that a fusion system is "centerfree" if it contains no nontrivial cent*
*ral sub-
group. Our main goal in this paper is to develop techniques for listing, for a*
* given
2-groups S, all centerfree nonconstrained saturated fusion systems over S (up t*
*o iso-
morphism). This restriction is motivated largely by the two results stated in *
*the
following proposition: they imply that any minimal exotic fusion system is cent*
*erfree
and nonconstrained.
Theorem 2.4. Let F be a saturated fusion system over a finite p-group S.
(a) If F is constrained, then there is a unique p0-reduced p-constrained finite*
* group
G _ a unique G such that Op0(G) = 1 and CG(Op(G)) Op(G) _ such that
F ~=FS(G).
(b) If A C S is central in F, then F is exotic if and only if F=A is exotic. H*
*ere,
F=A is the fusion system over S=A such that for all P, Q S containing A,
Hom F=A(P=A, Q=A) is the image of Hom F(P, Q) under projection.
Proof.See [BCGLO1 , Proposition C] and [BCGLO2 , Corollary 6.14]. In both *
*cases,
much more precise results are shown. In (a), one can choose G with normal p-
subgroup Q such that Q ~=Op(F) (the maximal normal p-subgroup of F) and G=Q ~=
SATURATED FUSION SYSTEMS OVER 2-GROUPS 11
AutF (Op(F)). Under the hypotheses of (b), if F=A is the fusion system of a fi*
*nite
group G, then F is the fusion system of a central extension of G by A.
One of the key problems when constructing fusion systems over a p-group S is *
*to
determine which subgroups of S can contribute automorphisms; i.e., for which P *
* S
the group AutF (P ) need not be generated by restrictions of automorphisms of l*
*arger
subgroups. This is what motivates the definition of F-essential subgroups. The *
*follow-
ing result was shown by Puig [Pg , Theorem 5.8] and Linckelmann [Li, Theorem 1.*
*11],
and was originally pointed out to us by Grodal.
Proposition 2.5. Let F be a saturated fusion system over a p-group S, and let P*
* S
be an F-centric subgroup which is fully normalized in F. Then P is F-essential *
*if and
only if Aut F(P ) is not generated by restrictions of morphisms between strictl*
*y larger
subgroups of S.
Proof.Write S0 = OutS(P ) for short. Thus S0 ~=NS(P )=P since P is F-centric, a*
*nd
S0 6= 1 since P S. We first claim that
G0 = <' 2 G | 'S0'-1 \ S0 6=>1. (1)
To see this, fix ' such that 'S0'-1 \ S0 6= 1, and consider the groups
N' def={g 2 NS(P ) | 'cg'-1 2 S0} and H def=OutN'(P ).
Then 'H'-1 = 'S0'-1 \ S0 6= 1, so N' P . By condition (II), ' extends to a
morphism in Hom F (N', S), and this proves that ' 2 G0. Conversely, if ' 2 G =
OutF (P ) extends to Q P , then 'Out Q(P )'-1 OutS(P ), and so
'S0'-1 \ S0 'Out Q(P )'-1 6= 1.
This proves (1). Hence by Lemma 1.4, G contains a strongly embedded subgroup *
*at
p (equivalently, P is F-essential) if and only if G0 G.
As a corollary, we get Alperin's fusion theorem stated for restriction to ess*
*ential
subgroups. Roughly, it says that every saturated fusion system is generated by *
*auto-
morphisms of S and of essential subgroups, and their restrictions.
Corollary 2.6. Fix a saturated fusion system F over a p-group S. Then for each
P, P 0 S and each ' 2 IsoF(P, P 0), there are subgroups P = P0, P1, . .,.Pk = *
*P 0,
subgroups Qi (i = 1, . .,.k) which are F-essential or equal to S, *
*and
automorphisms 'i 2 AutF (Qi), such that 'i(Pi-1) = Pi for all i and ' = ('k|Pk-*
*1) O
. .O.('1|P0).
Proof.By Alperin's fusion theorem in the form shown in [BLO2 , Theorem A.10], *
*this
holds if we allow the Qi to be any F-centric F-radical subgroups of S which are
fully normalized in F. So the corollary follows immediately from that together *
*with
Proposition 2.5.
3. Semicritical and critical subgroups
The following definition gives necessary conditions for subgroups of a p-grou*
*p to
possibly be F-radical or F-essential in some fusion system.
Definition 3.1. Let S be a finite p-group. A subgroup P S will be called semi*
*critical
if the following two conditions hold:
12 BOB OLIVER AND JOANA VENTURA
(a) P is centric in S; and
(b) OutS (P ) \ Op(Out (P )) = 1.
A subgroup P S will be called critical if it is semicritical, and if
(c) there are subgroups
OutS(P ) G0 G Out(P )
such that G0 is strongly embedded in G at p and OutS (P ) 2 Sylp(G).
The importance of (semi)critical subgroups lies in the following proposition.
Proposition 3.2. Fix a p-group S, a saturated fusion system F over S, and a sub*
*group
P S. If P is F-centric and F-radical, then it is a semicritical subgroup of S*
*. If P
is F-essential, then P is a critical subgroup of S.
Proof.Set G = OutF (P ). If P is F-centric and F-radical, then
OutS(P ) \ Op(Out (P )) G \ Op(Out (P )) Op(G) = 1,
and so P is a semicritical subgroup of S.
If P is F-essential, then by definition, P is F-centric (hence centric in S),*
* fully
normalized in F, and G def=OutF(P ) contains a strongly embedded subgroup G0 G
at p. Since any strongly embedded subgroup at p contains a Sylow p-subgroup, we*
* can
assume (after replacing G0 by a conjugate subgroup if necessary) that G0 OutS*
*(P ) 2
Sylp(G). Since Op(G) gG0g-1 \ G0 for all g 2 G, this shows that Op(G) = 1, he*
*nce
that P is F-radical, and thus a semicritical subgroup of S. This proves that P*
* is
critical in S.
The following lemma is an easy consequence of Proposition 1.1, and is given h*
*ere
because it will be used frequently to prove that certain subgroups are not (sem*
*i)critical.
It will often be applied with P the characteristic subgroup Z2(P ) (the sub*
*group
such that Z2(P )=Z(P ) = Z(P=Z(P ))).
Lemma 3.3. Fix a prime p, a p-group S, a subgroup P S, and a subgroup P
characteristic in P . Assume there is g 2 NS(P )r P such that
(a) [g, P ] .Fr(P ), and
(b) [g, ] Fr(P ).
Then cg 2 Op(Aut (P )), and hence P is not semicritical in S.
Proof.Point (a) implies that cg is the identity on P= .Fr(P ), and (b) implies *
*it is the
identity on .Fr(P )=Fr(P ). Hence cg 2 Op(Aut (P )) by Proposition 1.1, and so*
* P is
not semicritical in S.
For the above definition to be useful, simple criteria are needed which imply*
* that most
subgroups are not critical. This works best when p = 2. The following propositi*
*on is a
start at doing this. For example, point (a) implies that P is not critical in S*
* if OutS(P )
contains a subgroup isomorphic to D8 _ since D8 contains noncentral involutions.
Recall that when V is a vector space and ff is a linear automorphism of V , w*
*e write
[ff, V ] = Im[V -ff-Id---!V ].
Proposition 3.4. Fix a critical subgroup P of a 2-group S, and set S0 = NS(P )=*
*P ~=
OutS(P ). Then the following hold.
SATURATED FUSION SYSTEMS OVER 2-GROUPS 13
(a) Either S0 is cyclic, or Z(S0) = {g 2 S0| g2 = 1}. If rk(Z(S0)) > 1, then |S*
*0| =
|Z(S0)|m for m = 1, 2, or 3.
(b) All involutions in S0 are conjugate in Out (P ), and hence in Aut (P=Fr(P )*
*). In
fact, there is a subgroup R Out(P ) (or R Aut(P=Fr(P ))) of odd order, *
*which
normalizes S0 and permutes its involutions transitively.
(c) Set |S0| = 2k. Then rk(P=Fr(P )) 2k. If k 2, then rk([s, P=Fr(P )]) 2*
* for
all 1 6= s 2 S0.
(d) Assume Z(S0) ~=Cn2with n 2, and fix 1 6= s 2 Z(S0). Then rk([s, P=Fr(P )])
n.
Proof.Fix subgroups
OutS(P ) G0 G Out(P )=O2(Out (P ))
such that G0 is strongly embedded in G and Out S(P ) 2 Syl2(G). By Proposition
1.1, Out (P )=O2(Out (P )) injects into Aut (P=Fr(P )), so we can also consider*
* G as a
subgroup of this group.
By Bender's theorem ([Be ] or Theorem 1.5), either S0 is cyclic or quaternion*
*, or
0 n n
O2 (G=O20(G)) is isomorphic to one of the simple groups P SL2(2 ), P SU3(2 ), or
Sz(2n) (where n 2, and n is odd in the last case).
(a) Condition (a) is clear if S0 is cyclic or quaternion. If not, let L be the *
*simple group
0 n n n
L = O2 (G=O20(G)). If L ~=P SL2(2 ), then S0 ~=C2. If L ~=Sz(q), where q = 2 f*
*or
odd n 3, then by Suzuki's description of the Sylow 2-subgroups [Sz, x4, Lemma*
* 1]
(see also [Sz, x9]), |S0| = q2, Z(S0) ~=Cn2, and all involutions in S0 are in Z*
*(S0). So
(a) holds in both of these cases.
If L ~=P SU3(q), where q = 2n for odd n 3, then we can identify
*
* i1 srj
S0 = {V (r, s) | r, s 2 Fq2| r + ~r= s~s} where ~r= rq and V (r, s) =*
* 001~s01.
Also, V (r, s).V (u, v) = V (r + u + s~v, s + v). Thus |S0| = 23n, Z(S0) = {V (*
*r, 0) | r 2
Fq} ~=Cn2, and V (r, s)2 = V (s~s, 0) = 1 only if s = 0. Thus also in this case*
*, S0 satisfies
(a).
(b) By [Sz2, Lemma 6.4.4], all involutions in G0 are conjugate to each other. *
*Since
the involutions in S0 are all central, they must be conjugate to each other by *
*elements
in NG0(S0); and we can write NG0(S0) = S0o R where |R| is odd.
(c,d) These follow immediately from Lemma 1.6, applied with V = P=Fr(P ).
We can now outline the general procedure which will be used to determine all *
*of
the critical subgroups of a given 2-group S. We first try to find a normal sub*
*group
S0 C S, as large as possible, which we can show is contained in all critical su*
*bgroups.
For example, in many cases, we do this for S0 = Z2(S) (i.e., the subgroup such *
*that
Z2(S)=Z(S) = Z(S=Z(S))). We then search for critical subgroups P S such that
|NS(P )=P | = 2, by first applying Lemma 1.8 (when possible) to list all subgro*
*ups
of index 2 in their normalizer, and then applying Lemma 3.3 to eliminate most of
them. Afterwards, we search for subgroups P S such that |NS(P )=P | = 2k 4,
rk(P=Fr(P )) 2k, and rk([s, P=Fr(P )]) 2 for all s 2 NS(P )r P , and check *
*(using
Proposition 3.4) which of them could be critical. In practice, this seems to w*
*ork
surprisingly well on groups of order 210, at least on those where we have tes*
*ted it.
