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=<z> over S=<z>. 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 = <g 2 G | gSg-1 \ S 6=>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 <g, h> 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 <bg,>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   <Pi-1, Pi> (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 <c*ij> =

{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 <OE,>`,
                                     __               __
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<OE,>`. 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 <OE>.

  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 = <E12, E13> =    010001fifia, b 2 F4 Q0 =    01~a001fifia, b 2 F4
                        ni 10a jfi            o
      A2 = <E13, E23> =    01b001fifia, b 2 F4 Q  = <Q0, `> .


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 = <e113>, 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 = <A1, A2> 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) = <e113>.

  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  = <Q0, X`> 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 ) = <e113>, and this is possible only if a = 0.  So th*
 *e only

possibility is P = <Q0, `> = 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 = <Q0, ea12, eb12`> 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 ) = <E13, [ea12, `], [eb12,>`]= 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  = <Q0, ea12,>`.  Then Fr(P *
 *) =

<E13, [ea12,>`]~=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`) = <e113>   *
 *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`] = <e1*
 *13>, 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 = <e113>, P = <e113,>h~=C22, CS`(P ) = <h, CS0(*
 *h)> 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 = <E13, e112Y1, e!12Y*
 *2,>ea12ea23for

some Yi 2 <g>. 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 = <E13, ea12eb23, eb12ea23>
                  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 = <E13, ea12eb23, eb12ea23,>`= <P0, `>.


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 =_<E13, ea12, ea23,>`for some 0 6= a 2 F4.  Then Z(P ) = <e*
 *113>,

Z2(P ) = <E13, ea12ea23>, 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=<fl0, c`> ~= 3,      1 def=<fl1,>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=<c*12>) x (R2=<c*23>).( 0 x  1) ~=C42o ( 3 x  3).

(b) If ' 2 Aut (A1) commutes with all elements of <c*23>, 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 <o>. By Lemma 1.2, there is a short exact sequence



   1 ---! H1(<c*23>; A1) -----! Out 0(S0) --Res---!NGL4(2)(<c*23>)=<c*23> ---!(*
 *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)(<c*23>) =     0  I   fifiX 2 M2x2(F2)  .         (2)


Also,
                    ae`       ' fi                                          oe
                        A  X    fi                                 -1
  NGL4(2)(<c*23>) =     0  B    fifiX 2 M2x2(F2), A, B 2 GL2(F2), A  B 2 <Z>  .


Thus, the elements in NGL4(2)(<c*23>) are all represented by automorphisms of S*
 *0 in the

subgroup generated by R2, fl0, fl1, and cOE.

  Consider the spectral sequence

               Eij2= Hi(<c!23>; Hj(<c123>; A1))  =)  Hi+j(<c*23>; 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(<c*23>; A1) ~=H1(<c!23>; H0(<c123>; A1)) = H1(<c!23>; 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

                            = <fl1, fl0,>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`) = <f`l1>.


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)(<c`>)=<c`>(*
 *3)
                  =

is an isomorphism. By Lemma 4.6(a),

                     COut(S0)(<c`>)=O2(COut(S0)(<c`>)) ~= 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`) = <f`l1> 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`) = <f`l1>.

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<OE> ~=(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) = <c*23> 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  (<c*23>) 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 <c*23, fl0|A1> ~=A4.  A*
 *lso,

any isomorphism between two subgroups of SL2(4) isomorphic to A4 is conjugation*
 * by

some element of SL2(4) o <OE> ~= 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`)= <f`l1>  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`)= <f`l1> Out F3(S0)=     Out F3(Q)= A5

   OutF4(S`)= <f`l1> 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`) = <f`l*
 *1>,

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= <e113, e!13>,     Z(SOE) = <e113>,     and    [SOE, SOE] = <E13,*
 * e112,>e123.


  The following subgroups will play an important role in this section:

         A1 = <E12, E13>       H1 = <A1, OE>       N1 = <A1, e123,>OE

         A2 = <E13, E23>       H2 = <A2, OE>       N2 = <A2, e112,>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)   <E13, e112,>e123, and CS0(g) \ E1*
 *3 =

    <e113>.

(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 <e112, e123>, this shows*
 * that

                         X 2 <E13, e112,>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  <e1*
 *13>. 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] = <E*
 *13, e112,>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]) = <e113>.

  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 <e113>, and again P is n*
 *ot critical.

  This leaves only the possibility

      P = <E13, e112, e123,>OE= <e!13,>OEx<e113><e112, e123> ~=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 ) = <e!12, e!23> ~*
 *=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) = <E13, e112e123> ~= 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) = <e113>*
 *   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] = <e*
 *113>, 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   <E13, e112,>e123. Since e113=2Fr(P ), this in turn implies P0   A1*
 * or A2. Also,

P0   <e113>, since P is centric. So up to symmetry, we can assume that P0 = <e1*
 *13, e112>

or P0 = <e113, e112e!13>.  Since P = <P0, h> 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 = <e113, e112,>OE  or    P = <e113, e112e!13,>e123OE.


In either case, P   Q def=<[SOE, SOE],>OE~=D8 xC2 D8, and P   [Q, Q] = <e113>. *
 * 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 )   <e112, e123>. 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 <g>.

  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 =

<E13, e112Y1, e!12Y2,>e!23Y3for some Yi2 <g> = <e123>. 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=<E13, e112>) = <e123, e!12>, <e123, e!12>   N(P ), a*
 *nd so e!12Y 2 P for

some Y 2 <g>. Thus P0 = <E13, e112, e!12Y>. 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 = <E13, e112, e!12Y,>OEfor some Y 2 <g>. 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 <g>, *
 *and (again

up to conjugacy) P = <E13, e112e123, e!12e!23Y,>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 = <E13, e112e123, e!12e!23,>OE,     Z(P ) = <e113>,     Z2(P ) = <E13, *
 *e112e123>;

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 = <fl0, cOEo> and  1 = <fl1,>o. By Lemma 4.6,


          Out(S0) = (R1=<c*12>) x (R2=<c*23>).( 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(<cOE*
 *>; 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)(<cOE>)=<cOE> = COut(S0)(cOE)=<cO*
 *E>

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)   <fl0, fl1,>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) = <cOE> 2 Syl2(Out F(S0)) (each F-essential subgroup is fully normal*
 *ized);

and so  0  <fl0, fl1,>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  <fl0, fl1,>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) = <ji, OutNi(Hi)> ~= 3    or    Out F(H1) = <j01, OutNi(Hi)> ~= 3 .

(b) If OutF (S0)   <fl0, cOE>, and H1 is F-essential, then there is ' 2 Aut(SOE*
 *) such that

    Out'F'-1(S0) = OutF (S0) and Out'F'-1(H1) = <j1, c123>. If in addition, H2 *
 *is F-

    essential, then ' can be chosen such that we also have Out'F'-1(H2) = <j2, *
 *c112>.


Proof.Since cOEacts freely on the basis {e!13, e~!13, e!12, e~!12} of A1, we ha*
 *ve Hi(<cOE>; 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)(<cOE>)=<cOE> = COut(A1)(cOE)=<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 <Z, J> = GL2(2)),


                       Out (H1) = O2(Out (H1)).<j1, c123>.



(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,

  = <ffj1ff-1, c123> 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 <j1, c123> or <j01, c123>.

(b)  Now assume that Out F(S0)   <fl0, cOE>, and H1 is F-essential.  If Out F(H*
 *1) =

<j1, c123>, there is nothing to prove, so assume that Out F(H1) = <j01, c123>. *
 * 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) = <j1, c123>. S*
 *imi-

larly, if H2 is F-essential, we can arrange that OutF (H2) = <j2, c112> (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 <cOE, c123> acts freely on the basis


                            {e!12, e!12e!13, e~!12, e~!12e~!13}           (1)


of A1 over F2, so Hi(<cOE, c123>; A1) = 0 for i > 0. Hence the restriction map

                                     ~=                  1           1
            Out(N1) = Out(N1, A1) -----! NOut(A1)(<cOE, c23>)=<cOE, c23>


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)(<cOE, c123>) =  ff 2 Aut(A1) fifiM(ff) =   0100b01asome a, b, c 2*
 * F2
                                                         000 1

and so CAut(A1)(<cOE, c123>)=<cOE, c123> ~=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 <cOE, c123>, 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 <cOE, c123>, 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)   <fl1, cOE>.



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 <cOE> 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) = <A1, e123>*
 *. 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)   <fl1, cOE>. 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 )=<v   v | v 2>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)=<diag> 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)   <Out SOE(A1),>j1~=<Out SOE(A1),>j01~= 5 and fl0|A1 belongs to*
 * both

    of these groups of automorphisms;

(b) <Out SOE(A1), j1,>fl1= <Out SOE(A1), j1, fl0,>fl1~=(C3 x A5) o C2;

(c) <Out SOE(A1), j01,>fl1= <Out SOE(A1),  1, j01,>fl0,~fl1=A7;

(d) <Out SOE(A1),  1,>fl0~=A6;

(e) <Out SOE(A1),  1, fl0fl1> = <Out SOE(A1),  1, fl0,>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=<Out SOE(A1),>j1= <c!23, c123, cOE,>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=<Out SOE(A1),>j01= <c!23, c123, cOE,>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=<Out SOE(A1), j1,>fl1= <Ha, fl1> = <A12;45678, (1 3 2)(5>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=<Out SOE(A1), j01,>fl1= <A12;35678, (1 3 2)(5>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=<Out SOE(A1), fl0,> 1= <(1 2)(5 6), (5 8)(6 7), (5 6)(7 8), (5 6 7),>*
 *(2 5 8)(1 6 7)

      = <A12;5678, (2 5 8)(1>6=7)A125678~=A6.



(e):  Consider the subgroup


          He def=<Out SOE(A1),  1, fl0fl1> = <Out SOE(A1), (2 5 8)(1 6 7),>(1.3*
 * 2)


Then  -11(1 3 2) 1 = (7 3 8) = j012 He, and so

     He = <H0a, (2 5 8)(1 6 7), (1>3=2)<A12;35678, (2 5 8)(1 6 7),>(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*
 * <OE>

and P  Ln(q) = P SLn(q)o <OE>, 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)   <fl0, fl1,>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) = <ji, OutSOE(Hi)>.


(e) By Lemma 5.10 again, if N1 and N2 are both F-essential, then OutF(S0)   <fl*
 *1, cOE>.

    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) = <fl0, cOE>, H1 is F-essential, and Out F(H1) =

<j01, c123>, then there is an automorphism ' 2 Aut (SOE) such that Out 'F'-1(S0*
 *) =

OutF (S0) and Out 'F'-1(H1) = <j1, c123>. If, furthermore, H2 is also F-essenti*
 *al, then

' can be chosen such that Out 'F'-1(H2) = <j2, c112>.  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) = <ji, OutSOE(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         |
  |___________|__________|_______________|______________|_________________|__
  |              |        |             |              |                   |
  |  <fl0, cOE> H|1, H2   |     5       |      5       |     P  L3(4)      |
  |______________|________|_____________|______________|___________________||||*
 *|||

  | <fl0, fl1,>cOE|H1, H2(C3|x A5) o C2 |(C3 x A5) o C2 |    P  L3(4)      |
  |_____________|___________|___________|_______________|__________________||||*
 *|||

  |  <fl0, cOE> N|1, H2   |    A6       |      5       |       M22        |
  |______________|________|_____________|______________|__________________|_|||*
 *|||

  | <fl0, fl1,>cOE|N1, H2   |  A7       |(C3 x A5) o C2 |      M23        |
  |_____________|___________|___________|_______________|_________________|_|||*
 *|||

  |  <fl0, cOE> N|1, N2   |    A6       |     A6       |P SL4(5) ~=P  +(5) |
  |______________|________|_____________|______________|______________6____||||*
 *|||

  | <fl0, fl1,>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) = <e14, e15> 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 *
 *) =

<e15>, and Z2 = [T, T 0] = Z2(T ) = <e15, e14, e25>. 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 = <e15>   P . Also, for x 2 {e14, e25}, *
 *[x, P ]

[x, T ] = <e15>, 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 ] = <e14, *
 *e15>, 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 ] = <e13, e14, e15>   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 ] =

<e13, e14> and e142=Fr(P ). Hence e142=[e13, P ], which implies P   <T 0, e12, *
 *e23,>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   <T 0, e12,*
 *>e45.