14 BOB OLIVER AND JOANA VENTURA
4. Fusion systems over the Sylow 2-subgroup of J2 and J3
We are now ready to begin working with some concrete examples. In the next fo*
*ur
sections, we list all nonconstrained centerfree saturated fusion systems over e*
*ach of
four different 2-groups S. In each case, this procedure can be broken up into *
*three
steps: first determine the critical subgroups of S (or at least a list of subgr*
*oups which
includes all critical subgroups), then determine the automorphism group of each*
* critical
subgroup, and finally work out all possible combinations of which critical subg*
*roups
can be F-essential for any given F and what their F-automorphism groups can be.
The last step is carried out only up to isomorphism, in the sense that we make *
*a list
of fusion systems over S and show that for each F, there is some ' 2 Aut (S) su*
*ch
that 'F'-1 is on the list (see Definition 2.3). If we did find a candidate for *
*an exotic
fusion system, then there would be the additional step of proving that it is sa*
*turated,
but otherwise this is done by identifying it with the fusion system of some fin*
*ite group.
In this section and the next, S0 = UT3(4) denotes the group of 3x3 upper tria*
*ngular
matrices over F4 with 1's in all diagonal entries. Let exij2 S0 (for i < j) be*
* the
elementary matrix with entry x 2 F4 in the (i, j) position, and set Eij= {exij|*
* x 2 F4}.
Thus, for example,
Z(S0) = [S0, S0] = E13= {ex13| x 2 F4}.
We also let cxijdenote conjugation by exij, as an automorphism of S0 and also a*
*s a
homomorphism between subgroups of S0 or groups containing S0, and write =
{cxij| x 2 F4}.
_ 2 __
Let x 7! x = x denote the field automorphism on F4, and let M 7! M denote t*
*he
induced field automorphism on S0. Let o 2 Aut(S0) be the automorphism "transpose
inverse" which sends eaijto ea4-i,4-j. Consider the semidirect product
SOE`= UT3(4) o `,
__ __
where for M 2 S0 = UT3(4), OEMOE-1 = M and `M`-1 = o(M ). Thus SOE`is a Sylow
2-subgroup of the full automorphism group Aut(P SL3(4)) = P GL3(4) o`. In *
*this
section, we determine the nonconstrained saturated fusion systems over the group
S` def=UT3(4) o <`>,
while in the next section we work with the group SOEdef=UT3(4) o .
Let ! denote an element in F4r F2, so that F4 = {0, 1, !, !2}. The following*
* sub-
groups will play an important role throughout this section:
ni 1a bjfi o ni 1a bj fi o
A1 = = 010001fifia, b 2 F4 Q0 = 01~a001fifia, b 2 F4
ni 10a jfi o
A2 = = 01b001fifia, b 2 F4 Q = .
Thus A1 and A2 are the "rectangular subgroups", both isomorphic to C42; while Q*
*0 ~=
C2 x Q8 and Q ~=Q8 xC2 D8.
We start with some elementary facts about S` and its subgroups.
Lemma 4.1. (a) For each g 2 S`r S0, CS0(g) Q0, CS0(g) \ E13 = , and
|CS0(g)| 4.
(b) All involutions in S0 are in A1 or in A2, while all involutions in S`r S0 a*
*re in Q.
SATURATED FUSION SYSTEMS OVER 2-GROUPS 15
(c) A1 and A2 are the only subgroups of S` isomorphic to C42.
(d) The subgroups S0 and Q are both characteristic in S`.
Proof.(a) Fix g 2 S`r S0, and write g = M` where M 2 S0. Then
__
CS0(M`) = {X 2 S0| Mo(X )M-1 = X}.
Thus e1132 CS0(MOE)_and e!13=2CS0(MOE). Also, for X 2 CS0(MOE), X is_S0_= UT3(4*
*)-
conjugate to o(X ), and since S0=E13is abelian, this implies that o(X ) X (mo*
*d E13).
Since ` acts on S0=E13 ~=E12x E23 fixing Q0=E13, this shows that X 2 Q0. Finall*
*y,
for X, Y 2 Q0,
[g, XY ] = [g, X]. X[g, Y ]X-1 ,
so X 7! [g, X] defines a homomorphism from Q0 to E13, and thus its kernel CS0(g*
*) has
order at least 4.
i 1abj 2 i1 0acj
(b) The first statement holds since 01c001= 001001. If M` has order 2 for s*
*ome
__
M 2 UT3(4), then oMi o-1j= M-1, so M is invariant under conjugate transpose and
1ab
hence has the form 01~a001for some a 2 F4 and b 2 F2. Thus M 2 Q0.
(c) Assume A S` is isomorphic to C42. If A S0, then A (A1 [ A2) by (b), *
*and
A = A1 or A2 since no element of A1r E13commutes with any element of A2r E13. So
assume A S0, and fix g 2 Ar S0. By (a), A \ S0 CS0(g) Q0 ~=C2x Q8; and th*
*is
is impossible since A \ S0 ~=C32.
(d) The subgroup S0 = is characteristic by (c). By (b), Q is generate*
*d by
the involutions in S`r S0, and so it is also characteristic.
4.1___Candidates_for_critical_subgroups_
The following proposition is the main result of this subsection.
Proposition 4.2. If P is a critical subgroup of S`, then P is one of the subgro*
*ups Q,
S0 = UT3(4), A1, or A2.
Proposition 4.2 follows immediately from Lemmas 4.3 and 4.5. We first deal w*
*ith
the normal critical subgroups.
Lemma 4.3. If P C S` is a normal critical subgroup of S`, then P = Q or P = S0 =
UT3(4).
Proof.By Proposition 3.4(c), rk(P=Fr(P )) 2k if |S`=P | = 2k. Thus |S`=P | *
*4.
Assume first that |S`=P | = 4. Then S`=P is abelian, so P [S`, S`] = Q0, a*
*nd
|P=Q0| = 2. Also, |P | = 25 and rk(P=Fr(P )) 4, so |Fr(P )| 2. Thus, if P i*
*s critical,
Fr(P ) = Fr(Q0) = .
If P S0, then P contains some element of E12r1, and hence Fr(P ) E13 which
we saw to be impossible. This leaves only the possibility P = for so*
*me
X 2 S0; and since S0 = Q0E12, we can assume X = ea12for some a 2 F4. Also,
(X`)2 = [ea12, `] 2 Fr(P ) = , and this is possible only if a = 0. So th*
*e only
possibility is P = = Q.
Now assume |S`=P | = 2, and fix g 2 S`r P . Since S`=Fr(S`) ~=C32, there are *
*seven
subgroups of index 2 in S`. Assume P 6= S0. Then P = for some*
* a, b 2 F4
16 BOB OLIVER AND JOANA VENTURA
where a 6= 0. Also, [Q0, ea12] = E13. If b =2{0, a}, then
Fr(P ) = `]= Q0 = Fr(S`),
so [g, P ] Fr(S`) for g 2 S`r P , and P is not (semi)critical by Lemma 3.3 (a*
*pplied
with = 1).
We are left with the case b 2 {0, a}, and thus P = `. Then Fr(P *
*) =
`]~=C2 x C4, and so E13 is characteristic in P since it is the 2-t*
*orsion
subgroup of Fr(P ). Thus CP(E13) = P0 def=P \S0 is characteristic in P . For g *
*2 S0r P ,
[g, P ] P0 and [g, P0] E13 Fr(P ); and thus P is not (semi)critical by Lem*
*ma 3.3
applied with = P0.
Lemma 4.4. Each critical subgroup of S` contains E13.
Proof.Assume P is critical in S`. Then P is centric in S`, so Z(S`) = *
*P . It
remains to show that e!132 P .
Assume otherwise. Then e!132 N(P )r P , and so by Lemma 3.3, e!13acts nontriv*
*ialy
(by conjugation) on P=Fr(P ); i.e., [e!13, P ] Fr(P ). Since [e!13, S`] = , this implies
e11362 Fr(P ). Furthermore, if we let denote the 2-torsion subgroup of Z(P )*
*, then
e1132 , and thus [e!13, P ] . Since P is critical, Lemma 3.3 now implies t*
*hat
[e!13, ] Fr(P ).
Thus there is some h 2 Z(P ) such that h2 = 1 and [e!13, h] 6= 1. Since e!132*
* Z(S0), we
have h 2 S`r S0. Also, since h 2 Z(P ), P0 def=P \ S0 CS0(h), and by Lemma 4.*
*1(a),
P0 Q0. Since e113=2Fr(P ) (and e113is the square of each element in Q0r E13),*
* this in
turn implies P0 E13. Hence P0 = , P = h~=C22, CS`(P ) = has
order 8 by Lemma 4.1(a), and this contradicts the assumption that P is centri*
*c.
It remains to show the following:
Lemma 4.5. If P S` is a critical subgroup and not normal, then P = A1 or P = *
*A2.
Proof.Fix such a P . Assume first that |N(P )=P | 4. Since S` has order 27, |*
*N(P )|
26, and so |P | 24. Since P is critical, we must have rk(P=F r(P )) 4 by Pr*
*oposition
3.4(c). This can only happen if P ~=C42, and by Lemma 4.1(c), P = A1 or P = A2.
Now assume |N(P )=P | = 2. Then Q0 = Fr(S`) P because P not normal in S`,
and E13 P by Lemma 4.4. Also, Q0 N(P ) since Q0=E13 = Z(S`=E13). Since
|N(P )=P | = 2, this implies that N(P ) = Q0P , and that |Q0 \ P | = 23.
_
Fix x = ea12ea232 (Q0\P )r E13. By symmetry with respect to the field automor*
*phism
__ _ def _
M 7! M , we can assume that a = 1 or a = !. Thus g = e!12e!23is not in P , and *
*hence
g generates N(P )=P .
Assume first that P S0. Then [P, S0] E13 P , and S0 N(P ). Thus N(P ) *
*= _
S0 (since P is not normal in S`), and [S0 : P ] = 2. So P = ea12ea23for
some Yi 2 . Then [g, P ] E13 = Fr(P ), and so P is not (semi)critical by L*
*emma
3.3 applied with = 1.
Now assume P S0, and set P0 = P \ S0. By Lemma 1.8 (applied to the group
S`=E13 ~=C42o C2), P0 has index 4 in S0. Since P0 is normalized by any element *
*of
P rP0 (and P0 6= Q0), this means that
_ _
P0 =
SATURATED FUSION SYSTEMS OVER 2-GROUPS 17
_
for some a, b 2 F4 such that b 6= a. Furthermore, S`=P0 ~=D8, and P=P0 is a sub*
*group
of order 2 not contained in S0=P0. There are only two such subgroups, and they *
*are
conjuate in D8. We can thus assume that
_ _
P = `= .
For any g 2 NS0(P )r P , [g, P ] P0 and [g, P0] [S0, S0] = E13 Fr(P ). So *
*if P0 is a
characteristic subgroup of P , then by Lemma 3.3, P cannot be critical.
_ _ a b _b _a
If ab 6= 0, then aa = bb = 1, so [e12e23, e12e23] = 1, and hence P0 is abelia*
*n. For
all g 2 P rP0, CP0(g).E13 P0 \ Q0 P0 (Lemma 4.1(a)) and e!13=2CP0(g); thus
|CP0(g)| 4, and g is not contained in any abelian subgroup of P of order 16. *
*So P0 is
the unique abelian subgroup of index 2, thus characteristic in P , and P is not*
* critical.
_
If ab = 0, then P =_`for some 0 6= a 2 F4. Then Z(P ) = ,
Z2(P ) = , E13 is the 2-torsion subgroup in Z2(P ) (hence charac*
*teristic in
P ), and so P0 = CP(E13) is characteristic in P . Hence again in this case, P *
*is not
critical.
4.2___Automorphisms_of_critical_subgroups___
Before describing the automorphism group of S0, we need to give names to some
automorphisms. For each f 2 Hom F2(F4, F4), define aef1, aef22 Aut(S0) by setti*
*ng aei|Ai =
Id, and
aef1(ex23) = ex23ef(x)13 and aef2(ex12) = ex12ef(x)13.
0 f+f0 f
Note that aefiO aefi= aei , and hence Ri = {aei | f 2 Hom F2(F4, F4)} is a su*
*bgroup of
Aut(S0) isomorphic to C42. One easily sees that R1 and R2 commute in Aut(S0), a*
*nd
that they generate the group of all automorphisms of S0 which induce the identi*
*ty on
E13 and on S0=E13. Thus R1 x R2 is a normal subgroup of Aut(S0), and is contain*
*ed
in O2(Aut (S0)).
Next define fl0, fl1 2 Aut(S0) by letting fl0 be conjugation by diag(!, 1, ~!*
*), and letting
fl1 be conjugation by diag(!, 1, !). Then fl0 and fl1 both have order 3,
0 def= ~= 3, 1 def=o~= 3,
and [ 0, 1] = 1 in Aut(S0).
Lemma 4.6. (a) Aut(S0) = (R1 x R2).( 0 x 1) ~=C82o ( 3 x 3), and hence
Out(S0) = (R1=) x (R2=).( 0 x 1) ~=C42o ( 3 x 3).
(b) If ' 2 Aut (A1) commutes with all elements of , then ' = j|A1 for some
j 2 R2. If ' also commutes with fl0|A1, then ' = Id.
Proof.The elements we have defined clearly define subgroups of Aut(S0) and Out(*
*S0);
it remains to show that these are the only automorphisms.
Set Out 0(S0) = Out (S0, A1). Since A1 and A2 are the only subgroups of S` i*
*so-
morphic to C42, an automorphism of S0 either preserves each Ai or switches them*
*. So
Out(S0) = Out0(S0) o . By Lemma 1.2, there is a short exact sequence
1 ---! H1(; A1) -----! Out 0(S0) --Res---!NGL4(2)()= ---!(*
*1.1)
18 BOB OLIVER AND JOANA VENTURA
Fix the ordered basis {e113, e!13, e112, e!12} for A1 ~=C42as a vector space *
*over F2. With
respect to this basis, conjugation by e123and e!23take the form
` ' ` '
I I ! I Z
c123= 0 I and c23= 0 I ,
where matrices are written in 2 x 2 blocks, and Z = (0111). From this, one sees*
* that
ae` ' fi oe
I X fi
CGL4(2)() = 0 I fifiX 2 M2x2(F2) . (2)
Also,
ae` ' fi oe
A X fi -1
NGL4(2)() = 0 B fifiX 2 M2x2(F2), A, B 2 GL2(F2), A B 2 .
Thus, the elements in NGL4(2)() are all represented by automorphisms of S*
*0 in the
subgroup generated by R2, fl0, fl1, and cOE.
Consider the spectral sequence
Eij2= Hi(; Hj(; A1)) =) Hi+j(; A1).
Since c123acts freely on the basis {e112, e112e113, ex12, ex12ex13} of A1 over *
*F2, the spectral
sequence collapses, and leaves us with an isomorphism
H1(; A1) ~=H1(; H0(; A1)) = H1(; E13) ~=(Z=2)2
(where the action of c!23on E13 in the last cohomology group is trivial). Sinc*
*e this
represents the image in Out(S0) of those automorphisms which restrict to the id*
*entity
on A1, we see that all such automorphisms are represented by elements of R1.
The description of Out (S0) (and hence of Aut(S0)) now follows upon comparing*
* it
with the short exact sequence (1).
The first statement in (b) follows from (2). Also, since fl0|A1 is conjugati*
*on by
diag(!, 1, !-1), it is represented by the matrix Z-100Z, and centralizes (IX0I*
*)only for
X = 0. This finishes the proof of (b).
Now set
= c`~=C3 x 3.
Also, let f`l12 Aut (S`) be such that f`l1|S0 = fl1 (conjugation by diag(!, 1, *
*!)) and
`fl1(`) = `.
Lemma 4.7. Let F be any saturated fusion system over S`.
(a) Out(S`)=O2(Out (S`)) ~= 3, and hence OutF (S`) has order 1 or 3.
(b) If S0 is F-essential, then there is some ' 2 Aut(S`) such that either
o Out'F'-1(S0) = 0 ~= 3 and Out'F'-1(S`) = 1; or
o Out'F'-1(S0) = ~=C3 x 3 and Out'F'-1(S`) = .
Proof.By Lemma 4.1(d), S0 is characteristic in S`. Also, Z(S0) = E13 is free a*
*s a
F2[<`>]-module, so Hi(<`>; Z(S0)) = 0 for i > 0. Hence by Lemma 1.2, the restri*
*ction
map
Out(S`) ---Res--!~NOut(S0)(Out S`(S0))=Out S`(S0) = COut(S0)()=(*
*3)
=
is an isomorphism. By Lemma 4.6(a),
COut(S0)()=O2(COut(S0)()) ~= 3
SATURATED FUSION SYSTEMS OVER 2-GROUPS 19
(represented by 1); and this finishes the proof of (a).
Now assume S0 is F-essential. Then OutF (S0) is isomorphic to 3 or C3 x 3, *
*and
is generated by 3-torsion and c`. By Lemma 1.7, there is some '0 2 O2(Aut (S0))*
* such
that ['0, c`] = 1 in Out(S0) and '0Out F(S0)'-10is equal to 0 or . By (3), '0*
* extends
to some ' 2 Aut(S`), and thus Out'F'-1(S0) is equal to 0 or .
If Out 'F'-1(S0) = , then by the extension axiom, fl1 extends to an element *
*of
Aut'F'-1(S`), and so Out 'F'-1(S`) = since this extension is unique in O*
*ut (S`).
Conversely, if Out'F'-1(S`) has order 3, then a generator of this group restric*
*ts to an
automorphism of S0 of order 3 which commutes with c` (since S0 is characteristi*
*c in
S`); and thus Out'F'-1(S0) = .
It remains to check the possibilities for AutF (Ai) when the Ai are essential.
Lemma 4.8. Let F be any saturated fusion system over S`, and assume that A1 and*
* A2
are F-essential. Then S0 is also F-essential. There is an automorphism ' 2 Aut(*
*S`)
such that either
o Out'F'-1(Ai) = SL2(4), Out'F'-1(S0) = 0 ~= 3 and OutF (S`) = 1; or
o Out'F'-1(Ai) = GL2(4), Out'F'-1(S0) = ~=C3 x 3 and Out'F'-1(S`) = .
Here, SL2(4) and GL2(4) are the subgroups of Aut (Ai) defined with respect to t*
*he
natural F4-vector space structures on A1 and A2.
Proof.Set = Out F(A1). Thus is a subgroup of Aut(A1) ~=GL4(2) ~=A8 which
has Aut S`(A1) ~= C22as Sylow 2-subgroup, and which contains a strongly embed-
0
ded subgroup. By Bender's theorem (Theorem 1.5), O2 ( =O20( )) is isomorphic to
SL2(4) ~=A5. The only nontrivial odd order subgroup of GL4(2) which has A5 in i*
*ts
normalizer is C3, with normalizer GL2(4) o ~=(C3x A5) oC2. If H GL4(2) ~=*
*A8
and H ~=A5, then since the only proper subgroups of A5 of index 8 have index *
*5 and
6, each orbit of H acting on {1, . .,.8} has length 1, 5, or 6. Thus H is in on*
*e of two con-
jugacy classes: either it acts as SL2(4) for some F4-vector space structure, or*
* it acts via
the permutation action on F52=diag. Since the fixed set of AutS`(A1) = a*
*cting on
A1 is 2-dimensional, this last action cannot occur. Thus must be Aut(A1)-conj*
*ugate
to SL2(4) or GL2(4).
By the extension axiom, all elements in N () extend to elements of Aut*
*F (S0);
and conversely all elements of Aut F(S0) leave A1 invariant and hence restrict *
*to el-
ements of . Thus ~= SL2(4) implies Aut F(S0) ~= 3 and ~= GL2(4) implies
AutF (S0) ~=C3x 3. By Lemma 4.7, we can assume (after replacing F by 'F'-1 for
some appropriate ' 2 Aut(S`)) that for some _ 2 Aut(A1), either AutF (S0) = 0 *
*and
= AutF (A1) = _SL2(4)_-1, or AutF (S0) = and = AutF (A1) = _GL2(4)_-1.
Now, [ , ] ~=A5 and SL2(4) both contain the subgroup ~=A4. A*
*lso,
any isomorphism between two subgroups of SL2(4) isomorphic to A4 is conjugation*
* by
some element of SL2(4) o ~= 5. Hence upon composing _ by such an element, *
*we
can arrange that _ centralizes c*23and fl0|A1. Then _ = IdA1 by Lemma 4.6(b), *
*and
thus AutF (A1) is equal to SL2(4) or GL2(4). Finally, the same holds for AutF *
*(A2),
either by repeating the same argument, or because AutF (A2) = c`Aut F(A1)c-1`.
4.3___Fusion_systems_over_S`_
Theorem 4.9. Let F be any nonconstrained saturated fusion system over the group
S` = UT3(4) o <`>, where ` acts on UT3(4) P GL3(4) by sending a matrix M to
20 BOB OLIVER AND JOANA VENTURA
__
o(M ). Then F is isomorphic to the fusion system of one of the groups P SL3(4) *
*o <`>,
P GL3(4) o <`>, J2, or J3.
Proof.Since Q ~=D8xC2Q8, and Q=Z(Q) ~=C42contains exactly five involutions which
lift to elements of order 2 in Q, Inn(Q) is the group of automorphisms which in*
*duce
the identity on Q=Z(Q), and Out (Q) ~= 5 is the group which permutes those five
involutions. Hence if Q is F-essential, then OutF (Q) = A5: this is the only su*
*bgroup
which contains Out S`(Q) as Sylow 2-subgroup and which has a strongly embedded
subgroup.