  We have now shown that


(a) either e122 P or P   <T 0, e12, e23,>e45;

(b) either e242 P or P   <T 0, e12,>e45; and (by symmetry)

(c) either e452 P or P   <T 0, e12, e34,>e45.


Assume e242=P . Then |P |   27 by (b), and e14, e252=Fr(P ). Set P0 = <Z2, e13,*
 * e35>

P . Since [e24, P ] = <e14, e25>, there are elements e12x, e45y 2 P for x, y 2 *
 *<e24> (by (b)

again), and <P0, e12x, e45y> = 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 = <T 0, e12,>e45, Fr(P ) = Z2, rk([e23, V ]) = 1;

 o P = <T 0, e12,>e34, Fr(P ) = <e14, e15>, rk([e23, V ]) = 3, rk([e45, V ]) = *
 *2;

 o P = <T 0, e12, e34e45>, Fr(P ) = <Z2, e35>, 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 ] = <e15>, 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) = <Z2, e24>   P . Set H = <Z2, e24> 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 <e23, e34>   T=H, and hence P is

T -conjugate to <H, e23, e34>. But for P = <H, e23, e34>, [e13, P ] = <e14> 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 )   <A1A2, e45> 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(<H, e35>) = 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 = <H, e23x, e34y,>e35for so*
 *me

x, y 2 <e13>.  Also, [e13, P ] = [e13, T ] = <e14, e15> 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  = <H, e23, e34,>e35.  In this case, Fr(P ) = <*
 *e24, e25>,

Z(P ) = H = <Fr(P ), e14,>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] = <e15>   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(<Z2, e24>).  There are three cases to consider:

(a) P \ Z2 = <e15, e14>, g = e25, h 2 e12.A1A2; (b) P \ Z2 = <e15, e25>, g = e1*
 *4,

h 2 e45.A1A2; (c) P \ Z2 = <e15, e14e25>, 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) = <Z2, e13e24, e35>.


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= <Z2, e13, e24*
 *,>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,
                 =<e13,e24,e35,e12,e45>            =<e23,e34>
                  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] = <e13>, [e34, S0] = <e35>, and [e23e34, S0] = <e13, e35>.  Fin*
 *ally, since

Z(S) = [S, S] = T 0=Z2, we must in all cases have rk((P \ T 0)=Z2) = 2.

  By Lemma 1.8, we must consider the following cases, where g 2 NT0(P )r P :


rk(T0=P0) = 1  :  If P0 6C T , then P   [T, T0] = <e13, e35,>Z2, and so e24z 2 *
 *P for

some z 2 <e13, e35>. Also, there are elements e12x, e45y 2 P0 for some x, y 2 T*
 * 0. Thus

e14= [e24, e45] and e25= [e24, e45] are in Z2, and Z2   Fr(P ) in this case.

  If P0 C T , then P0   [T, T0] = <Z2, e13, e35>.  So we can take g = e24.  Als*
 *o,

e23e34x =2P for any x 2 T0 (since (e23e34x)2 2 e24.[T, T0]); and so up to symme*
 *try we

can assume P=P0 = <e23>. Thus P = <Z2, e13, e35, e12x, e45y,>e23zfor some x, y,*
 * z 2 <g>.

In all cases, Z(P ) = Z, Z2(P )   <T 0, e45>, and [e24, Z2(P )]   <e15, e25>   *
 *Fr(P ).


rk(T0=P0) = 2 :  If rk(P=P0) = 1, then by symmetry, we can assume P=P0 = <e23>

or <e23e34>.  Also, since P0   [x, T0] for each x 2 T rT0, in particular for x *
 *= e23

or e34, we have e13, e35 =2P0.  Hence e13e35 2 P , and e15 = (e13e35)2 2 Fr(P )*
 *.  So

we can take g = e13.  If P=P0 = <e23>, then [e13, P ]   <e15>   Fr(P ).  Otherw*
 *ise,

P = <Z2, e13e35, e24x, e12e45,>e23e34for some x 2 <g>, Z(P )   Z2, Z2(P )   T 0*
 *, and so

[e13, Z2(P )]   <e15>   Fr(P ).

  If rk(P=P0) = 2, then up to symmetry, we can assume that [e23, T0] = <e13>   *
 *P0,

while [e34, T0] = <e35> =2P0.  Then P = <Z2, e13, e24x, e12y, e23z,>e34and g = *
 *e35, for

some x, y, z 2 <g>.  In all cases, Z(P )   <e15, e14> and Z2(P )   T 0, so [g, *
 *Z2(P )]

<e15>   Fr(P ).


rk(T0=P0) = 3, 4  :  Since rk((P \ T 0)=Z2) = 2, we have P0   T 0, and P0   [x,*
 * T0]

for all x 2 {e23, e34, e23e34}.  Thus e13, e35 =2P0, and so P0 = <Z2, e13e35, e*
 *24x> for

some x 2 <e13>.  Also, e23y, e34z 2 P for some y, z 2 T0, and [e13e35, e23y]   *
 *e25 and

[e13e35, e34z]   e14(mod <e15>). Thus Fr(P )   Z2 in all cases.


  We have now checked that in all cases, (1)holds with   = 1 or   = Z2(P ), and*
 * so

P is not semicritical.
6.2___Automorphisms_of_critical_subgroups___

  Recall that o 2 Aut(T5) denotes the transpose along the "back" diagonal compo*
 *sed

with A 7! A-1; i.e., the automorphism o(eij) = e6-j,6-i. Define `1, `2, _1, _2 *
 *2 Aut(T5)

by setting

           `1(e23) = e23e15,     _1(e12) = e12e35,   _1(e13) = e13e15e25,

letting `1 and _1 send the other generators eijto themselves, and setting `1 = *
 *o`2o-1

and _1 = o_2o-1.  Note that [e12e35, e23] = e13e15e25, which is the only hard s*
 *tep

when checking that _1 is an automorphism. These automorphisms can be visualized

pictorially as follows:
        |________________|||||||||____                        |________________*
 *|||||||||____                        |________________|||||||||____          *
 *       |_________-_______|||||||||____
        |________________|||||||||||||||||_|||||_||||_||||||||_|||*____________*
 *____||||||||||||||||||_|||||_||||_||||||||_|||________________||||||||||||||_*
 *|||||_||_||||_|||Q_H________________||||||||||||||_|||||_||_||||_||||
                                                                     ~         *
 *                                       |||||||||||||||||||||QHHj             *
 *         |||||||||||||||||||||JJ]AAK|6
   `1 =   |____________|||||||||||||_||_||||_||||||,     `2 =   |____________||*
 *|||||||||||_||_||||_||||||,     _1 =   _____________||||||_||_||||_|||QQs,   *
 *  _2 =   _____________||||||_||_||||_|||JA|.
            |________|||||||||_||||_|||                           |________||||*
 *|||||_||||_|||                           |________|||||||||_||||_|||         *
 *           |________|||||||||_||||_|||J
              |____|||_||                                           |____|||_||*
 *                                           |____|||_||                       *
 *             |____|||_||


  Let Aut0(T5)   Aut(T5) be the subgroup of automorphisms which send A1 to itse*
 *lf,

and set Out0(T5) = Aut0(T5)=Inn(T5). By Lemma 6.1, any automorphism of T5 either

sends the Ai to themselves or switches them, so Aut(T5) = Aut0(T5) o <o>.
36                     BOB OLIVER AND JOANA VENTURA

  Finally, for each i = 1, 2, 3, 4, set  i= OutGL5(2)(Qi) ~= 3.


Proposition 6.5. (a) Out 0(T5) = <`1, `2, _1,>_2~=C42. Hence |Out (T5)| = 25.

(b) For each i = 1, 2, 3, 4, Out(Qi) = O2(Out (Qi)). i, and O2(Out (Qi)) is ele*
 *mentary

    abelian.  If F is a saturated fusion system over T5 and Qi is F-essential, *
 *then

    Out(Qi) = ' i'-1 for some ' 2 O2(Out (Qi)) which extends to an automorphism

    of T5.


Proof.Set

   Q14= Q1 \ Q4 = E123;345= A1A2    and    R = E12;45= A1 \ A2 = Z(Q14)


for short. Let Out 0(Q14)   Out(Q14) be the subgroup of (classes of) automorphi*
 *sms

which send A1 to itself. In Steps 1 and 2, we describe Out(T5), Out(Q1), and Ou*
 *t(Q4)

by comparison with Out(Q14). Then in Step 3, we prove (b) for Q2 and Q3.


Step 1:   Each ff 2 Aut0(Q14) induces automorphisms of A1=R and A2=R, and any

automorphism which induces the identity on Q14=R lies in O2(Aut (Q14)) by Propo*
 *sition

1.1. We thus have a short exact sequence

 1 ---! O2(Out (Q14)) -----! Out 0(Q14) -----! Aut (A1=R) x Aut(A2=R) ---! 1,

where Aut(Ai=R) ~= 3, and where the last map is onto since Aut(A1=R) x Aut(A2=R)

is generated by the image of OutGL5(2)(Q14).

  If ff 2 Aut(Q14) induces the identity on Q14=R, then it also restricts to the*
 * identity

on R: since (ei3e3j)2 = eijfor i = 1, 2 and j = 4, 5.  Hence each ff 2 O2(Aut (*
 *Q14))

has the form ff(g) = g.bff(gR) for some bff2 Hom (Q14=R, R).  Thus O2(Aut (Q14)*
 *) ~=

Hom (Q14=R, R) ~=C162. Since Inn(Q14) ~=Q14=R has rank 4, O2(Out (Q14)) ~=C122.

  Now, R = Z(Q14) is a free F2[T5=Q14]-module. So if U is any of the groups T5,*
 * Q1,

or Q4, then Hi(U=Q14; Z(Q14)) = 0 for all i   1.  By Lemma 1.2, restriction to *
 *Q14

defines an isomorphism


               Out (U) ---Res--!~NOut(Q14)(Out U(Q14))=Out U(Q14).        (2)
                          =

This proves that Out(Qi) = O2(Out (Qi)). i for i = 1, 4, and that O2(Out (Qi)) *
 *is ele-

mentary abelian. If Qiis F-essential, then by Proposition 1.7, OutF (Qi) = ' i'*
 *-1 for

some ' 2 Out(Qi) which centralizes OutT5(Qi). Since Z(Qi) ~=C22is a free F2[T5=*
 *Qi]-

module, Lemma 1.2 again applies to show that ' extends to an automorphism of T5.


Step 2:  When U = T5, (2)restricts to an isomorphism


                   Out 0(T5) ---Res--!~CO2(Out(Q14))(<c12, c45>).
                                =

We now prove point (a) by describing this normalizer explicitly. Write O2(Aut (*
 *Q14)) =

A1 x A2, where A1 ~= Hom (A1=R, R) is the subgroup of automorphisms which are

the identity on A2 and on A1=R, and A2 ~=Hom  (A2=R, R) is the subgroup of auto-

morphisms which are the identity on A1 and on A2=R.  Set Ab1= A1=<c34, c35> and
bA2= A2=<c13, c23>; thus O2(Out (Q14)) = Ab1x bA2. The actions of c12 and c45 c*
 *learly

preserve this decomposition.