By Lemma 4.1(d), Q and S0 are both characteristic subgroups of S`. If Q were *
*the
only F-essential subgroup, then all morphisms in F would be composites of restr*
*ictions
of automorphisms of Q and S`, and hence Q would be normal in F. Similarly, if
S0 were the only F-essential subgroup, then it would be normal in F. Since F is
nonconstrained, neither Q nor S0 can be the unique F-essential subgroup.
By Lemma 4.8, if the Ai are F-essential, then so is S0. Upon putting all of *
*this
together, we see that either Q is not essential and S0 and the Ai are; or Q and*
* S0 are
essential and the Ai are not; or all of these subgroups are essential.
Case 1: Assume first that Q is not F-essential, and hence that S0 and the Ai*
* are
F-essential. Let F1 and F2 be the fusion systems over S` generated by the follo*
*wing
automorphism groups and their restrictions:
OutF1(S`)= 1 Out F1(S0)= 0 ~= 3 Out F1(Ai)= SL2(4)
OutF2(S`)= Out F2(S0)= ~=C3 x 3 Out F2(Ai)= GL2(4) .
Here, the groups SL2(4) and GL2(4) are defined with respect to the natural F4-v*
*ector
space structures on A1 and A2.
By Lemma 4.8, we can assume (after replacing F by 'F'-1 for appropriate ') th*
*at
either Out F(Ai) = SL2(4) (for i = 1, 2) and Out F(S0) = 0 ~= 3, or Out F(Ai) =
GL2(4) and Out F(S0) = ~= C3 x 3. Furthermore, by Lemma 4.7, Out F(S`) is
determined (exactly) by OutF (S0). Since F is generated by automorphisms of S`,*
* S0,
and the Ai and their restrictions, this proves that F = F1 or F = F2.
The fusion systems of P SL3(4)o <`> and P GL3(4)o <`> clearly fit these descr*
*iptions.
Note in particular that Q is not G-essential when G is one of these groups: all*
* elements
of OutG (Q) must leave Q \ S0 ~=C2 x Q8 invariant, and hence this cannot be the*
* full
group A5. Thus FS`(P SL3(4)o <`>) ~=F1 and FS`(P GL3(4)o <`>) ~=F2.
Case 2: Now assume Q and S0 are both F-essential. Let F3 and F4 be the fusion
systems over S` generated by the following automorphism groups and their restri*
*ctions:
OutF3(S`)= Out F3(S0)= Out F3(Q)= A5
OutF4(S`)= Out F4(S0)= Out F4(Q)= A5 OutF4(Ai)= GL2(4) .
For F = F3 or F4, all involutions in E13, and all involutions in S0r E13, are F*
*-conjugate
via automorphisms of S0; while all noncentral involutions in Q are F-conjugate *
*via
automorphisms of F. Since this includes all involutions in S` (Lemma 4.1(b)), w*
*e see
that S` contains two F3-classes of involutions and one F4-class.
For arbitrary F of this type, OutF (Q) = A5 has index 2 in Out(Q), and so Aut*
*F (Q)
contains all automorphisms of Q of odd order. Since `fl1|Q has order 3 in Aut *
*F(Q),
it must extend (by the extension axiom) to some automorphism in Aut F(S`). Thus
OutF (S`) has order 3 by Lemma 4.7. By Lemma 4.7 or 4.8, we can assume (after
SATURATED FUSION SYSTEMS OVER 2-GROUPS 21
replacing F by 'F'-1 for appropriate ') that Out F(S0) = and Out F(S`) = ,
and also that OutF (Ai) = GL2(4) if the Ai are F-essential. Thus F = F3 or F = *
*F4.
By Janko's original characterization of the sporadic simple groups J2 and J3 *
*[J], both
contain involution centralizers of odd index isomorphic to (D8xC2Q8)o A5, and J*
*2 has
two conjugacy classes of involutions while J3 has only one class. Also, S` is i*
*somorphic
to the Sylow 2-subgroups of these groups; this is shown explicitly in [GH , p.3*
*31], and
also follows since S` is a Sylow 2-subgroup of (D8 xC2 Q8) o A5. Thus FS`(J2) ~*
*=F3
and FS`(J3) ~=F4.
In fact, the main result of [GH ] is that if G is a finite group with Sylow 2*
*-subgroup
isomorphic to S`, then either G=O20(G) is isomorphic to one of the groups P SL3*
*(4) o
<`>, P GL3(4) o <`>, J2, or J3, or G=O20(G) ~=CG(x) for some involution x.
5. Fusion systems over the Sylow 2-subgroup of M22
Again in this section, S0 = UT3(4) denotes the group_of 3x3 upper triangular *
*matri-
ces over F4 with_1_in all diagonal entries, x 7! x = x2 denotes the field autom*
*orphism
on F4, and M 7! M_ denotes the induced field automorphism on S0. Set SOE= S0o <*
*OE>,
where OEMOE-1 = M for all M 2 S0. We want to list all nonconstrained centerfr*
*ee
saturated fusion systems over SOE, up to isomorphism.
Recall exij2 S0 (for i < j) is the elementary matrix with entry x 2 F4 in the*
* (i, j)
position, Eij= {exij| x 2 F4}, and cxijdenotes conjugation by exij. Also, ! den*
*otes an
_
element in F4r F2, so that F4 = {0, 1, !, !}. Then
Z(S0) = E13= , Z(SOE) = , and [SOE, SOE] = e123.
The following subgroups will play an important role in this section:
A1 = H1 = N1 = OE
A2 = H2 = N2 = OE.
Thus A1 and A2 are the "rectangular subgroups", both isomorphic to C42. Also, N*
*i=
NSOE(Hi).
Lemma 5.1. (a) If g 2 SOErS0, then CS0(g) e123, and CS0(g) \ E1*
*3 =
.
(b) A1 and A2 are the only subgroups of SOEisomorphic to C42.
Proof.(a) Fix g 2 SOErS0, and write g = MOE where M 2 S0. Then
__
CS0(MOE) = {X 2 S0| MX M-1 = X}.
Thus e1132 CS0(MOE)_and e!13=2CS0(MOE). Also, for X 2 CS0(MOE),_X_is S0 = UT3(4*
*)-
conjugate to X , and since S0=E13is abelian, this implies that X X (mod E13).*
* Since
the field automorphism acts on S0=E13~= E12x E23fixing , this shows*
* that
X 2 e123= [SOE, SOE].
(b) By Lemma 4.1, A1 and A2 are the only subgroups of S0 isomorphic to C42. So
assume P S0 and P ~=C42. Set P0 = P \S0, and fix g 2 P rP0. Then P0 is contai*
*ned
in A1 or A2. Also, P0 CS0(g), and so by (a), P0 [SOE, SOE] and P \ E13 . Since
22 BOB OLIVER AND JOANA VENTURA
|P0| = 23, this shows that P0 is contained in neither A1 nor A2, which we have *
*already
shown is impossible.
5.1___Candidates_for_critical_subgroups_
Our main result here is the following:
Proposition 5.2. If P is a critical subgroup of SOEthen P is one of the subgrou*
*ps
S0 = UT3(4), N1, N2, H1, or H2.
Proof.In Lemma 5.3, we show that if P is normal, then P is one of the subgroups*
* S0,
N1, or N2. In Lemma 5.5, we show that if P is not normal, and has index 2 in i*
*ts
normalizer, then P = H1 or H2.
Now assume P is not normal, and |N(P )=P | 4. Since SOEhas order 27, |N(P )*
*| 26,
and so |P | 24. Since P is critical, we must have rk(P=F r(P )) 4 by Propos*
*ition
3.4(c). This implies P ~=C42, and by Lemma 5.1(b), any such P is normal.
Lemma 5.3. If P C SOEis a normal critical subgroup of SOE, then P is one of the*
* three
subgroups S0, N1, or N2.
Proof.By Proposition 3.4(c), rk(P=Fr(P )) 2k if |SOE=P | = 2k. Thus |SOE=P | *
* 4.
Assume first that |SOE=P | = 4. Then SOE=P is abelian, so P [SOE, SOE] = e123,
and |P=[SOE, SOE]| = 2. Also, |P | = 25 and rk(P=Fr(P )) 4, so |Fr(P )| 2. *
*Thus, if P
is critical, Fr(P ) = Fr([SOE, SOE]) = .
Set X = {e!12, e!23, e!12e!23} (as a set of elements of S0). If P = <[SOE, SO*
*E],>Xfor some
X 2 X, then Fr(P ) E13, and P is not critical. If P = <[SOE, SOE],>XOEfor som*
*e X 2 X,
then (XOE)2 = [X, OE] 2 Fr(P ), this element is not in , and again P is n*
*ot critical.
This leaves only the possibility
P = OE= OEx ~=D8 xC2 D8 ~=Q8 x*
*C2 Q8.
Then Out(P ) ~= 3o C2. If P were critical, then by Proposition 3.4(b), there wo*
*uld be
an odd order subgroup of Out (P ) which normalizes Out SOE(P ) = ~*
*=C22and
permutes its involutions transitively, and this is not the case. Thus P is not *
*critical;
and SOEcontains no normal critical subgroups of index 4.
Now assume |SOE=P | = 2, and fix g 2 SOErP . Since SOE=Fr(SOE) ~=C32, there a*
*re seven
subgroups of index 2 in SOE. If Fr(P ) = [SOE, SOE], then [g, P ] Fr(P ), and*
* so P is not
(semi)critical by Lemma 3.3 (applied with = 1). This leaves only the subgroup*
*s S0,
N1, N2, and N3 = <[SOE, SOE], e!23e!12,>OE. It remains only to check that this *
*last subgroup
is not critical.
Now, Fr(N3) = ~= C2 x C4, and hence its 2-torsion subgroup E1*
*3 is
characteristic in N3. Let N3 be such that =E13= Z(N3=E13). Then = [SOE, *
*SOE]
is characteristic in N3, [g, N3] , and [g, ] E13 Fr(N3) for g 2 SOErN3. *
*So also
in this case, P is not (semi)critical by Lemma 3.3.
Lemma 5.4. Each critical subgroup of SOEcontains E13.
Proof.Assume P is critical in SOE. Then P is centric in SOE, so Z(SOE) = *
* P . It
remains to show that e!132 P .
Assume otherwise. Then e!132 N(P )r P , and so by Lemma 3.3, e!13acts nontriv*
*ialy
(by conjugation) on P=Fr(P ); i.e., [e!13, P ] Fr(P ). Since [e!13, SOE] = , this implies
e11362 Fr(P ). Furthermore, if we let denote the 2-torsion subgroup of Z(P )*
*, then
SATURATED FUSION SYSTEMS OVER 2-GROUPS 23
e1132 , and thus [e!13, P ] . Since P is critical, Lemma 3.3 now implies t*
*hat
[e!13, ] Fr(P ).
Thus there is some h 2 Z(P ) such that h2 = 1 and [e!13, h] 6= 1. Since e!132*
* Z(S0),
we have h 2 SOErS0. Also, since h 2 Z(P ), P0 def=P \ S0 CS0(h), and by Lemma
5.1(a), P0 e123. Since e113=2Fr(P ), this in turn implies P0 A1*
* or A2. Also,
P0 , since P is centric. So up to symmetry, we can assume that P0 =
or P0 = . Since P = where h 2 Z(P ), we must have h 2 *
*OEA1 if
e1122 P , and h 2 e123OEA1 if e112e!132 P . If we add to this the condition tha*
*t h2 = 1, we
are left with the two possibilities
P = OE or P = e123OE.
In either case, P Q def=<[SOE, SOE],>OE~=D8 xC2 D8, and P [Q, Q] = . *
* Hence
P C Q, so |N(P )=P | |Q=P | = 4.
Since P ~=C32in both cases, this contradicts Proposition 3.4(c).
We are left with the following case.
Lemma 5.5. Let P SOEbe a critical subgroup with index 2 in NSOE(P ) and not n*
*ormal
in SOE. Then P = H1 or P = H2.
Proof.Obviouly [SOE, SOE] P because P is not normal in SOE, and E13 P by Lem*
*ma
5.4. Hence at least one of the matrices e112, e123or e112e123is not in P . Sinc*
*e [e112, SOE] =
E13 P and [e123, SOE] = E13 P , N(P ) . So exactly one of the*
* matrices
e112, e123or e112e123is in P because |N(P )=P | = 2. By symmetry, we can assum*
*e that
e123=2P (hence that g def=e123generates N(P )=P ), and hence that e112X 2 P for*
* some
X 2 .
Assume first that P S0. Then [P, S0] E13 P , and S0 N(P ). Thus
N(P ) = S0 (since P is not normal in SOE), and [S0 : P ] = 2. It follows that *
*P =
e!23Y3for some Yi2 = . Then [g, P ] E13 Fr(P*
* ), and so
P is not (semi)critical by Lemma 3.3 applied with = 1.
Now assume P S0, and set P0 = P \ S0. Then |P0| 24, since |P | 1_4|S| =*
* 25. If
e1122 P , then since Z(SOE=) = , N(P ), a*
*nd so e!12Y 2 P for
some Y 2 . Thus P0 = . Furthermore, SOE=P0 ~=D8, and D8 co*
*ntains
exactly two conjugacy classes of subgroups which are not normal. Since P S0, *
*this
proves that up to conjugacy, P = OEfor some Y 2 . If Y = *
*1, then
P = H1. If Y = g = e123, then [g, P ] = E13 Fr(P ), and again P is not (semi)c*
*ritical
by Lemma 3.3.
By a similar argument, if e112e1232 P , then e!12e!23Y 2 P for some Y 2 , *
*and (again
up to conjugacy) P = OE._ If Y = 1, then Fr(P ) = <*
*E13, e112e123>
(note in particular that (e!12e!23)2 = e!13); and so [g, P ] Fr(P ). If Y = e*
*123, then
_
P = OE, Z(P ) = , Z2(P ) = ;
so [g, Z2(P )] Fr(P ) and [g, P ] Z2(P ); and P is not (semi)critical by Le*
*mma 3.3
applied with = Z2(P ).
5.2___Automorphisms_of_critical_subgroups___
By Proposition 5.2, the only critical subgroups of SOE, and hence the only es*
*sential
subgroups in a saturated fusion system over SOE, are S0, Hi, and Ni (i = 1, 2).*
* The
24 BOB OLIVER AND JOANA VENTURA
automorphism group of S0 was computed in Section 4. In this subsection, we fir*
*st
compute Out (H1) and Out (N1), and then determine all possibilities for Out F(S*
*0),
OutF (Hi), and OutF (Ni) when F is a saturated fusion system over SOE.
We first recall some of the notation used for automorphisms of S0. For each *
*f 2
Hom F2(F4, F4), we defined aef1, aef22 Aut(S0) by setting
ii 1a bjj i1 ab+f(c)j ii 1a bjj i1 ab+f(a)j
aef1 010c01 = 0 1 c and aef2 01 c = 0 1 c ;
0 0 1 00 1 0 0 1
and set Ri = {aefi| f 2 Hom F2(F4, F4)} ~=C42. Also, we defined fl0, fl1, o 2 A*
*ut(S0) by
setting
ii 1a bjj i1 !a~!bj ii 1a bjj i1 !a bj i i1 abjj i1 cbj-1
fl0 010c01 = 0010!c1 , fl1 010c01 = 0010~!c1, o 001c01 = 001a01 *
* ;
and defined 0 = and 1 = o. By Lemma 4.6,
Out(S0) = (R1=) x (R2=).( 0 x 1) ~=C42o ( 3 x 3).
Lemma 5.6. Out (SOE) is a 2-group. If ff 2 Aut(S0) commutes with cOEas elements*
* of
Out(S0), then ff extends to an automorphism of SOE.
Proof.Since cOEacts freely on the basis {e!13, e~!13} of Z(S0), we have Hi(; Z(S0)) = 0
for i = 1, 2. So by Lemma 1.2, and since S0 is a characteristic subgroup of SO*
*E, the
restriction map
~=
Out(SOE) = Out(SOE, S0) -----! NOut(S0)()= = COut(S0)(cOE)=
is an isomorphism. This proves the last statement. Since the centralizer of cOE*
*in
Out (S0)=O2(Out (S0)) ~= 3 x 3
has order 4, COut(S0)(cOE) is a 2-group, and hence Out(SOE) is a 2-group.
We next check the possibilities for OutF (S0) when F is a saturated fusion sy*
*stem.
Lemma 5.7. If F is a saturated fusion system over SOE, then there is an automor*
*phism
' 2 Aut(SOE) such that
Aut'F'-1(S0) cOE.
Proof.Set = OutF (S0) and Q = O2(Out (S0)) for short. Then \ Q = 1 since S0
is F-radical. So there is a unique subgroup 0 0 x 1 such that Q = Q 0. Als*
*o,
OutSOE(S0) = 2 Syl2(Out F(S0)) (each F-essential subgroup is fully normal*
*ized);
and so 0 cOE.
By Proposition 1.7, there is some ff 2 CQ(cOE) such that 0 = ff ff-1. Then *
*ff
extends to an automorphism ' 2 Aut(SOE) by Lemma 5.6, and
Aut'F'-1(S0) = ff ff-1 = 0 cOE.
We next describe Out(P ) for P = Hi and Ni, and list the possibilities for Ou*
*tF (P )
when F is a saturated fusion system over SOE. When doing this, it will be helpf*
*ul to
translate automorphisms of A1 to matrices.
Throughout the rest of the section, for any ff 2 Aut(A1), M(ff) denotes the m*
*atrix
for ff with respect to the basis {e113, e!13, e112, e!12}. When possible, matr*
*ices will be
written as 2 x 2 blocks, where
I = (1001), J = (1101), and Z = (0111).
SATURATED FUSION SYSTEMS OVER 2-GROUPS 25
Thus, for example,
` ' ` ' ` '
J 0 1 I I ! I Z
M(cOE) = 0 J , M(c23) = 0 I and M(c23) = 0 I .
Define ae*i2 Aut(S0) by setting
ii 1a bjj i1 ab+~cj ii 1a bjj i1 ab+~aj
ae*1 010c01 = 0 1 c and ae*2 01 c = 0 1 c .
0 0 1 00 1 0 0 1
Thus ae*iis the identity on Ai, ae*1= oae*2o-1, and M(ae*2|A1) = (IJ0I). The ae*
**icommute
in Aut(S0) with cOE, and hence extend to automorphisms `ae*i2 Aut(SOE) by sendi*
*ng OE to
itself. Similarly, we let `o2 Aut(SOE) be the extension of o which sends OE to *
*itself.
Let j1 2 Aut(H1) be the automorphism such that j1(OE) = OE and
` '
0 I
M(j1|A1) = I I .
Define j012 Aut (H1) by setting j01= `ae*2j1`ae*2-1. Finally, let j2, j022 Aut*
* (H2) be the
automorphisms j2 = `oj1`o-1and j02= `oj01`o-1.
Lemma 5.8. The following hold for any saturated fusion system F over SOE.
(a) If Hi is F-essential (i = 1, 2), then
Out F(Hi) = ~= 3 or Out F(H1) = ~= 3 .
(b) If OutF (S0) , and H1 is F-essential, then there is ' 2 Aut(SOE*
*) such that
Out'F'-1(S0) = OutF (S0) and Out'F'-1(H1) = . If in addition, H2 *
*is F-
essential, then ' can be chosen such that we also have Out'F'-1(H2) = .
Proof.Since cOEacts freely on the basis {e!13, e~!13, e!12, e~!12} of A1, we ha*
*ve Hi(; A1) = 0
for i = 1, 2. So by Lemma 1.2, and since A1 is a characteristic subgroup of H1*
*, the
restriction map
~=
Out (H1) = Out(H1, A1) -----! NOut(A1)()= = COut(A1)(cOE)=
is an isomorphism.
Let M0(ff) denote the matrix for ff 2 Aut (A1) with respect to the ordered ba*
*sis
{e113, e112, e!13, e!12}. Thus M0(ff) is obtained from M(ff) by exchanging the *
*middle two
rows and columns, and it will be more convenient in this proof to work with M0(*
*ff) than
with M(ff). For example, M0(cOE) = (II0I)and M0(c123) = (J00J). By direct compu*
*tation,
fi 4
CGL4(2)(II0I)= (A0BA)fiA 2 GL2(2), B 2 M2x2(F2) ~=C2 o GL2(2).
Hence Out (H1) ~= C32o GL2(2) ~= C32o 3. Also, since M0(j1|A1) = ( Z00Z)and
M0(c123|A1) = (J00J)(and = GL2(2)),
Out (H1) = O2(Out (H1))..
(a) We prove this for H1; the case H2 then follows by symmetry. Assume H1 is F-
essential in some saturated fusion system F, and set = OutF (H1) for short. T*
*hen
\ O2(Out (H1)) = 1, O2(Out (H1)). = Out(H1), and c1232 . By Proposition 1.7,
= for some ff 2 O2(Out (H1)) which centralizes c123. Thus M0*
*(ff|A1) =
(IB0I)for some B 2 M2x2(F2) such that JBJ-1 = B or B + I, and hence B is in the
additive group generated by I, Z, and (0100). Since we are working modulo I, an*
*d since
(IZ0I)also commutes with (Z00Z)(the matrix of j1|A1), we can always choose B = *
*0 or
26 BOB OLIVER AND JOANA VENTURA
B = Y def=(0100). Since (IY0I)= (J0YJ)(J00J)where (J0YJ)= M0(ae*2|A1), this sho*
*ws that
we can take ff = Idor ff = (a`e*2c123)|H1. Also,
1 1 -1 -*1 * -1 -*1 0-1
(a`e*2c123)j1(a`e*2c123)-1 = `ae*2c23j1c23 `ae2= `ae2j1 a`e2 = j1 *
* ,
and thus must be one of the two groups or .