  We must identify the centralizer in Abiof Out T5(Q14) = <c12, c45>.  The acti*
 *on of

c45 on A1 fixes only those ff 2 A1 such that [ff, A1] = Im (bff)   <e15, e25>. *
 * Thus

CA1(c45) ~= Hom (A1=R, <e15, e25>) ~= C42.  Since the action of c45 on <c34, c3*
 *5> is free,

each element of Ab1fixed by c45lifts to an element of CA1(c45). Thus CAb1(c45) *
 *~=C32is
                  SATURATED FUSION SYSTEMS OVER 2-GROUPS                 37

the group (modulo <c35>) of all ff 2 Aut(Q14) such that ff|A2 = Id and ff(ei3) *
 *= ei3gi

for some gi2 <e15, e25> (i = 1, 2).

  Let flij2 CA1(c45) (for i, j 2 {1, 2}) be the automorphism which sends ei3to *
 *ei3ej5

and the other generators to themselves. Thus the following elements
         __________||                   __________||_-                _________*
 *_||        __________||_-

   fl21= ________||||||||__||*fl11fl12= ________||||||||__||HHHjfl22= ________|*
 *|||||||__||_-fl11fl22=_||||||||____||_-
           ______||||                     ______||||                    ______|*
 *|||  =c35    ______||||


form a basis for CA1(c45), and the first three form a basis for CAb1(c45). Also*
 *, the action

of c12fixes fl21, fixes fl11fl12modulo c35, and sends fl22to fl21fl22. Thus CAb*
 *1(Out T5(Q14))

has rank two, and is generated by the restrictions of `1 and _1 as defined abov*
 *e. To-

gether with the corresponding argument for A2, this finishes the proof that Out*
 *0(T5) ~=

C42is generated by the automorphisms `i and _i for i = 1, 2.


Step 3:   It remains to prove (b) for Q2 and Q3. We do this for Q3, and the res*
 *ult

for Q2 then follows via conjugation by o.  Consider automorphisms fii 2 Aut (Q3)

(i = 1, 2, 3, 4, 5) as described by the following diagrams:
    |________________|||||||||___                 ________________||||||||||___*
 *                 ________________||||||||||___________________||||||||||_____*
 *______________|-|||||||||___                                           -
    |________________|||||||||||||||||_|||||_||||||||_||||*________________||||*
 *||||||||||||||_|||||_||||||||_||||`________________||||||||||||||||||_|||||_|*
 *|||||||_||||@@@R_________________||||||||||||||||||_|||||_||||||||_||||Q_HHHj*
 *________________||||||||||||||||||_|||||_||||||||_|||||
                                                                               *
 *                                                    JJAAK]|6
fi1 = |____________||||||||||||||||_||||_||||fi2 =  ____________|||||||||||||||*
 *||_||||_||||fi3 =  ____________|||||||||||||||||_||||_||||@fi4_=___________||*
 *|||||||||||||||_||||_||||QQQsfi5_=___________|||||||||||||||||_||||_||||AJ|.
=`1|Q3  |________||||||_||                            ________|||||||_||       *
 *                     ________|||||||_||@@R=_1|Q3________|||||||_||=_2|Q3_____*
 *___|||||||_||J
          |____|||_||                                   ____||||_||            *
 *                       ____||||_||     ____||||_||     |____|||_||

Thus, for example, fi3(e12) = e12e45, fi3(e14) = e14e15e25, and fi3 sends all o*
 *f the

other generators eijto themselves.  We will show that these elements form a bas*
 *is

of O2(Out (Q3)) ~=C52.

  By Lemma 1.2, there is a short exact sequence


1 ---! H1(Q3=A1; A1) ---! Out (Q3) -Res--!NAut(A1)(Aut Q3(A1))=Aut Q3(A1) ---! *
 *1.


Automorphisms of A1 will be represented here as 6 x 6 matrices with respect to *
 *the

ordered basisi{e15,je25, e14, e24,ie13,je23}.iThus,jfor example, AutQ3(A1) is g*
 *enerated by
               J0 0        11      II0         I0I
the matrices   0J 0  (J = (01)),   0I0 , and   0I0 . By an easy computation, the
               00 J                00I         00I           i      j
                                                               AX Y
centralizer in Aut(A1) of AutQ3(A1) is the group of matrices   0A 0   for A, X,*
 * Y 2
                                                               0 0A
{(a0ba)| a, b 2 F2} and A invertible. After dividing out by AutQ3(A1), this giv*
 *es a group

of order 4 generated by the restrictions of fi1 and fi2.

  For g 2 Q3r A1, [g, A1] = <e15, e25> ~= C22if g 2 <A1, e35, e45>, while [g, A*
 *1] ~=

C32otherwise.  Hence for any fi 2 Aut (Q3), the induced automorphism of Q3=A1 =

<e12, e35, e45> ~=C32leaves <e35, e45> invariant. This shows that
                                                                     4
       NAut(A1)(Aut Q3(A1))=Aut Q3(A1) = Res <fi1, fi2, fi3,>fi4,~=3C2o  3.


Furthermore,


            Ker(ResA1) ~=H1(Q3=A1; A1) ~=H1( C22 ;   C32  ) ~=Z=2;
                                            e35,e45e13,e14,e15

this follows upon checking that there is a short exact sequence


              1 ---! <e13, e14, e15> -----! F2[C22] -----! F2 ---! 0


of F2[C22]-modules. Hence Ker(ResA1) is generated by fi5.

  In particular, <fi5> is a normal subgroup of Out(Q3) of order 2, and hence ce*
 *ntral. One

easily sees that the other fii commute in Out(Q3), and hence that O2(Out (Q3)) *
 *~=C52

with the fii as basis.
38                     BOB OLIVER AND JOANA VENTURA

  Now assume that Q3 is F-essential.  Then Out F(Q3) ~=  3 and contains c34.  By

Proposition 1.7, Out F(Q3) = fi 3fi-1 for some fi 2 Out F(Q3) which commutes wi*
 *th

c34; i.e., such that

                     fi 2 COut(Q3)(Out T5(Q3)) = <fi1, fi4,>fi5.

All of these extend to automorphisms of T5, and this finishes the proof of (b) *
 *for Q3.
6.3___Fusion_systems_over_T5_

  We are now ready to begin studying nonconstrained saturated fusion systems ov*
 *er

T5. We begin by looking at automorphisms of Q2 and Q3 in such fusion system.

Proposition 6.6. Let F be a nonconstrained saturated fusion system over T5. Then

Q2 and Q3 are both F-essential. Also, F is isomorphic to a fusion system F0 ove*
 *r T5

such that OutF0(Q2) =  2 and OutF0(Q3) =  3.


Proof.If Q3 is not F-essential, then by Proposition 6.4, F is generated by rest*
 *rictions of

automorphisms of Q1, Q2, Q4, all of which send A2 to itself. Hence each morphis*
 *m in F

extends to a morphism between subgroups containing A2 which sends A2 to itself,*
 * and

so A2 is normal in F. But A2 is centric in T5, and so this contradicts the assu*
 *mption

that F is nonconstrained.  Thus Q3 is F-essential; and by a similar argument, Q*
 *2 is

also F-essential.

  By Proposition 6.5(b), Out F(Q3) = ('|Q3) 3('|Q3)-1 for some ' 2 Out (T5).  So

upon replacing F by '-1F', we can assume that Out F(Q3) =  3. Then, by Propo-

sition 6.5(b) again, Out F(Q2) = (_|Q2) 2(_|Q2)-1 for some _ 2 Out (T5); and si*
 *nce

`1|Q2 = Id and _1 centralizes  2, we can assume that '2 2 <`2, _2>.  In particu-

lar, (_|Q3) 3(_|Q3)-1 =  3.  So if we set F0 = _-1F_, then Out F0(Q2) =  2 and

OutF0(Q3) =  3.


  We now study how the automorphisms of Q1 and Q4 fit with those of Q2 and Q3. *
 *In

the following proposition, 3 6 denotes a nonsplit extension with kernel of orde*
 *r 3 and

quotient group  6.

Proposition 6.7. Fix a nonconstrained saturated fusion system F over T5, and as*
 *sume

that OutF (Q3) =  3 and OutF (Q2) =  2. Then Q1 and Q4 are both F-essential. Al*
 *so,

when (i, j) = (1, 2) or (4, 1), either

 o OutF (Qi) =  i, AutF (Aj) ~= 3 x GL3(2), and AutF (Uj) ~=GL4(2); or

 o OutF (Qi) = (`j_j) i(`j_j)-1, AutF (Aj) ~=3 6, and AutF (Uj) ~=C32oGL3(2).


Proof.By Proposition 6.5(a), Out (T5) is a 2-group, and hence Out F(T5) = 1. So*
 * by

Proposition 6.4, F is generated by Inn(T5) together with AutF (Qi) for i = 1, 2*
 *, 3, 4.

  If neither of the subgroups Q1 nor Q4 is F-essential, then F is generated by *
 *Inn(T5)

together with  2, and  3, all of which leave U1 and U2 invariant. Thus U1 and U*
 *2 would

both be normal in F, which contradicts our assumption that F is nonconstrained.*
 * So

Q1 or Q4 is F-essential.

  For each i = 1, 4, if Qi is F-essential, then by Proposition 6.5(b), Aut F(Qi*
 *) =

'i i'-1ifor some 'i 2 Aut 0(T5).  (We drop "restricted to Qi" to simplify the n*
 *ota-

tion.) Since `1_1 commutes with  1 (mod c35), we can assume that '1 2 <`1, `2, *
 *_2>.

Similarly, we can assume that '4 2 <`1, `2, _1>.

  Step 1  We first prove that if Q1 is F-essential, then '1 2 <`2_2>; while if *
 *Q4 is

F-essential, then '4 2 <`1_1>.
                  SATURATED FUSION SYSTEMS OVER 2-GROUPS                 39

  First assume Q1 is F-essential, and consider OutF (Q1). As was done earlier, *
 *auto-

morphisms of A1 will be represented as 6 x 6 matrices with respect to the order*
 *edibasisj
                                                                              I*
 *0X
{e15, e25, e14, e24, e13, e23}. Let X be the 2 x 2 matrix such that '1|A1 =   0*
 *I0  . Set
                                                                              0*
 *0I
Y = (0110). Then Out F(Q3) and Out F(Q1) contain elements whose restrictions to*
 * A1

are
 iI 00j        i I0X j  iY 0 0j  iI 0X j  iY 0 XY +Y Xj  iI 0Zj  i Y 00 j
  0 0I    and    0I 0  . 0 Y 0  . 0 I0  =   0Y    0    =  0 I0  .  0Y 0   ,
  0 I0           00 I    0 0 Y    0 0I      00   Y        0 0I     0 0Y

respectively, where Z = X +Y XY -1. So if we set Q13= Q1\Q3 and let ff 2 AutF (*
 *Q13)

be the commutator of these elements, then
                           hiY 0 ZY j  i I00ji   iI ZZ j
                   ff|A1 =    0Y  0  ,   00I   =  0 I 0  .
                              00 Y       0I0      0 0 I

Thus ff is the identity on Fr(Q13) = <e15, e25>, on A1=Fr(Q13), and on Q13=A1. *
 *Since

these are characteristic subgroups of Q13, ff 2 O2(Aut F(Q13))   OutT5(Q13) by *
 *Propo-

sition 1.1. Hence Z = 0 or Z = I. But X = 0 if '1 2 <`2, _2> and X = (0100)othe*
 *rwise;

and in the second case we get Z = (0110).  This proves that '1 2 <`2, _2> if Q1*
 * is

F-essential; and also (by symmetry) that '4 2 <`1, _1> if Q4 is F-essential.