(b) Now assume that Out F(S0) , and H1 is F-essential. If Out F(H*
*1) =
, there is nothing to prove, so assume that Out F(H1) = . *
* Set ' =
`ae*22 Aut (SOE). Then 'j01'-1 = j1, '|S0 commutes with fl0, and '|H2 = Id. T*
*hus
Out'F'-1(P ) = Out F(P ) for P = S0 and H2, while Out 'F'-1(H1) = . S*
*imi-
larly, if H2 is F-essential, we can arrange that OutF (H2) = (withou*
*t changing
OutF (S0) or OutF (H1)) by conjugating F with `ae*1if necessary.
We now turn our attention to the remaining critical subgroups N1 and N2. Let
1 2 Aut(N1) be the automorphism such that
` 100 0'
M( 1|A1) = 01101100, 1(e123) = e123OE, and 1(OE) = e123.
000 1
Set 2 = `o 1`o-1. These will be shown to be well defined homomorphisms as part*
* of
the proof of the following lemma.
Lemma 5.9. If F is a saturated fusion system over SOE, then
o N1 F-essential implies OutF (N1) = < 1, c!23> ~= 3, while
o N2 F-essential implies OutF (N2) = < 2, c!12> ~= 3.
Proof.We prove this for N1. The group acts freely on the basis
{e!12, e!12e!13, e~!12, e~!12e~!13} (1)
of A1 over F2, so Hi(; A1) = 0 for i > 0. Hence the restriction map
~= 1 1
Out(N1) = Out(N1, A1) -----! NOut(A1)()=
is an isomorphism by Lemma 1.2.
Since M(c123) = (II0I), its centralizer is the group of matrices of the form *
*(A0BA)for
A 2 GL2(2) and B 2 M2x2(F2). Such a matrix commutes with M(cOE) = (J00J)exactly
when A and B commute with J; i.e., when they have the form (ab0a)for a, b 2 F2.*
* Thus
ae fifi ` 1ab c' *
*oe
CAut(A1)() = ff 2 Aut(A1) fifiM(ff) = 0100b01asome a, b, c 2*
* F2
000 1
and so CAut(A1)()= ~=C2.
Set 0= 1|A1, as defined above by its matrix. By inspection, 0 permutes cyc*
*lically
the first three elements of the basis in (1), and fixes the fourth element. Upo*
*n com-
paring this action with those of cOEand c123on the basis, we see that 0cOE 0-1*
*= c123and
0c123 0-1= cOEc123. Thus 0is in the normalizer of , and extends to*
* 1 2 Aut(N1)
by setting 1(OE) = e123and 1(e123) = e123OE. Also, this shows that 0 and c!2*
*3generate all
automorphisms of , and hence
Out (N1) = O2(Out (N1)) x < 1, c!23> ~=C2 x 3.
In particular, < 1> is the unique subgroup of Out (N1) of order 3. So if N1 *
*is F-
essential for some saturated fusion system F, then OutF (N1) = < 1, c!23>.
SATURATED FUSION SYSTEMS OVER 2-GROUPS 27
We next describe some restrictions on which combinations of subgroups can be *
*es-
sential in a centerfree nonconstrained saturated fusion system.
Lemma 5.10. Let F be any centerfree nonconstrained saturated fusion system over
SOE. Then for each of i = 1 and 2, either Hi or Ni is F-essential, but not both*
*. If N1
and N2 are both F-essential, then OutF (S0) .
Proof.By Proposition 5.2 (and since Out (SOE) is a 2-group), F is generated by *
*auto-
morphisms in Inn(SOE), Out F(S0), Out F(Hi), and Out F(Ni) (for i = 1, 2), and *
*their
restrictions. Since 2 Syl2(Out F(S0)), each ff 2 AutF (S0) must send A1 a*
*nd A2 to
themselves.
If neither H1 nor N1 is F-essential, then all morphisms in F are composites of
restrictions of automorphisms of SOE, S0, H2, and N2, all of which send A2 to i*
*tself.
Hence A2 is normal in F, which contradicts the assumption that F is nonconstrai*
*ned.
Similarly, if neither H2 nor N2 is F-essential, then A1 C F, which again contra*
*dicts
our assumption.
Thus at least one subgroup in each pair (H1, N1) and (H2, N2) must be F-essen*
*tial.
If N1 is F-essential, then 1 2 OutF (N1) by Lemma 5.9, and 1(H1) = *
*. This
last subgroup is normal in SOE, while N(H1) = N1. Hence H1 is not fully normali*
*zed in
F, and so cannot be F-essential. Similarly, if N2 is F-essential, then H2 is no*
*t.
It remains to prove the last statement. Assume otherwise: assume N1 and N2 are
F-essential, and OutF (S0) . Then neither H1 nor H2 is F-essential,*
* so F is
generated by automorphisms of SOE, N1, and N2; as well as by fl1, cOE2 Aut(S0).*
* All of
these automorphisms fix e113(since SOE, N1, and N2 all have center e113). Thus *
*e113is in
the center of F, and this contradicts the assumption that F is centerfree.
5.3___Fusion_systems_over_SOE_
In order to better describe the subgroups generated by certain sets of elemen*
*ts of
the Aut(Ai), we define an explicit isomorphism from Aut(A1) to the alternating *
*group
A8. We first describe this on an abstract 4-dimensional F2-vector space V with *
*ordered
basis {v1, v2, v3, v4}.
Let 2(V ) = (V V )=Vbe the second exterior power of V , let [v*
* w] 2
2(V ) be the class of v w, and set vij = [vi vj]. Thus {vij| i < j} is a b*
*asis
for 2(V ). Define q: 2(V ) ---! F2 by setting q(x) = 0 if x = [v w] for so*
*me
v, w 2 V , and q(x) = 1 otherwise. Let b: V x V ---! F2 be the associated form
b(x, y) = q(x + y) + q(x) + q(y). Thus q(vij) = 0 for all i, j, and b(vij, vkl)*
* = 1 if i, j, k, l
are distinct and is zero otherwise. One can show that q is a quadratic form wi*
*th
associated bilinear form b by comparing them with the quadratic and bilinear fo*
*rms
which take the same values on the vij. Hence this defines an explicit isomorph*
*ism
from Aut (V ) ~= GL4(2) to ( 2(V ), q) ~= +6(2) (the commutator subgroup of t*
*he
orthogonal group O( 2(V ), q)), by sending ff to 2(ff).
We next construct an explicit isomorphism ( 2(V ), q) ~= A8. Let I(F82) be *
*the
subgroup of elements of even weight in F82(with an even number of 1's), and let*
* q be
the quadratic form q(x) = 1_2wt(x) (mod 2). This is still well defined after di*
*viding out
by the diagonal element (1, 1, 1, 1, 1, 1, 1, 1). To simplify notation, we also*
* identify I(F82)
with the group Pe(8_) of subsets of even order in 8_= {1, 2, . .,.8}. Thus und*
*er this
identification, if b is the bilinear form associated to q, then q(X) = 1_2|X| a*
*nd b(X, Y ) =
|X \ Y | (mod 2) for X, Y 8_. The symmetric group 8 clearly acts on I(F82)=<*
*diag> =
28 BOB OLIVER AND JOANA VENTURA
Pe(8_)=<8_> preserving the form, and this defines isomorphisms SO(Pe(8_)=<8_>, *
*q) ~= 8
and (Pe(8_)=<8_>, q) ~=A8.
~=
Define ~: 2(V ) ---! Pe(8_)=<8_> = I(F82)= by setting
~(v12)= {1234} ~(v13)= {1256} ~(v14)= {1357}
~(v34)= {1238} ~(v24)= {2356} ~(v23)= {1367}
This clearly preserves the quadratic forms on the two spaces. Let
2(-) 2 ~*
OV :Aut(V ) ------!~ ( (V ), q) ------! A8
= ~=
denote the isomorphism induced by 2(-) and ~.
We apply this here with V = A1, and with the ordered basis {e113, e!13, e112,*
* e!12}. The
following table describes the images under OA1 of some automorphisms, where M(f*
*f)
denotes the matrix of ff with respect to this basis:
ff = cOE c123 c!23 ae*2|A1
` ' ` ' ` ' ` '
J 0 I I I Z I J
M(ff) = 0 J 0 I 0 I 0 I (2)
OA1(ff) = (1 2)(5 6) (5 6)(7 8) (5 8)(6 7) (1 2)(3 4) .
Here, as usual, J = (1101)and Z = (0111). We also get the following values for *
*O(ff|A1),
for certain automorphisms ff 2 Aut(P ) of order 3 which can occur in AutF (P ):
(ff, P ) = (fl0, S0) (fl1, S0) ( 1, N1) (j1, H1) (j01, H1)
` ' ` ' ` ' ` ' ` '
Z-1 0 I 0 10000111 0 I J J
M(ff|A1) = 0 Z 0 Z 0 100 (3)
0 001 I I I I+ J
OA1(ff|A1) = (5 6 7) (1 3 2)(5 7 6)(2 5 8)(1 6 7)(4 8 7) (3 8 7) .
This is now applied in the following lemma, which identifies certain groups o*
*f auto-
morphisms of A1.
Lemma 5.11. (a) j1~=j01~= 5 and fl0|A1 belongs to*
* both
of these groups of automorphisms;
(b) fl1= fl1~=(C3 x A5) o C2;
(c) fl1= fl0,~fl1=A7;
(d) fl0~=A6;
(e) = fl1~=A7.
Here, we write 1, j1, j01, and fli, but mean their restrictions to A1.
~=
Proof.The proof will be based on the isomorphism O = OA1: Aut(A1) ---! A8 con-
structed above. To simplify notation, we identify these two groups, and omit "O*
*(-)"
where it would be appropriate.
Whenever I and J are disjoint subsets of 8_= {1, . .,.8} (m 1), we let AI,J*
* A8
(AI A8) denote the subgroups of permutations which leave I and J invariant (l*
*eave
I invariant), and fix all other elements in 8_. Elements of the subsets are lis*
*ted without
brackets or commas. Thus, for example, A125678(~= A6) is the subgroup of (even)
SATURATED FUSION SYSTEMS OVER 2-GROUPS 29
permutations which fix 7 and 8, while A12;5678contains those permutations which*
* fix 7
and 8 and leave the subset {1, 2} invariant.
We refer to (2)and (3)for the images in A8 of certain elements of Aut(A1).
(a): Consider first
Ha def=j1= j1= <(5 8)(6 7), (5 6)(7 8), (1 2)*
*(5 6),>(4.8 7)
Then Ha A12;45678. Also, the image of Ha under projection to 5 (permutations*
* of
{4, 5, 6, 7, 8}) contains the 2-cycle (5 6) and the 5-cycle c!23j1 = (5 8 6 7 4*
*) (where we
compose from right to left). Thus the projection is surjective, and this prove*
*s that
Ha = A12;45678~= 5. In particular, fl0 = (5 6 7) 2 Ha.
Simiarly, if we set
H0adef=j01= j1= <(5 8)(6 7), (5 6)(7 8), (1 2*
*)(5 6),>(3,8 7)
then H0a= A12;35678~= 5, and fl0 = (5 6 7) 2 H0a.
(b): By (a),
Hb def=fl1= = 7=6)A123;4*
*5678.