  Next assume that Q1 and Q4 are both F-essential;iwejshow in this case that '4*
 * 2
                                                 I0X               00    01
<`1_1>. Set Q14= Q1 \ Q4 = A1A2. Write '4|A1 =   0I0  , where X = (00), (00 ),
                                                 00I
(1010), or (1110), depending on whether '4 = Id, `1, _1, or `1_1.  Set Y  = (01*
 *10), as

before. Since '1 2 <`2, _2> where `2|A1 = _2|A1 = Id, OutF (Q1) and OutF (Q4) c*
 *ontain

the elements
           iY 0 0j             iI 0X j i 0I0 j i I0X j   i0 IX j
             0Y 0      and      0 I0  .  I00  .  0I 0  =  I 0X   .
             00 Y               0 0I     00I     00 I     0 0I

Hence their commutator
                    hi Y 0 0j   i 0IX ji  iI 0X+Y XY -1j
                       0 Y 0  ,   I0X   =  0 IX+Y XY -1
                       0 0Y       00I      0 0    I

is the restriction of an automorphism fi 2 AutF (Q14) which is the identity on *
 *Fr(Q14) =

A1 \ A2 and on Q14=Fr(Q14). Hence fi 2 O2(Aut F(Q14))   AutT5(Q14), and so X +

Y XY -12 {I, 0}. If X = (0100)or (1010), then X +Y XY -1= (0110), which is impo*
 *ssible;

and so we conclude that '4 2 <`1_1> in this situation.

  Now assume Q4 is F-essential and Q1 is not.  If '4 = `1, then F is generated *
 *by

Inn(T5) and AutF (Qi) for i = 2, 3, 4, and all of these automorphism groups lea*
 *ve U1

invariant. Thus U1 is normal in F in this case, which contradicts the assumptio*
 *n that

F is nonconstrained.

  Assume '4 = _1.  Set V  = <e23, e24, e25>   A1, and let A   Aut F(A1) be the

subgroup of those elements which leave V invariant. Let


              : A -----! Aut (V ) x Aut(A1=V ) = GL3(2) x GL3(2)


be the homomorphism induced by restriction and projection, where we identify Au*
 *t(V )

and Aut(A1=V ) with GL3(2) via the restrictions of the bases used above. Restri*
 *ctions

of elements of  3 and of c35 and c45 show that (M, M) 2 Im ( ) forieachjM 2 H,
                                                                 1
where H   GL3(2) is the subgroup of matrices with first column   00.  The above
                                                  ii 01 0j   i 011jj
computations (with X = (1100)) show that the pair    100001,   101001also lies *
 *in

Im( ). We claim that these elements generate GL3(2)2, which would imply 26 divi*
 *des

|Aut F(A1)|, contradicting the Sylow axiom.
40                     BOB OLIVER AND JOANA VENTURA

  To see this, note first that since Im( ) surjects onto each factor GL3(2), th*
 *e sub-

group of all g 2 GL3(2) such that (1, g) 2 Im( ) is normal. So if Im( ) is stri*
 *ctly con-

tained in GL3(2)2, then it must be of the form {(g, ff(g))} for some ff 2 Aut(G*
 *L3(2)).

Furthermore, ff|H = Id, while ff 6= Id. But this is impossible: ff cannot be an*
 * inner au-

tomorphism since CGL3(2)(H) = 1 (H ~= 4), while no outer automorphism of GL3(2)

can be the identity on a Sylow 2-subgroup.

  This finishes the proof that '4 2 <`1_1> in both cases (Q1 F-essential or not*
 *). As

usual, it then follows by symmetry that '1 2 <`2_2> if Q1 is F-essential.


  Step 2 Assume Q4 is F-essential, and AutF(Q4) =  4. If Q1 is not F-essential,*
 * then

F is generated by automorphisms of Q2, Q3, and Q4, all of which leave U1 invari*
 *ant.

Hence U1 is normal in F, which contradicts the assumption that F is nonconstrai*
 *ned.

  If Q1 is F-essential, then since '1|A1 = Id, the restriction to A1 of AutF (Q*
 *1) is equal

to that of  1.  So Aut F(A1) is generated by restrictions of automorphisms of  *
 *i for

i = 1, 3, 4. Thus AutF (A1) ~= 3 x GL3(2), where the first factor acts on the r*
 *ows of

A1 and the second factor on the columns.

  Similarly, if Q1 is F-essential and AutF (Q1) =  1, then Q4 is also F-essenti*
 *al and

AutF (A2) ~= 3 x GL3(2).


  Step 3  Now assume Q4 is F-essential and AutF (Q4) = (`1_1) 4(`1_1)-1. We will

show that AutF (A1) ~=3 6, and that Q1 is also F-essential. The corresponding r*
 *esult

for when AutF (Q1) = (`2_2) 1(`2_2)-1 then follows by symmetry.

  We first identify the subgroup of Aut (A1) generated by the restrictions of  *
 *3,

(`1_1) 4(`1_1)-1, and <c12> = Aut T5(Q1).  This time, we identify A1 with F34. *
 * Set

F4 = F2[!], and give A1 the structure of a F4-vector space by setting !e1j = e2j

and !e2j = e1je2j.  Fix the ordered F4-basis_{e15, e14, e13} for A1.  We thus i*
 *dentify

e15= (1, 0, 0), e25= (!, 0, 0), etc. Write ! = !2 = ! + 1 2 F4. Then  1 is gene*
 *rated

by diag(!, !, !) and the field automorphism OE2 2 Aut(F34);
                               Di 100j    i 100jE
                           3 =    011001,   001010;


while             Di    j    i    jE                  i   _j
                     110       010                     1 0!
              4 =    010001,   100001   and    `1_1 =  0 10   .
                                                       0 01
From this, we get
                                      Di 110j    i 01_!jE
                   (`1_1) 4(`1_1)-1 =    010001,   10_! .
                                                   001
                                                                           _
Consider the following six points in P_(F34):  ~1 = <(1,_!, 0)>, ~2 = <(1, !, 0*
 *)>, ~3 =

<(!, 0, 1)>, ~4 = <(!, 1, 1)>, ~5 = <(! , 0,>1), ~6 = <(! , 1,>1). These form a*
 *n "oval", in

the sense that no three of them lie in a projective line. By a direct check, th*
 *e above

generators permute these points in the following way:
                          i 10 0j   i 100j    i1 10j   i 01 _!j
                 OE2        010101    001010   001001    10 _!
                                                         00 1   .

            (1 2)(3 5)(4 6)(3 4)(5 6)(1 5)(2 3)(1 2)(4 6)(1 2)(3 6)


The first two and last two of these permutations generate the subgroup of eleme*
 *nts of

 6 which leave {1, 2} invariant, and hence this set of five permutations genera*
 *tes  6.

Since this extension of C3 by  6 is not split, this finishes the proof that Aut*
 *F(A1) ~=3 6

in this case.

  In particular, the element diag(!, !, !) is in AutF (A1). By the extension ax*
 *iom, this

extends to an element of AutF (Q1), and so Q1 must also be F-essential.
                  SATURATED FUSION SYSTEMS OVER 2-GROUPS                 41

  Step 4 It remains to look at AutF (Ui) for i = 1, 2. The groups  2 and  3 (to*
 *gether

with AutT5(U1)) generate the subgroup of all automorphisms of U1 which send e15*
 * to

itself.  If Out F(Q4) =  4, then this group sends U1 to itself, does not centra*
 *lize

e15, and hence we get Aut F(U1) = Aut (U1) ~= GL4(2).  Otherwise, if Out F(Q4) =

(`1_1) 4(`1_1)-1, then none of the elements in Out F(Q4)r OutT5(Q4) sends U1 to

itself, and so Aut F(U1) contains only the automorphisms preserving e15.  Simil*
 *arly,

OutF (U2) ~=GL4(2) if AutF (Q1) =  1, and OutF (U2) is the group of elements wh*
 *ich

preserve e15otherwise.


  We can now summarize these results in the following theorem.  The much more

difficult classification of simple groups with Sylow 2-subgroup UT5(2) is due t*
 *o Held

[He ], and is also shown in [A , Chapter 14].


Theorem 6.8. Every nonconstrained saturated fusion system over T5 is isomorphic*
 * to

the fusion system of one of the simple groups GL5(2), M24, or He.


Proof.The four subgroups Qi are all F-essential by Proposition 6.7. By Proposit*
 *ion

6.6, we can assume that Out F(Qi) =  i for i = 2, 3. Then by Proposition 6.7, t*
 *here

are just four possibilities for F, of which two are isomorphic via o. We refer *
 *to [He ],

and to [A , x40], for a description of the groups Aut G(Ai) when G = GL5(2), M2*
 *4,

or Held's group.  They show that F is isomorphic to the fusion system of GL5(2)*
 * if

AutF (A1) ~=AutF (A2) ~= 3x GL3(2) (if AutF (Qi) =  ifor i = 1, 4); F is isomor*
 *phic

to the fusion system of M24 if AutF (A1) ~= 3 x GL3(2) and AutF (A2) ~=3 6 or v*
 *ice

versa (if Aut F(Qi) =  i for i = 1 or i = 4 but not both); and F is isomorphic *
 *to

the fusion system of Held's group if Aut F(A1) ~= AutF (A2) ~= 3 6 (if Aut F(Qi*
 *) =

(`j_j) i(`j_j)-1 for (i, j) = (1, 2) and (4, 1)).



           7. Fusion systems over the Sylow subgroup of Co3
  We follow here the notation used in [LO ]. Fix elements Y, B 2 SL2(9) such th*
 *at Y

has order 8 and <Y, B> ~=Q16, and set A = Y 2.  In particular, Y 4= B2 = -I, and

<A, B> ~=Q8. Consider the groups

                                                               (12)
             S0 def=<Y, B>3=<(-I, -I, -I)>    and     S def=S0 o  C2,
                                                                   o

and let [[C1, C2, C3]] denote the class of (C1, C2, C3) in S0. Thus


                 o2 = 1  and   o[[C1, C2, C3]]o -1= [[C2, C1, C3]].


Write a1 = [[A, I, I]], a2 = [[I, A, I]], a3 = [[I, I, A]], b 1= [[B, I, I]], b*
 * 2= [[I, B, I]],

b3 = [[I, I, B]], c = [[Y, Y, Y ]], and zi= a2i. Finally, set


                      T *= <a1, a2, a3, c, b1, b2,>b3, oS :


a group of order 210. For later reference, we list the following relations in T*
 * *:


 zi= a2i= b2i= [ai, bi],     z1z2z3= 1,     [ai, bj] = 1 = [ai, aj] = [b i, bj]*
 *  (i 6= j);


     c2= a1a2a3,       [c, ai] = 1,       cbic-1= aibi,       bicb-1i= a-1ic;


     oco -1= c,     oaio-1 = aoe(i),   obio-1 = boe(i)(where oe = (1 2) 2  3).
42                     BOB OLIVER AND JOANA VENTURA

  The embedding of T *as a Sylow 2-subgroup of Spin7(3) is described in [LO , x*
 * 2]. It

follows essentially from the identifications Spin3(q) ~=SL2(q) and Spin4(q) ~=S*
 *L2(q) x

SL2(q) (applied with q = 3 or q = 9); together with the inclusions

               Spin3(3) xC2 Spin4(3)   H   Spin3(9) xC2 Spin4(9)

for some H   Spin7(3) of odd index which contains Spin3(3) xC2 Spin4(3) with in*
 *dex

2. Note that Q8 2 Syl2(SL2(3)) and Q162 Syl2(SL2(9)).

  We now list some of the subgroups of T *which play an important role in what

follows. First consider the following subgroups:

                        R0 = <a1, a2, a3, b1, b2,>b3

                        R1 = <a1, a2, a3, b1, b2,>b3, c

                        R2 = <a1, a2, a3, b1, b2,>b3, o

                        R3 = <a1, a2, a3, b1, b2, b3,>co.


Thus T *=R0 ~=C22, and R1, R2, and R3 are the three subgroups of index two in T*
 * *

which contain R0. Also, Z(R0) = Z(R1) = <z1, z2>, while Z(Ri) = <z3> = Z(T *) f*
 *or

i = 2, 3.

  Next consider the following family of subgroups:

     Q = <z1, a1a2, a3, b1b2, b3,>o= <a1a2, b1b2> xC2 <a3, b3> xC2 <z1, o> ~=21*
 *+6+

    H1 = <Q , c>        H2 = <Q , a1c>        H3 = <Q , a1>

    R4 = <a1, a2, a3, b1b2, b3,>c,.o

Thus H1, H2, and H3 are the three subgroups of index two in R4 which contain Q .

Also, H3 C T *, while H1 and H2 are conjugate in T *and NT*(H1) = NT*(H2) = R4.

These three subgroups will be seen to be permuted transitively by Out(R4).

  Geometrically, R4 is the subgroup of elements of Spin7(3) which leave all sub*
 *spaces

invariant under some decomposition of F73as an orthogonal direct sum V1 V2 V3 V4

where dim(Vi) = 2 for i = 1, 2, 3. Similarly, Q is the subgroup of elements of *
 *Spin7(3)

which fix the elements of an orthonormal basis up to sign.

  We will show later that the only critical subgroups of T *are R1, R2, R3, R4,*
 * H1,

and H2.

  The following abelian subgroup of T *will also play a role in what follows:

                           A  = <a1, a2, a3,>c~=C34.


Lemma 7.1. Assume P   T *, |P | = 27, and P=[P, P ] ~=C62(i.e., |Fr(P )| = 2). *
 *Then

P = Q .


Proof.Set A  = <a1, a2, a3,>c~= C34, and let A 0 = <z1, z2, a1a2a3> be its 2-to*
 *rsion

subgroup. By the above relations, A  is normal in T *, and T *=A  = <b 1, b2, b*
 *3,>o~=

D8 x C2. Since |T *=A | = 24, P \ A is a normal subgroup of P of order at least*
 * 23. If

|P \ A| = 23, then P=(P \ A) ~=T *=A  ~=C2x D8, so Fr(P ) \ A = 1, and P \ A = *
 *A 0

since it cannot have 4-torsion.  Hence [P, P ]   CT*(A 0) = <A , b1b2b3>, and t*
 *his is

impossible since the class of b1b2b3 is not in the commutator subgroup of T *=A*
 * .

  It follows that |P \A |   24. In particular, [P, P ]   A (since P \A  is not *
 *elementary

abelian); and P   A 0and |P \ A| = 24 since P contains no subgroup C24. So P A=A

is an elementary abelian subgroup of order 23 in T *=A  ~=D8x C2. So either P A*
 *=A  =

<b 1, b2,>b3(and P A = R1), or P A=A  = <b 1b2, b3,>o(and P A = R4). In either *
 *case,

b3 2 P , and so [P, P ] = [b 3, a1a2a3] = <z3>, and P=A 0~= C42.
                  SATURATED FUSION SYSTEMS OVER 2-GROUPS                 43

  If P A = R1, then b1g 2 P for some g 2 A , so [b 1g, a1a2a3] = z12 [P, P ], a*
 *nd this

is impossible. Thus P A = R4.

  Consider the quotient group

R4=A 0= R4=<z1, z2, a1a2a3> = <a1, a2,>ox<a3><c, b3> x <b 1b2b3> ~=D8 xC2 D8 x *
 *C2.

We have already seen that P=A 0~= C42. Hence a3, b1b2b3 2 P , and P intersects *
 *each

of the two subgroups D8 with order 4. But a12=P (since a21= z12=[P, P ]) and c *
 *=2P

(since c2= a1a2a32=[P, P ]), so P=A 0= <o , b3, b1b2>, and thus P = Q .
7.1___Determining_the_critical_subgroups__

  We start as usual by reducing to the case of subgroups of index 2 in their no*
 *rmalizers.


Lemma 7.2. 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, rk([g, V ])   2 for each g 2 N(P )r P with g2*
 * 2 P .

Moreover, rk([g, V ]) is independent of the choice of g 2 N(P )r P with g2 2 P .

  Since P is centric in T *, Z(T *) = <z3>   P . Also, since [x, P ]   [x, T *]*
 *   <z3> for

x 2 Z2(T *) = <z1, a3> ~=C2 x C4, Z2(T *)   P . In particular, z3= a232 Fr(P ).

  Since [a1a2, P ]   [a1a2, T *] = <z1, z3>, rk([a1a2, V ])   1, and hence a1a2*
 * 2 P .

Similarly, [b 3, P ]   [b 3, T *] = <a3> implies b3 2 P .  Set T0 = <a1a2, a3, *
 *z1,>b3  P .

Then |T0| = 25, T0 C T *, and

                    T *=T0 = R4=T0 o <b 1> ~=   C42   o C2.
                                             <a1,b1b2,c,o>b1


  By Proposition 3.4(a), all involutions in N(P )=P are central. Since N(P ) = *
 *R4 or

T *, and b1P will be an involution if N(P ) 6= R4, this shows that

           either N(P ) = R4,     or P   R4,     or     a1, b1b22 P .     (1)


  If a1 2 P , then z1 = a212 Fr(P ), and so b 1b2 2 P  and P  C T *.  Set T1 =

<T0, a1, b1b2>; thus |T1| = 27 and so [P :T1]   2.  If b1 2= P , then since [T1*
 *, b1]

<z1, z3>   Fr(P ), rk([b 1, V ])   1.  Then P  = <T0, b1> = R0, Fr(P ) = <z1, z*
 *3>,

rk([o , V ]) = 2, and rk([c, V ]) = 3. So also in this case, P cannot be critic*
 *al.

  Thus a1 2 N(P )r P .  Now, Z(T *=T0) = <a1, b1b2>   N(P )=T0, and [a1, T *] =

[b 1b2, T *] = <z1, a1a2>.  Since rk([a1, V ])   2, we must have [a1, P ] = <z1*
 *, a1a2> and

Fr(P ) \ <z1, a1a2> = <z3>.

  Consider the subgroup R4 = <T0, a1, b1b2, c,>o. Since [a1a2, b1] = z1, [a1a2,*
 * R4] =

<z3>   Fr(P ), and z12=Fr(P ), we have P   R4. So by (1), N(P ) = R4.

  If b 1b2 2 P , then P    <T0, b1b2, o>, since otherwise z1, a1a2, or z1a1a2 i*
 *s in

[b 1b2, P ]   Fr(P ).  Hence either P = <T0, b1b2> or P = <T0, b1b2, o> = Q .  *
 *Both of

these are normal in T *, which contradicts (1).

  If a1b1b2 2 P , then P   <T0, a1b1b2, a2co>, since otherwise

                   [a1b 1b2, P ] \ <z1, a1a2>   Fr(P ) \ <z1, a1a2>

is not contained in <z3>.  Hence either P = <T0, a1b1b2> or P = <T0, a1b1b2, a2*
 *co>.

Both of these are normal in T *, which again contradicts (1).

  Thus (P=T0) \ Z(T *=T0) = 1, and Z(T *=T0)   N(P )=T0. Also, N(P )=P ~=Ck2im-

plies rk([a1a2, P=Fr(P )])   k (Proposition 3.4(d)), which we already know is i*
 *mpossible
44                     BOB OLIVER AND JOANA VENTURA

for k   3. Hence

                   P = <R0, cx, oy> = <a1a2, a3, z1, b3, cx,>oy

for some x, y 2 <a1, b1b2>. Also, since b1cb-11= a-11c, it suffices to consider*
 * the case

where x 2 <b 1b2>. Then one of the following happens:


 o y 2 <b 1b2>, [b 1b2, P ] = <a1a2>, so rk([b 1b2, V ])   1 and P is not criti*
 *cal.

 o y = a1, (o a1)2 = a1a22 Fr(P ), and so P is not critical.

 o y = a1b1b2, (o a1b1b2)2 = a1a2z22 Fr(P ), and so P is not critical.


This finishes the proof.


  It remains to handle the subgroups of T *of index two in their normalizer.


Proposition 7.3. The only critical subgroups in T *are R1, R2, R3, R4, H1, and *
 *H2.


Proof.The subgroups Ri (i = 1, 2, 3, 4) and Hi (i = 1, 2) are seen to be critic*
 *al by the

descriptions of their automorphism groups in Lemmas 7.4, 7.6, and 7.7.

  Fix a critical subgroup P   T *. By Lemma 3.3, for any g 2 N(P )r P , there i*
 *s no

characteristic subgroup     P such that

                  [g, P ]    .Fr(P )    and     [g,  ]   Fr(P ).          (2)


  By Lemma 7.2, |N(P )=P | = 2.  Also, z3 2 P since P is centric.  Since [z1, T*
 * *] =

[a3, T *] = <z3>   P , z1, a32 N(P ). Thus at least one of the elements z1, a3,*
 * or z1a3is

in P . If z12=P , then z3= a23= (z1a3)2 2 Fr(P ), so [z1, P=Fr(P )] = 1, and so*
 * z12 P .

  Thus <z1, z3>   P . Also, a1a2, a32 N(P ) since they are central in T *=<z1, *
 *z3>, and

hence at least one of the elements a1a2, a3, a1a2a3is in P .

  In Step 1, we show that if P   R1, then <z1, z3>   Fr(P ). In Step 2, we show*
 * that

a1a2, a32 P . We then handle the remaining cases where P is not normal in T *in*
 * Step

3, and the cases P C T *(hence [T *:P ] = 2) in Step 4.


Step 1:  Assume P   R1 = <ai, bi,>c; we want to show that a1, a2, a32 P , and h*
 *ence

<z1, z3>   Fr(P ). This is clear if P = R1, so we assume P   R1. Then NR1(P )=P*
 * 6= 1,

so N(P )   R1 since we are assuming |N(P )=P | = 2. Thus P is also critical in *
 *R1. We

will show, for all critical subgroups P   R1 with |NR1(P )=P | = 2, that <z1, z*
 *3>   Fr(P ).

  The aiare all central in R1=<z1, z3>, so <a1, a2, a3>   N(P ), and P \<ai> ha*
 *s index at

most two in P . If a1a2, a1a32 P , then (a1a2)2 = z3and (a1a3)2 = z2are both in*
 * Fr(P ),

and we are done. So we can assume one of the aiis in P . Since Aut(R1) permutes*
 * the

aitransitively, we can assume without loss of generality that a1, a2a32 P .

  Assume h 2 P is such that [h, a3] 6= 1; thus [h, a3] = z3. Then

                     [h, a2a3] = [h, a2].a2[h, a3]a-122 <z2>.z3.

In particular, no such h can be central in P , and so [a3, Z(P )] = 1.  Since [*
 *a3, P ]

<z3>   Z(P ), (2) is satisfied (with g = a3 and   = Z(P )), and so P  cannot be

semicritical.

  Thus z1, z32 Fr(P ) whenever P   R1. Hence [ai, P ]   <zi>   Fr(P ) for each *
 *i; and

thus ai2 P since it normalizes P .


Step 2:  If P   R1, then z1, z32 Fr(P ) by Step 1. If P   R1, then ox 2 P for s*
 *ome

x 2 R1 = CT*(z1), and [o x, z1] = z3 2 Fr(P ).  Thus in all cases, z3 2 Fr(P ).*
 *  Since

[a3, P ]   [a3, T *] = <z3>, this implies that a32 P .
                  SATURATED FUSION SYSTEMS OVER 2-GROUPS                 45

  Set T1 = <z1, z3, a3>   P , and consider the quotient group
                                  i            j
            T *=T1 = R0=T1 o C22~=    C42  o C22 x C2.           (c0= b1b2c)
                                   a1,a2,b1,b2c0,o b3


We want to apply Lemma 1.8, with S = T *=T1 and S0 = R0=T1 = <ai, bi> ~= C52.

Set P0 = P \ R0. Note that [c, S0] = <a1, a2>, [o , S0] = <a1a2, b1b2>, and [co*
 * , S0] =

<a1a2, a1b1b2> all have rank 2. Also, since P0 cannot contain all of them (sinc*
 *e P 6C T *),

either b3 or a1a2b3 is in P0.

  By Lemma 1.8, we must consider the following cases, where g 2 NR0(P )r P :


rk(T0=P0) = 1  :  If P   R0, then a1, a2, a32 P by Step 1. Hence [c, R0]   P0, *
 *so

c2 N(P ), and |N(P )=P |   4.