Thus Hb ~=(C3 x A5) o C2, and fl0 = (5 6 7) 2 Hb.
(c): By (a) again,
Hc def=fl1= 7=6)A1235678~=A7.
In particular, 1 = (2 5 8)(1 6 7) and fl0 = (5 6 7) are both in Hc.
(d): We have
Hd def= 1= <(1 2)(5 6), (5 8)(6 7), (5 6)(7 8), (5 6 7),>*
*(2 5 8)(1 6 7)
= 6=7)A125678~=A6.
(e): Consider the subgroup
He def= = (1.3*
* 2)
Then -11(1 3 2) 1 = (7 3 8) = j012 He, and so
He = 3=2)(1=3A2)12356*
*78.
Thus He ~=A7, and fl0, fl1 2 He.
We are now ready to list fusion systems over SOE. In the statement and the pr*
*oof of the
following theorem, we follow the usual notation by writing P Ln(q) = P GLn(q)o*
*
and P Ln(q) = P SLn(q)o , where OE is a generator of Aut (Fq) (extended to*
* an
automorphism on matrix groups).
Theorem 5.12. If F is a nonconstrained centerfree saturated fusion system over *
*SOE,
then it is isomorphic to the fusion system of one of the following groups: M22,*
* M23,
McL , P L3(4), P L3(4), or P SL4(5) ~=P +6(5).
Proof.Let F be a saturated fusion system over SOE. Assume F is nonconstrained a*
*nd
centerfree. By Lemma 5.7, upon replacing F by 'F'-1 for some ' 2 Aut(SOE), we c*
*an
assume that
OutF(S0) cOE. (4)
30 BOB OLIVER AND JOANA VENTURA
We first list the different choices for the set of F-essential subgroups, the*
*n we list the
different combinations for AutF (P ) (or OutF(P)) for each F-essential subgroup*
* P . We
then show that F is isomorphic to one of a list of six explicitly defined fusio*
*n systems
over SOE, which we then compare with those in the statement of the theorem.
Consider the following additional restrictions which we know hold for F:
(a) By Lemma 5.6, Out(SOE) is a 2-group. Hence OutF (SOE) = Inn(SOE).
(b) By Proposition 5.2, the only possible F-essential subgroups are S0, Ni, and*
* Hi
(i = 1, 2). By Lemma 5.10, exactly one of the subgroups H1 or N1 is essent*
*ial,
and exactly one of the subgroups H2 or N2 is essential.
(c) By Lemma 5.11(a), if H1 is F-essential (so j1 or j01is in OutF (H1)), then *
*fl0|Ai 2
AutF (Ai), and so fl0|Ai must extend to an automorphism in AutF (S0). Thus *
*by
(4), fl0 2 OutF (S0). Similarly, if H2 is F-essential, then ofl0o-1 = fl0 2*
* OutF (S0).
(d) By Lemma 5.11(c), if j012 AutF(H1) and fl0, fl1 2 AutF(S0), then 1|A1 2
AutF (A1), which implies by the extension axiom that 1|A1 extends to an au-
tomorphism in Aut F(N1). Thus N1 is F-essential (and so H1 is not) in this
case. In other words, if fl0, fl1 2 Aut F(S0) and Hi is essential (i = 1, *
*2), then
OutF (Hi) = .
(e) By Lemma 5.10 again, if N1 and N2 are both F-essential, then OutF(S0) .
Thus at least one of the automorphisms fl0, fl0fl1, or fl0fl-11must be in O*
*utF (S0).
Putting together points (a)-(e), one concludes that S0 must be F-essential, a*
*nd that
the choices for the set of F-essential subgroups are the following:
{H1, H2, S0} , {H1, N2, S0} , {N1, H2, S0} and {N1, N2, S0} .
Upon combining this with the restrictions on the automorphism groups OutF (P ) *
*im-
posed by points (c)-(e), we are reduced to the following list of candidates (up*
* to
symmetry by `o2 Aut(SOE)) for "extra" automorphisms which generate F:
{H1, H2, S0} :{j1, j2, fl0} , {j1, j2, fl0, fl1} , {j01, j2, fl0} , {j01, *
*j02, fl0} ;
{H1, N2, S0} :{j1, 2, fl0} , {j1, 2, fl0, fl1} , {j01, 2, fl0} ;
{N1, N2, S0} :{ 1, 2, fl0} , { 1, 2, fl0, fl1} , { 1, 2, fl0fl1} , { 1,*
* 2, fl0fl-11} .
Thus, up to isomorphism, there are at most eleven saturated fusion systems over*
* SOE.
By Lemma 5.11(e), if N1 is F-essential and fl0fl1 2 Out F(S0), then fl0|A1, f*
*l1|A1 2
AutF (A1). So by the extension axiom (and (4)), fl0, fl1 2 Out F(S0) in this c*
*ase. By
symmetry, if fl0fl-11= o(fl0fl1)o-1 2 Out F(S0) and N2 is F-essential, then fl0*
*, fl1 2
OutF (S0). In other words, the sets { 1, 2, fl0fl1} and { 1, 2, fl0fl-11} lis*
*ted above cannot
occur.
By Lemma 5.8(b), if Out F(S0) = , H1 is F-essential, and Out F(H1) =
, then there is an automorphism ' 2 Aut (SOE) such that Out 'F'-1(S0*
*) =
OutF (S0) and Out 'F'-1(H1) = . If, furthermore, H2 is also F-essenti*
*al, then
' can be chosen such that Out 'F'-1(H2) = . In other words, we can e*
*limi-
nate all of the cases which involve j01or j02, since the corresponding fusion s*
*ystems are
isomorphic to others in the list.
Thus F is isomorphic to one of the six fusion system listed in Table 5.1, whe*
*re in
all cases, Out F(Hi) = if Hi is F-essential. The descriptions*
* of the
groups Aut F(Ai) follow from Lemma 5.11. By inspection, these six fusion syste*
*ms
are distinguished by the groups AutF (A1) and AutF (A2) as described in the tab*
*le. It
SATURATED FUSION SYSTEMS OVER 2-GROUPS 31
_________________________________________________________________________
| | | | | |
| Out F(S0) |F-essential| AutF (A1) | Aut F(A2) | G |
|___________|__________|_______________|______________|_________________|__
| | | | | |
| H|1, H2 | 5 | 5 | P L3(4) |
|______________|________|_____________|______________|___________________||||*
*|||
| cOE|H1, H2(C3|x A5) o C2 |(C3 x A5) o C2 | P L3(4) |
|_____________|___________|___________|_______________|__________________||||*
*|||
| N|1, H2 | A6 | 5 | M22 |
|______________|________|_____________|______________|__________________|_|||*
*|||
| cOE|N1, H2 | A7 |(C3 x A5) o C2 | M23 |
|_____________|___________|___________|_______________|_________________|_|||*
*|||
| N|1, N2 | A6 | A6 |P SL4(5) ~=P +(5) |
|______________|________|_____________|______________|______________6____||||*
*|||
| cOE|N1, N2 | A7 | A7 | McL |
|_____________|___________|___________|______________|__________________|_
Table 5.1
remains to prove that the groups G listed in there have Sylow 2-subgroups isomo*
*rphic
to SOE, and have automorphism groups Aut G(Ai) as described. This is clear for*
* the
groups P L3(4) and P L3(4) using the well-known isomorphisms P L2(4) ~= 5 and
P L2(4) ~=(C3 x A5) o C2 (or by directly determining AutG (Hi) and AutG (S0)).
The group GL4(2) ~=A8 contains unique conjugacy classes of subgroups isomorph*
*ic
to A6 and A7. Hence SOEis a Sylow subgroup of any semidirect product C42oA6 or
C42oA7 (which is not a product).
When q 5 (mod 8), then P 6(q) is the commutator subgroup of the projective
orthogonal group of a quadratic form on V = F6qwith orthogonal basis {v1, . .,*
*.v6}.
This group contains two conjugacy classes of subgroups C42oA6: the groups of au*
*tomor-
phisms which preserve up to sign one of the two bases {vi} or {v1 v2, v3 v4, *
*v5 v6}.
(These two orthogonal bases are inequivalent, since 2 is always a nonsquare for*
* such
q.) Since these are subgroups of odd index, P 6(q) has Sylow 2-subgroups isom*
*or-
phic to SOE, and its fusion system is the one with these automorphism groups (a*
*nd is
independent of q).
Finally, M22 contains subgroups C42o 5 (the quintet subgroup) and C42o A6 (t*
*he
hexad subgroup); while M23contains subgroups (C42oC3) o 5 (the quintet subgrou*
*p)
and C42o A7 (the heptad subgroup). See [Co , Table 3] for more detail. Also, *
*by
[Fi, Theorem 1], McLaughlin's group McL contains two conjugacy classes of subgr*
*oups
C42oA7. So all three of these groups have the fusion systems described in Table*
* 5.1.
Note also that McL contains M22, P -6(3), and P L3(4) as subgroups of odd i*
*ndex,
while M23contains M22and P L3(4) as subgroups of odd index.
6. Fusion systems over UT5(2)
Throughout this section, T5 = UT5(2) denotes the group of 5 x 5 upper triangu*
*lar
matrices over F2. We let eij2 T5 (for i < j) be the elementary matrix with nont*
*riv-
ial entry in the (i, j) position. Also, cijdenotes conjugation by eij, regarde*
*d as an
automorphism of T5 or as a homomorphism between subgroups of T5.
For any pair of sets of indices I, J {1, 2, 3, 4, 5}, let EI;J T5 denote t*
*he subgroup
generated by all eijfor i 2 I and j 2 J (and i < j). In particular, we focus at*
*tention on
the "rectangular" subgroups A1 = E12;345, A2 = E123;45, U1 = E1;2345, and U2 = *
*E1234;5.
32 BOB OLIVER AND JOANA VENTURA
These can be described pictorally as follows:
|________________|||||||||_ |________________|||||||||_ *
* |________________|||||||||_ |________________|||||||||_
|________________|||||||||||||||||_||||||||________________||||||||||*
*||||||||_|||________________||||||||||||||||||_|||||________________|||||||||*
*|||||||||_||||
A1 = |____________|||||||||||||_||||A2 = |____________|||||||||||||_||*
*||||U1 = |____________|||||||||||||U2 = |____________|||||||||||||_||
|________||||||||| |________|||||||||_||| *
* |________||||||||| |________|||||||||_||||
|____||| |____||| *
* |____||| |____|||_||
We also need to consider the following index two subgroups Qi:
|________________|||||||||___________________||||||||||________________*
*___||||||||||___________________||||||||||___
|________________|||||||||||||||||_||||_||||||||_|||________________|||*
*|||||||||||||||_|||||_||||_|||________________||||||||||||||||||_|||||_||||||*
*||_||||________________||||||||||||||||||_|||||_||||||||_|||
Q1 = |____________||||||||||||||||_||||_||||_||||||Q2_=___________||||||||*
*|||||||||_||||_||||||Q3_=___________|||||||||||||||||_||||_||||Q4_=__________*
*_|||||||||||||||||_||||_||||||.