  If rk(P=P0) = 1, then P0 6C T *, since [S, S] = <[S, S0], [c,>o]= [S, S0].  H*
 *ence

P0   [g, R0] for g 2 P rP0 while P0   [g, R0] when g 2 T *rP R0. This leaves us*
 * with

the following possibilities:


 o P = <T1, b3x, a1, a2, b1,>cyand g = b1b2, for some x, y 2 <g>. In all cases,*
 * z1 =

      a 212 Fr(P ).  Also, Z(P ) = <zi, a2> if x = y = 1 and Z(P ) = <zi> other*
 *wise.

      Hence Z2(P )   <zi, ai, b3>, so [g, P ]   Z2(P ), and [g, Z2(P )]   <zi> *
 *  Fr(P ).

 o P = <T1, b3x, a1a2, b1b2, b1,>oyand g = a1, for some x, y 2 <g> where (x, y)*
 * 6=

      (1, 1). In all cases, z1= b212 Fr(P ). Since (o a1)2 = a1a2and [o , b3a1]*
 * = a2a-11,

      [g, P ]   <z1, a1a2>   Fr(P ) when x = a1or y = a1.

 o P  = <zi, a1a2, a3, b1, b2,>b3,aond g = a1.  Then Z(P ) = <z3>, and Z2(P ) =

      <zi, a3, b1b2,>b3.  Hence Z(Z2(P )) = <zi, b1b2> is also characteristic, *
 *and so

      is its 2-torsion subgroup <zi>.  Hence   = CP(<zi>) = <zi, a1a2, a3,>biis*
 * also

      characteristic, and [g, P ]    Fr(P ) and [g,  ]   Fr(P ).

 o P = <T1, b3x, a1a2, a1b1b2, b1,>coyand g = a1, for some x, y 2 <g> where (x,*
 * y) 6=

      (1, g).  Then Z(P ) = <z3>, and Z2(P ) = <zi, a3> if y  = 1, and Z2(P ) =

      <zi, a3, b1a2b2> if y = a1.  Also, Fr(P ) always contains b21= z1, and [g*
 *, P ] =

      <z1, a1a2>.  So [Z2(P ), g]   <z1> in all cases.  By direct computations,*
 * a1a2 2

      Fr(P ) if x = y, while (x, y) = (a1, 1) implies Fr(P ) = <zi, a1a2a3, a2b*
 *1b2>. Thus

      [g, P ]   Z2(P ).Fr(P ) in these cases.

 o P = <zi, a1a2, a3, b1, a2b2, b3,>coa1and g = a1.  Then Fr(P ) = <zi, a3, b1a*
 *2b2>

      has 2-torsion subgroup  0 = <zi, b1a2b2a3>. Set  = 0 = Z(P= 0); then   =

      <zi, a1a2, a3, a1b1b2, b1b3> is characteristic in P , [g, P ]    , and [g*
 *,  ] = <z1>

      Fr(P ).


rk(T0=P0) = 2 :  If rk(P=P0) = 1, then for each x 2 {c, o, co}, P0   [x, R0]. T*
 *his

implies a1a2 =2P , and so again P   R1 = <R0, o> by Step 1. This leaves us with*
 * the

following possibilities, where in all cases, g = a1a2and x 2 <g>:


 o P = <T1, b1, b2, b3x,>oor P = <T1, b1, a2b2, b3x, b1b2co>

 o P = <T1, b1a1a2, b2a1a2, b3x,>oor P = <T1, b1a1a2, a1b2, b3x, b1b2co>.


In all cases, [a1a2, P ]   <z1, z3>   Z2(P ). When b1 2 P , then z1= b212 Fr(P *
 *), and so

[a1a2, P ]   Fr(P ). In the other two cases, Z(P ) = <z3> and Z2(P )   <ai, b1b*
 *2, b3>, so

[a1a2, Z2(P )] = <z3>   Fr(P ).

  If rk(P=P0) = 2, then P0   [x, R0] for some x = c, o, or co. This leaves the *
 *following

possibilities:
46                     BOB OLIVER AND JOANA VENTURA

 o P = <T1, a1, a2, b3x, cy,>ofor some x, y 2 <g>, g = b1b 2.  In all of these *
 *cases,

      z1 = a21and a1a-12= [a1, o] are both in Fr(P ), and so [g, P ]   Fr(P ).

 o P = <T1, a1a2, b1b2, b3x, c,>oyfor some x, y 2 <g>, g = a1.  Then Fr(P ) alw*
 *ays

      contains [c, b1b2] = a1a2.  Since (b 3a1)2 = z2 and [o a1, b1b2] = z2, [g*
 *, P ]

      <z1, a1a2>   Fr(P ) if either x = a1or y = a1. If x = y = 1, then P = H1.

 o P = <T1, a1a2, a1b1b2, b3x, cy,>ofor some x, y 2 <g>, g = a1. In all cases, *
 *Z(P ) =

      <z3> and Z2(P ) = <zi, a3, a1a2>. Hence [g, P ]   Z2(P ) and [g, Z2(P )] *
 *= 1.


rk(T0=P0) = 3, 4  :  This can only occur when rk(P=P0) = 2 and P0 = <T1, b3x>

for some x 2 <a1a2>; and hence when P = <T1, b3x, b1b2c,>oup to conjugacy. Then

[a1a2, P ] = <z3>   Fr(P ).


  Thus in all cases when P is not conjugate to H1, the hypotheses of Lemma 3.3 *
 *are

satisfied (with   = Z2(P ) or   = 1 in most cases), and so P is not semicritica*
 *l.


Step 3:   It remains to handle the case where P C T *; i.e., where P has index *
 *2 in

T *. Thus P contains

                          [T *, T *] = <a1, a2, a3, b1b2>.

Also, Fr(P )   L3(T *) = [[T *, T *], T *] = <z1, a1a2>:  z1= a212 Fr(P ), and *
 *a1a2

[a1, o]   [a1, oc]   [b 1b2, c] (mod <z1, z3>) (and one of the elements c, o, o*
 *r oc must

be in P ). If Fr(P ) = [T *, T *], then any g 2 T *rP acts trivially on P=Fr(P *
 *), and so P

is not (semi)critical. We claim that this is the case for all of the index two *
 *subgroups

in T *, except for the four subgroups Ri for i = 1, 2, 3, 4.

  Write x1 = b3, x2 = c, x3 = b1, and x4 = o. This forms a basis for T *=[T *, *
 *T *], ar-

ranged in such a way that [xi, xj] = 1 if |j-i|   2, and such that the [xi, xi+*
 *1] form a ba-

sis for [T *, T *]=L3(T *). From this, one checks easily that Fr(P ) = [T *, T *
 **], except possi-

bly when P=[T *, T *] is generated by three of the xi, or when P=[T *, T *] = <*
 *x1, x3, x2x4>

or <x1x3, x2, x4>. Of these six cases, <x1, x2, x3>.[T *, T *] = R1, <x1, x2, x*
 *4>.[T *, T *] = R4,

<x1, x3, x4>.[T *, T *] = R2, and <x1, x3, x2x4>.[T *, T *] = R3. In the remain*
 *ing two cases,

P=[T *, T *] = <x2, x3, x4> and P=[T *, T *] = <x1x3, x2, x4>, [x2, x1] = a3 = *
 *c2(a1a2)-1 is

in Fr(P ), and hence Fr(P ) = [T *, T *].
7.2___Automorphisms_of_critical_subgroups___

  Define fl 2 Aut(R1) of order 3 by setting fl([[R, S, T ]]) = [[T, R, S]] for *
 *R, S, T 2 Q16.

Thus fl(ai) = ai+1and fl(b i) = bi+1 (with indices taken mod 3), and fl(c) = c.*
 *  Let

fii (i = 1, 2, 3, 4) and fi0i(i = 1, 2, 3) be the automorphisms of R1 which act*
 * via the

identity on <a1, a2, a3> and are otherwise as described in the following table,*
 * where we

also describe the extension to T *of each fii:
           ______________________________________________________
           |  g f|i|1(g)f|i2(g)f|i3(g) f|i4(g) f|i0(g)f|i0(g)f|i0(g) |
           |_____|_|_____|______|_______|_______|_1____|_2____|_3__  |

           | b1 | |b1 | b1  |  z3b1  | z 3b1  | b1  | b 1  |z3b1 |
           |____|_|___|_____|________|________|_____|______|_____|
           | b2 | |b2 | b2  |  z1b2  | z 1b2  | b2  |a 2b2 | b2  |
           |____|_|___|_____|________|________|_____|______|_____|
           | b3 | |b3 |a3b3 |   b3   | z 2b3  | b3  | b 3  |z2b3 |
           |____|_|___|_____|________|________|_____|______|_____|
           |  c |z|3c | c   |   c    |   c    | z2c |  c   | c   |
           |____|_|___|_____|________|________|_____|______|_____|
           | o  | |o  | o   |z1a1a2o |a1a2a3o ||
           |____|_|___|_____|________|________||

Note that flfi0ifl-1 = fii for all i = 1, 2, 3, while [fl, fi4] = 1.
                  SATURATED FUSION SYSTEMS OVER 2-GROUPS                 47

Lemma 7.4. Consider the subgroups Qi = <ai, bi>   T *(i = 1, 2, 3).  Then Q1 ~=

Q2 ~=Q3 ~=Q8, R0 = Q1Q2Q3, the Qi commute pairwise with each other, and the

inclusions Qi  R0 define an isomorphism R0 ~=(Q8)3=C2. Furthermore, the followi*
 *ng

all hold.


(a) If P   T *and P ~=R0, then P = R0.

(b) Each automorphism of R0 permutes the three subgroups QiZ(R0).

(c) Out(R0) ~= 4o  3, where the subgroup ( 4)3 is the group of automorphisms wh*
 *ich

    leave each QiZ(R0) invariant. The identification of  4 with the group of au*
 *tomor-

    phisms of QiZ(R0) ~=Q8 x C2 which are the identity on its center is induced*
 * by

    the action of this automorphism group on the set of the four subgroups of Q*
 *iZ(R0)

    isomorphic to Q8.

(d) Out(R1) = <fi1, fi2, fi3, fi4, fi01,>fi02,ofi03<fl, co> ~= C72o 3.  The con*
 *jugation action

    in Out(R1) of <fl, co> ~= 3 on <fii, fi0i> ~=C72is the following: [co, fii]*
 * = 1 for all i;

    [fl, fi4] = 1; and for i = 1, 2, 3, fl-1fiifl-1 = fiifi0i. flfi0ifl-1 = fii.

(e) For i = 2, 3, the restriction map
                                                        (
                  ~=                      ffi              4 x  4  if i = 2
      Out (Ri) -----! NOut(R0)(Out Ri(R0)) Out Ri(R0) ~=
                                                           4 x C22 if i = 3


    is an isomorphism.

(f) The restriction map

                          ~=                      ffi            4
             Out (T *) -----! NOut(R1)(Out T*(R1)) Out T*(R1) ~=C2

    is an isomorphism. Hence

                        Out(T *) = <ff1, ff2, ff3,>ff4~=C42

    where ffi2 Aut(T *) are automorphisms such that ffi|R1 = fii.


Proof.The first statements _ that Q1 ~=Q2 ~=Q3 ~=Q8, [Qi, Qj] = 1 for i 6= j, a*
 *nd

R0 = Q1Q2Q3 ~=(Q8)3=C2 _ are clear from the presentations Qi = <ai, bi> of these

subgroups.


(a,b)  Let P   T *be any subgroup isomorphic to Q8 x Q8.  Set A  = <ai, c> ~=C3*
 *4.

The image P A=A  of P in T *=A  ~=<b 1b2b3> x D8 is elementary abelian, and can*
 *not

contain b1b2b 3since all elements of the coset b1b2b 3.A have order 2.  Thus P *
 *A=A

has rank 2, and P \ A ~= C24.  Also, P A=A  contains b1b2.A or b3A  (the two ot*
 *her

central involutions in T *=A ), CA (b 1b2) = <z1, z2, a3> and CA (b 3) = <a1, a*
 *2> both have

Z(R0) = <z1, z2> as their 2-torsion subgroups, and this shows that Z(R0)   Z(P *
 *) (and

hence Z(R0) = Z(P )). So P   CT*(Z(R0)) = <A , b1, b2,>b3= R1. Furthermore,

      R1=Z(R0) = <a1, a2, a3, c, b1, b2,>b3=Z(R0) = (R0=Z(R0)).<c> ~=C62oC2,

where c acts on R0=Z(R0) ~=C62with fixed subgroup <a1, a2, a3> of rank 3. In ot*
 *her

words, any subgroup C42in R1=Z(R0) is contained in R0=Z(R0), and this shows that

P   R0. In particular, this finishes the proof of (a): R0 is the unique subgrou*
 *p of T *

with its isomorphism type.

  Among the 63 involutions in R0=Z(R0) ~=C62, the 9 cosets represented by the e*
 *le-

ments ai, bi, and aibi(i = 1, 2, 3) all lift to elements g 2 R0 of order 4 with*
 * centralizer

CR0(g) ~=C4xC2(Q8x Q8), the 27 cosets represented by elements xiyjfor i 6= j li*
 *ft to

elements g 2 R0 of order 4 with centralizer CR0(g) ~=C4xC2Q8x C2, and the 27 co*
 *sets
48                     BOB OLIVER AND JOANA VENTURA

represented by elements x1y2z3 lift to elements g 2 R0 of order 2.  Hence any a*
 *uto-

morphism of R0 preserves each of these three sets, and thus permutes the subgro*
 *ups

QiZ(R0). This proves (b).


(c)  By (b), any automorphism of R0 permutes the three subgroups Qi.Z(R0) for

i = 1, 2, 3. The image of Out(R0) under the projection to Aut(R0=Z(R0)) is thus*
 * the

group of automorphisms of C22x C22x C22which permute the three factors C22, and*
 * this

is isomorphic to  3 o  3.  The group of automorphisms which induce the identity*
 * on

R0=Z(R0) (and hence also on Z(R0)) is isomorphic to Hom (R0=Z(R0), Z(R0)) ~=C12*
 *2,

and this group contains Inn(R0) ~=R0=Z(R0) ~=C62. We thus have an extension

               1 ---! C62-----! Out (R0) -----!  3 o  3 ---! 1;

and thus Out (R0) ~=(C22o 3)3 o  3 ~= 4 o  3.  The identification of each facto*
 *r  4

with a subgroup of Aut(QiZ(R0)) can be seen via its permutation action on the s*
 *et of

all four subgroups of QiZ(R0) ~=Q8 x C2 which are isomorphic to Q8.


(d,e)  Assume R0   H   T *and [H:R0] = 2, and write H = R0.<x> where x2 2 R0.

By Lemma 1.2, there is an exact sequence
                                                            ffi
  1 ---! H1(<x>; Z(R0)) -----! Out (H) -----! NOut(R0)(<cx>) <cx>

                                                     -----!  H2(<x>; Z(R0)).


  When x = c, then [x, Z(R0)] = 1, and hence H1(<x>; Z(R0)) ~= (Z=2)2.  The

corresponding elements of Out (H) are fi1 and fi01, as described above.  Also, *
 *cx 2

Out(R0) is the diagonal element (t, t, t) 2 ( 4)3 for some transposition t 2  4*
 *.  So

NOut(R0)(cx) ~= C22o  3, and NOut(R0)(cx)=<cx> ~= C52o 3.  These are the automo*
 *r-

phisms fii (i = 2, 3, 4), fi0i(i = 2, 3), fl, and co.  The subgroup of Out (R1)*
 * generated

by the seven automorphisms fii and fi0iis abelian: it is isomorphic to a subgro*
 *up of

H1(R1=<a1, a2, a3>; <a1, a2, a3>) by Lemma 1.2 (applied with H = <a1, a2, a3>).*
 *  Also,

it has exponent two since (fi2)2 = ca3, (fi02)2 = ca2, and the others have orde*
 *r 2 in

Aut(R1).

  When x = o, then cx(z1) = z2and cx(z2) = z1. Thus Hi(<x>; Z(R0)) = 0 for i = *
 *1, 2.

Also, cx 2 Out(R0) is the element which exchanges two of the factors  3, so

                    Out (R2) ~=NOut(R0)(cx)=<cx> ~= 4 x  4

by Lemma 1.2 again.

  When x = co, then cx|Z(R0)= co|Z(R0), and so again Hi(<x>; Z(R0)) = 0 for i =*
 * 1, 2.

Also, cx 2 Out(R0) is the product of the element (s, s, s) (for s 2  3 of order*
 * 2) with

the transposition of two of the factors  3. This shows that

                    Out (R3) ~=NOut(R0)(cx)=<cx> ~= 4 x C22.


(f)  We have T *=R1 = <o > ~= C2, where o acts on Z(R1) = Z(R0) = <z1, z2> by

exchanging z1 and z2.  Thus Hi(T *=R1; Z(R1)) = 0 for i = 1, 2; and by Lemma 1.*
 *2,

the restriction map induces an isomorphism

                  Out (T *) ~=NOut(R1)(Out T*(R1))=Out T*(R1).

Furthermore, NOut(R1)(Out T*(R1))=Out T*(R1) = <fi1, fi2, fi3,>fi4~=C42, fii = *
 *ffi|R1 by

definition, and so Out(T *) = <ff1, ff2, ff3,>ff4~=C42.


  Let Aut0(R0) C Aut(R0) be the subgroup of those elements which are the identi*
 *ty

on Z(R0).  Thus Out 0(R0) def=Aut0(R0)=Inn(R0) ~=  34by Proposition 7.4(c).  For
                  SATURATED FUSION SYSTEMS OVER 2-GROUPS                 49

i = 1, 2, 3, let ji2 Aut0(R0) denote the automorphism ji(ai) = bi, ji(b i) = ai*
 *bi, and

jifixes ajand bjfor j 6= i. Also, set fl0 = fl|R0: the automorphism which permu*
 *tes the

subgroups <ai, bi> cyclically. Thus <j1, j2, j3> ~=C33is a Sylow 3-subgroup of *
 *Out0(R0),

and <j1, j2, j3,>fl0~=C3 o C3 is a Sylow 3-subgroup of Out(R0).

  Let j(2)12and j(2)32 Aut (R2) be the extensions of j1j2 and j3, respectively,*
 * which

send o to itself. Let j(3)2 Aut(R3) be the automorphism such that j(3)|R0 = j1j*
 *-12,

and j(3)(co ) = cb1o.


Proposition 7.5. Let F be a saturated fusion system over T *for which R1 is F-

essential. Then up to isomorphism of T *, Out F(Ri) (i = 0, 1, 2, 3) are as des*
 *cribed in

the following table:
     __________________________________________________________________
     |         | |   Out F(R0)       |OutF(R1) | Out F(R2)  |Out F(R3) |
     |_________|_|___________________|_________|____________|__________|_

     | Type 1 | |    <cc, fl0,>co     |<fl, co>   | <cc>     |  <cc>    |
     |________|_|_____________________|___________|__________|_________ |
     | Type 2 | | <j1j2j3, cc, fl0,>co <|fl,>co |<j(2)j(2),>cc| <cc>    |
     |________|_|_______________________|_______|__12__3______|________ |
     | Type 3 |<|j1j-1, j2j-1, cc, fl0,>co|<fl, co><j(|2)j(2),>cc|<j(3), cc>  |
     |________|_|___2______3_____________|___________1|2__3______|_____       |
     | Type 4 | |<j1, j2, j3, cc, fl0,>co<|fl,>co<|j(2), j(2),>cc|<j(3), cc>  |
     |________|_|_________________________|_______|_12____3_____|______       |

Thus if F has type n, then Out0F(R0) ~=Cn-13o C2, and OutF (R0) ~=Out 0F(R0) o *
 * 3.

Also, if F has type 1 or 2, then V def=<z1, z2, a1a2a3, b1b2b3> is OutF (R0)-in*
 *variant.



Proof.Since R1 is F-essential, and Out (R1)=O2(Out (R1)) ~=  3, Out F(R1) must *
 *be

isomorphic to  3 (and intersect O2(Out (R1)) trivially). Hence by Proposition 1*
 *.7, to-

gether with the description of Out(R1) in Proposition 7.4(d), there is fi 2 <fi*
 *1, fi2, fi3,>fi4

such that [fi, co] = 1 and Out F(R1) = <fiflfi-1,>co.  Since any such fi extend*
 *s to an

automorphism of T *(Lemma 7.4(f)), we can assume that Out F(R1) = <fl, co>. Thus

fl0 2 OutF (R0).

  Set Q = O2(Out (R0)), H0 = <cc, co, fl0> ~=C2 x  3, H = <j1, j2, j3,>H0, and *
 *H0 =

OutF (R0)   H0 for short. Then H0Q   HQ, since Out(R0)=Q ~= 3 o  3 has a unique

Sylow 3-subgroup which contains the class of fl0.  Hence by Proposition 1.7, th*
 *ere is

fi 2 CQ(H0) such that fiH0fi-1   H. By Lemma 7.4(d,f), fi 2 <fi4|R0>, fi4 commu*
 *tes

with fl in Out(R1), and fi4 extends to an automorphism ff4 of T *. So up to iso*
 *morphism,

we can now assume that OutF (R0)   H.

  Thus OutF (R0) must be one of the four groups listed in the above table. Also*
 *, for

i = 2, 3, Out F(Ri) is determined by Out F(R0) as described in the above table:*
 * each

OutRi(R0)-invariant element fi 2 OutF (R0) extends to a unique element fi02 Out*
 *(Ri)

by Proposition 7.4(e), and fi02 OutF (Ri) by the extension axiom. The last stat*
 *ement

follows by inspection.



  We next_look_at automorphisms of Q , and_of_the subgroups R4 and Hiwhich cont*
 *ain

Q .  Set Q  =_Q_=Z(Q ) for short.  Let q: Q ---!  F2 be the quadratic form where

for any ~x2 Q  which is the class_of x 2 Q , q(~x) = 0 if x2 = 1 and q(~x) = 1 *
 *if

x2 6= 1. Then Out (Q ) ~=GO(Q , q). We will need to choose an explicit isomorph*
 *ism

Out(Q ) ~=GO+6(2) ~= 8. This last isomorphism follows upon giving to the subgro*
 *up

I(F82) of elements of even weight in F82the quadratic form x 7! 1_2wt(x) (mod 2*
 *), and

dividing out by the diagonal element (1, 1, 1, 1, 1, 1, 1, 1).  To simplify not*
 *ation, we

identify I(F82) with the group Pe(8_) of subsets of even order in 8_= {1, 2, . *
 *.,.8}.
50                     BOB OLIVER AND JOANA VENTURA

  We now make this more explicit. Identify


                    Q = <a1a2, b1b2>xC2 <z1, o>xC2 <a3, b3>.
                            ~=Q8         ~=D8       ~=Q8

          __
Define ~: Q ---!  Pe(8_)=8_= I(F82)=diag by setting


       ~(a1a2)= {12},       ~(a3)= {78},      ~(z1) = {1234} = {5678},

      ~(b 1b2)= {13},      ~(b 3)= {68}        ~(o )= {1235} = {4678} .


Then ~ preserves the quadratic forms on these two vector spaces, and induces an

isomorphism

                           OQ :Out (Q ) ------!  8.


For example, the images via OQ of automorphisms in Out T*(Q ), and also of the *
 *re-

strictions of j(2)12, j(2)32 Aut(R2) defined above, are described in the follow*
 *ing table:
             ___________________________________________________
             | ff     | |ca    |  cc    |   cb    |j(2)|Q j|(2)|Q  |
             |________|_|__1___|________|_____1___|_12_____|3____  |

             | OQ (ff)(|1|2)(3 4)(|1 2)(7(8)1|3)(2 4)(|1 3(2)6|7 8) |
             |_________|_|________|__________|________|_______|_    |
  We now apply this to describe the automorphisms of R4 = <a1, a2, a3, c, b1b2,*
 *>b3,.o

In order to "see" better the symmetry of this subgroup, we give the following, *
 *alterna-

tive description.