=A1A2U2 |________||||||_||_|||=A2U1U2________|||||||_||_|||=A1U1U2________|*
*||||||_||=A1A2U1________|||||||_|||
____||||_|| ____||||_|| ____||||_|| ____||||
We will show in Proposition 6.4 that the Qi are the only critical subgroups of *
*T5.
The following lemma is very elementary and well known; we include it here for*
* the
sake of completeness.
Lemma 6.1. The only elementary abelian subgroups of rank 6 in T5 are A1 and A2.
Proof.Let A T5 be any elementary abelian subgroup of rank 6, and set k = rk(A*
* \
A1). Then k 3, since the largest abelian subgroups of T5=A1 ~=D8 x C2 have ra*
*nk
3. If k = 3, then AA1=A1 must contain the center of T5=A1, hence e12x, e35y 2 A*
* for
some x, y 2 A1, which is impossible since CA1(e12x, e35y) = has orde*
*r 4. Thus
k 4. If k = 6, then of course A = A1.
If k = 5, and g 2 Ar A1, then A \ A1 CA1(g) has order 25. By a direct check*
*, for
all g 2 T5r A1, CA1(g) has order at most 24. So this case is impossible.
If k = 4, then AA1=A1 is an abelian subgroup of rank two in T5=A1, and hence *
*must
contain some element in its center. Thus e12x, e35x, or e12e35x is in A for som*
*e x 2 A1.
Of these, only elements e35x have centralizer in A1 of order 24, so e35x 2 A *
*for some
x 2 A1. Thus CA1(e35) = A1 \ A2 A, and so A CT5(A1 \ A2) = A1A2. Since all
elements of order 2 in A1A2 lie in A1 or A2, this shows that A = A2.
6.1___Determining_the_critical_subgroups__
We use the following notation for subgroups of T = T5: T 0= [T, T ], Z = Z(T *
*) =
, and Z2 = [T, T 0] = Z2(T ) = . Also, o 2 Aut(T ) is the a*
*utomorphism
o(eij) = e6-j,6-i.
We start by reducing to the case of subgroups having index 2 in their normali*
*zers.
Lemma 6.2. If P is a critical subgroup of T , then |NT(P )=P | = 2.
Proof.Assume otherwise: let P be a critical subgroup of T with |N(P )=P | 4. *
*Set
V = P=Fr(P ). By Proposition 3.4(c), rk([g, V ]) 2 for each g 2 N(P )r P . Mo*
*reover,
by Proposition 3.4(b), rk([g, V ]) is independent of the choice of g 2 N(P )r P*
* with
g2 2 P .
Now, P is centric in T , and hence Z = P . Also, for x 2 {e14, e25}, *
*[x, P ]
[x, T ] = , so x 2 N(P ), [x, V ] has rank 1, and hence x 2 P by the abo*
*ve
remarks. Thus Z2 P . Then, since [T 0, P ] [T 0, T ] Z2, this in turn imp*
*lies that
T 0 N(P ). There are two cases to consider.
Case 1: Assume first that e15 2 Fr(P ). Since [e13, P ] [e13, T ] = , this
implies rk([e13, V ]) 1. Hence e132 P , and e352 P by the same argument.
SATURATED FUSION SYSTEMS OVER 2-GROUPS 33
Now, [e12, P ] [e12, T ] = P , and hence e12 2 N(P ). If*
* e12 =2P ,
then rk([e12, V ]) 2 by Proposition 3.4; and since e152 Fr(P ), this implies *
*[e12, V ] =
and e142=Fr(P ). Hence e142=[e13, P ], which implies P e45.
Likewise, if e242=P , then rk([e24, V ]) 2, and since [e24, T ] = Z2 this i*
*mplies Fr(P )\
Z2 = Z. Hence e142=[e13, P ] and e252=[e35, P ], and these imply P e45.
We have now shown that
(a) either e122 P or P e45;
(b) either e242 P or P e45; and (by symmetry)
(c) either e452 P or P e45.
Assume e242=P . Then |P | 27 by (b), and e14, e252=Fr(P ). Set P0 =
P . Since [e24, P ] = , there are elements e12x, e45y 2 P for x, y 2 *
* (by (b)
again), and = P since it has order 27. Also, (e12x)2 = [e12, x*
*] 2 Fr(P ),
and (e45y)2 = [e45, y] 2 Fr(P ). Since e14, e252=Fr(P ), this implies x = y = 1*
*, and thus
P = Q = U1U2. But Q is not critical since OutT (Q) ~=D8 has noncentral involuti*
*ons
(which would contradict Proposition 3.4(a)).
Thus e24 2 P , and hence T 0 P . By (a) and (c) above, P or o(P ) is one of*
* the
following three subgroups of T :
o P = e45, Fr(P ) = Z2, rk([e23, V ]) = 1;
o P = e34, Fr(P ) = , rk([e23, V ]) = 3, rk([e45, V ]) = *
*2;
o P = , Fr(P ) = , rk([e34, V ]) = 0.
Thus none of these satisfies the conditions to be critical.
Case 2: Now assume e15 =2Fr(P ). Since e14 2 P and [e14, T ] = , this im*
*plies
that e142 Z(P ), and similarly e252 Z(P ). So P CT(Z2) = A1A2. Since P is cen*
*tric
in T , this also implies that Z(A1A2) = P . Set H = for s*
*hort.
Now, P=H has index 4 in its normalizer in T=H ~=D8 x D8, and is contained in
the subgroup A1A2=H ~= C22x C22of this product. If (P=H) \ Z(T=H) = 1, i.e., if
P \ T 0= H, then P=H must be T=H-conjugate to T=H, and hence P is
T -conjugate to . But for P = , [e13, P ] = has*
* rank one, so
P is not critical.
Thus P \ T 0 H. Also, e13e352=P since (e13e35)2 = e152=Fr(P ). So either e13*
*2 P
or e35 2 P , but not both. By symmetry (with respect to o 2 Aut (T )), it suff*
*ices
to consider the case e35 2 P and e13 =2P . Then e45 2 N(P ) since [e45, T ] *
*P ,
and thus N(P ) has order 29. If |P | 26, then |N(P )=P | 8*
*, so
rk(P=Fr(P )) 6, P is elementary abelian of rank 6, and P = CT() = A2.*
* But
A2 is not critical, since N(A2)=A2 = T=A2 has order 16.
Thus |P | = 27, and P has index 2 in A1A2. Then P = e35for so*
*me
x, y 2 . Also, [e13, P ] = [e13, T ] = and e14 = (e13e34)2 =2 *
*Fr(P ), so
y = 1. Since the two remaining possibilities for P are T -conjugate (conjugate*
* by
e12), we are left to consider P = e35. In this case, Fr(P ) = <*
*e24, e25>,
Z(P ) = H = e15, conjugation by e13 is the identity on P=Z(P ) an*
*d on
Z(P )=Fr(P ), and hence is in O2(Aut (P )). So P is not semicritical.
We next show:
Lemma 6.3. All critical subgroups of T = T5 contain Z2.
34 BOB OLIVER AND JOANA VENTURA
Proof.Assume P is a critical subgroup which does not contain Z2. By Lemma 6.2,
|N(P )=P | = 2. Also, P Z since P is centric, and so Z2 N(P ). Hence |P \ Z*
*2| = 4
has index 2 in Z2, and N(P ) = P Z2.
Fix g 2 Z2r P . Then g 2 N(P )r P , and [P, g] = Z(P ). By Lemma
3.3 (applied with = Z(P )), [g, Z(P )] Fr(P ), since otherwise P would fail*
* to be
semicritical. So there is h 2 Z(P ) such that [h, g] = e15 =2Fr(P ). Since e15 *
*=2Fr(P ),
this also implies that P \ Z2 Z(P ); and also that an element of Z2 commutes *
*with
h if and only if it lies in P .
Recall that A1A2 = CT(Z2) = CT(). There are three cases to consider:
(a) P \ Z2 = , g = e25, h 2 e12.A1A2; (b) P \ Z2 = , g = e1*
*4,
h 2 e45.A1A2; (c) P \ Z2 = , g = e14, h 2 e12e45.A1A2. In all of *
*these
cases, [h, e24] 2 (P \ Z2)r Z Z(P ). Thus e242= P since h 2 Z(P ). Since [h, *
*P ] = 1
and [[h, e24], P ] = 1, the 3-subgroup lemma (cf. [G , Theorem 1.2.3]) implies*
* that
[[e24, P ], h] = 1. So [e24, P ] P , and e242 N(P ).
Since N(P ) = P Z2, this implies that e24g 2 P . But [h, e24g] = [h, e24].[h*
*, g] 6= 1
(since [h, e24] =2Z), which contradicts the assumption h 2 Z(P ).
We are now ready to finish the description of all critical subgroups of T , b*
*y handling
the subgroups of T which contain Z2 and have index 2 in their normalizer. This *
*still
requires a lot of case-by-case checks.
Proposition 6.4. The only critical subgroups of T5 = UT5(2) are the subgroups Qi
(i = 1, 2, 3, 4) of index 2.
Proof.The subgroups Qi are all critical, since they are all essential subgroups*
* of
GL5(2).
Let P be a critical subgroup of T = T5. By Lemmas 6.2 and 6.3, |N(P )=P | = 2
and P Z2. By Lemma 3.3, for any g 2 N(P )r P , there is no characteristic sub*
*group
P such that
[g, P ] .Fr(P ) and [g, ] Fr(P ). (1)
Assume first that P C T , and thus that P has index 2 in T . If Fr(P ) = [T, *
*T ], then
for any g 2 T rP , cg acts via the identity on P=Fr(P ) since T=Fr(P ) is abeli*
*an, and
so cg 2 O2(Aut (P )). Thus OutT (P ) \ O2(Out (P )) has order 2 in this case, s*
*o P is not
semicritical.
We can thus assume that Fr(P ) [T, T ]. By inspection, either P = Qi for s*
*ome
i = 1, 2, 3, 4; or P is one of the groups E4 or o(E4) where
E4 = {(aij) 2 T | a12= a34} and Fr(E4) = .
Also, Z3(E4) = Z3(T ) = [T, T ], so this subgroup is characteristic in E4; conj*
*ugation by
e12 acts via the identity on E4=[T, T ] and on [T, T ]=Fr(E4) ~=C2, and so OutT*
* (E4)
O2(Out (E4)). This shows that E4 and o(E4) are not semicritical.
Now assume P is not normal in T . We can always choose g 2 T 0= e35,
so [g, P ] [T 0, T ] = Z2 in all cases. Thus, for example, by (1), P cannot b*
*e semicritical
if Fr(P ) Z2.
We want to apply Lemma 1.8 to S = T=Z2, regarded as an extension
1 ---! T0=Z2 -----! T=Z2 -----! T=T0 ---! 1,
= =
SATURATED FUSION SYSTEMS OVER 2-GROUPS 35
where S0 = T0=Z2 ~=C52and S=S0 ~=C22. Using the notation of Lemma 1.8 (but with
P a subgroup of T and not of S = T=Z2), we set P0 = P \ T0. Also, as subgroups
of S, [e23, S0] = , [e34, S0] =