  Define

                                fi2      4       2          -1    -1
  S0=  z, r1, r2, r3, s1, s2, s3fiz= 1, ri = z, si = z, sirisi = ri ;
                                                                               *
 *    ff
                               [ri, rj] = 1, [si, rj] = 1, [si, sj] = z for all*
 * i 6= j


Thus S0is generated by the three subgroups <ri, si> ~=Q16(i = 1, 2, 3), which i*
 *ntersect

in Z(S0) = <z>, and whose cyclic subgroups of order 8 commute with each other. *
 *This

"twisted" product corresponds to the lifting to Spin7(9) of three copies of GO-*
 *2(3) ~=D8

in SO7(3).

  Define an embedding _ :R4 ---! S0by setting


      _(a1) = r-11r2     _(a2)= r1r2       _(a3) = r23        _(c) = r2r3

    _(b 1b2)= r21r22s1s2 _(b 3)= r21r-22s3  _(o )= r21r-22s1.


In particular, _([[Y i, Y j, Y k]]) = r(j-i)=21r(j+i)=22rk3whenever i   j   k (*
 *mod  2); and


   s1= _(z1o),     s2= _(z3b1b 2o) = _(b 1(z1o)b -11),     and    s3= _(z1b3).


Thus _(R4) = <rirj, si| i, j = 1,>2,.3

  To simplify notation in what follows, we identify elements of R4 with their i*
 *mages

under _. By the above relations, _(Q ) = <r2i, si| i = 1,>2,,3and


          ~(r21)= {34},         ~(r22)= {12},         ~(r23)= {78},

          ~(s1)= {45},          ~(s2)= {25},          ~(s3)= {75}.


Lemma 7.6. Out (R4) =  <"1, "2> o <,, cb1>  x <`1, `2> ~= 4 x C22, where these *
 *auto-

morphisms are described in the following table:
                  SATURATED FUSION SYSTEMS OVER 2-GROUPS                 51
    _____________________________________________________________________
    | ff      "|j|(j = 1, 2, 3) | `1      |  `2    |    ,      |   cb    |
    |__________|_|______________|_________|________|___________|_____1___|_

    | ff(ri)    | |zffiijri |     ri      |  ri    |   roe(i)  |  ro(i)  |
    |___________|_|_________|_____________|________|___________|_________|
    | ff(si)    | | si      |    r2si     |r2r2r2si|   soe(i)  |  so(i)  |
    |___________|_|_________|_____i_______|_12_3___|___________|_________|
    | OQ (ff|Q ) | |Id      |(1 2)(3 4)(7 8)(|5 6) (3|1 8)(4 2 7)(|1 3)(2 4) |
    |____________|_|________|________________|_______|____________|______    |

Here, oe, o 2  3 are the permutations oe = (1 2 3) and o = (1 2); and ffiij= 1 *
 *if i = j and

0 if i 6= j. Also, "1"2"3 = IdR4. If F is a saturated fusion system over T *and*
 * R4 is F-

essential, then OutF (R4) is one of the groups <,, cb1> or <"3,"-13, cb1>, both*
 * isomorphic

to  3, and the image under OQ of the restriction to Q  of Aut F(R4) is generate*
 *d by

OQ (Out T*(Q )) and (3 1 8)(4 2 7).


Proof.By Lemma 7.1, any automorphism of R4 leaves Q invariant. Hence by Lemma

1.2, there is an exact sequence


   1 ---! H1(R4=Q ; Z(Q )) ----! Out (R4) ----! NOut(Q()OutR4(Q ))=Out R4(Q ).


Also, since R4=Q  = <b, c> ~=C22and Z(Q ) = <a2> ~=C2, H1(R4=Q ; Z(Q )) ~=C22an*
 *d is

represented in Out(R4) by <"1, "2> as defined above.

  As seen earlier, OQ (Out R4(Q )) = <(1 2)(3 4), (1 2)(7>8), and hence


                   NOut(Q()OutR4(Q ))=Out R4(Q ) ~= 3 x C22.


From the above table, we see that `1 and `2 represent generators of the second *
 *factor

C22, while , and cb1 represent generators of the first factor. This proves that*
 * the last

homomorphism in the above exact sequence is onto, and shows that the automorphi*
 *sms

"1, "2, `1, `2, ,, and cb1 generate Out(R4).

  The description of the exact structure of this extension is immediate from th*
 *e above

table (and the relation "1"2"3 = Id). Namely, the subgroup <"1, "2, `1,>`2in Ou*
 *t(R4) is

elementary abelian, <,, cb1> ~= 3 permutes the three involutions "i, and acts t*
 *rivially

on <`1, `2>.

  If R4 is F-essential for some saturated fusion system F over T *, then OutF (*
 *R4) =

<ff, cb1> ~= 3 must be contained in the first factor (the one isomorphic to  4)*
 *.  Any

transposition in  4 normalizes exactly two subgroups of order 3, and in this ca*
 *se, these

are easily seen to be the subgroups <,> and <"3,"-13>.


  We now determine the outer automorphism groups of the Hi in a similar way.


Lemma 7.7. For i = 1, 2, 3, restriction to Q ~= 21+6+defines an epimorphism
                                              ffi            2
            Out(Hi) --- -i  NOut(Q)Out Hi(Q )  Out Hi(Q ) ~=C2 x  4


with kernel of order 2.


Proof.Since , 2 Aut(R4) permutes the Hi transitively, it suffices to prove this*
 * when

i = 3. Set H = H3 for short. Since Q is characteristic in H by Lemma 7.1, there*
 * is a

well defined restriction homomorphism
                                                   ffi
                 ae: Out(H) ----! NOut(Q )OutH(Q )  Out H(Q )


(Lemma 1.2), and Ker (ae) ~= H1(H=Q ; Z(Q )) ~= H1(C2; Z=2) has order 2.  Also,

OutH (Q ) = <ca1> and OQ (ca1) = (1 2)(3 4), and thus
                        ffi                                        2
       NOut(Q )OutH(Q )  Out H(Q ) ~=N 8(<OQ (ca1)>)=<OQ (ca1)> ~=C2 x  4.
52                     BOB OLIVER AND JOANA VENTURA

  Some automorphisms ff 2 Out(H3), and their images in  8, are listed in the fo*
 *llowing

table.
    ____________________________________________________________________
    | ff       | |0 | 1 = cb  |   2    |    3    |    4    |  oe   | 5  |
    |__________|_|__|_______1_|________|_________|_________|_______|____|_

    | ff(a1)   z|3|a1| z1a1  |   a 1   |   a1    |   a1    | a1   | a1  |
    |___________|_|_|________|_________|_________|_________|______|_____|
    | ff(a2)   z|3|a2|  a2   |   a 2   |   a2    |   a2    | a2   | a2  |
    |___________|_|_|________|_________|_________|_________|______|_____|
    | ff(a3)    |a|3|   a3   |   a 3   |   a3    |  z1a3   | b3   | a3  |
    |___________|_|_|________|_________|_________|_________|______|_____|
    | ff(b 1b2)b|1|b2|b 1b2  |a1b 1a2b2|  b1b2   |  b1b2   |b1b2  |b1b2 |
    |___________|_|_|________|_________|_________|_________|______|_____|
    | ff(b 3)   b|3||   b3   |   b 3   |  z1b3   |   b3    |a3b3  |a3b3 |
    |____________|_||________|_________|_________|_________|______|_____|
    | ff(o )    |o| |b 1b-1o |    o    |  z1a3o  | z1b3o   |  o   | o   |
    |___________|_|_|____2___|_________|_________|_________|______|_____|
    | OQ (ff|Q )I|d||(1 3)(2 4) |(1 2)  (|5 6)(7 8)(|5 7)(6 8)(|6 7(8)7|8) |
    |____________|_||___________|________|__________|__________|_______|_  |


Here,  0 is the generator of Ker(ae), while the other automorphisms restrict to*
 * non-

identity elements of Out (Q ) as described in the bottom row.  Note also that  *
 *2 5 is

conjugation by c. By inspection, all of these are automorphisms of H3, and thei*
 *r images

under OQ generate the normalizer of (1 2)(3 4) = ca1. This proves that ae is on*
 *to.
7.3___Fusion_systems_over_T_*_

  We are now ready to list the saturated fusion systems over T *.


Theorem 7.8. Every nonconstrained centerfree fusion system over T *is isomorphic

to the fusion system of Sol(3), Co3, or Aut(P Sp6(3)).


Proof.Let F be a nonconstrained fusion system over T *such that z3 is not centr*
 *al

in F.  By Lemma 7.4(f), Out (T *) is a 2-group, and hence Out F(T *) = 1.  So by

Proposition 7.3, all fusion in F is generated by fusion in R1, R2, R3, R4, H1, *
 *and H2.

Then R1 must be F-essential, since otherwise z3 is central in F.  So by Proposi*
 *tion

7.5, there are (up to automorphisms of T *) just four possibilities for Out F(R*
 *0), and

OutF (R0) determines OutF (Ri) for i = 1, 2, 3.

  If all F-essential subgroups contain R0, then R0 must be normal in F, which w*
 *ould

contradict the assumption that F is nonconstrained. Hence either R4, or H1 and *
 *H2,

are also F-essential.  (Recall that H1 and H2 are conjugate in T *.)  Also, if *
 *R4 is

essential, then H1, H2, and H3 are all F-conjugate by Lemma 7.6, so neither H1 *
 *nor

H2 is fully normalized in F, and hence neither is F-essential.

  The cases where R4 is F-essential will be handled in Step 1, and the cases wh*
 *ere

H1 and H2 are F-essential in Step 2. Afterwards, the distinct fusion systems fo*
 *und in

those two steps will be identified in Step 3.

Step 1:   Assume R4 is F-essential.  By Lemma 7.6, Out F(R4) leaves the subgroup

V = <z1, z2, a1a2a3, b1b2b3> invariant. Since we are assuming that F is not con*
 *strained

(hence that V  is not normal in F), this implies that Out F(R0) does not leave V

invariant. So F must be of type 3 or 4 in the notation of Proposition 7.5.

  By Lemma 7.6 again, Out F(R4) must be equal to one of the two groups <,, cb1>

or <"3,"-13, cb1>. We claim that only the second of these is possible (given ou*
 *r choice

of AutF (R1)). This is closely related to [LO2 , Lemma 1.2] (and to the error i*
 *n [LO ]

which made a correction necessary), but we give a more direct argument here. Co*
 *nsider

the subgroup A  = <a1, a2, a3,>c~= C34.  We regard automorphisms of A  as matri*
 *ces
                  SATURATED FUSION SYSTEMS OVER 2-GROUPS                 53

over Z=4, taken with respect to the basis {a1, a2, c}. Also, Aut F(A ) is gener*
 *ated by

restrictions of automorphisms in AutF (R4) and in AutF (R1) = <fl, co>. Then
     0          1           0        1        0        1         0          1
      -1   0 -1              1  2  -1          0  -1  0           0  -1  -1
 , = @-1   0  0 A, "3,"-13= @1  2   0A , fl = @1  -1  0A , ,fl = @0   1   0 A;
       1   1  1              1  1   1          0   2  1           1   0   1
             i -10 0 j
and (,fl)3 =   00120-1  Id (mod 2). Hence this must be conjugate in GL3(Z=4) to

some element of the Sylow 2-subgroup Aut T*(A ) = <cb1, cb2, cb3,>co.  But the *
 *only

element of AutT*(A) which is the identity modulo 2 is cb1b2b3= -Id, and so , an*
 *d fl

cannot both be in AutF (A ). (The analogous computation does show that ("3,"-13*
 *fl)3 =

Id.)

  Thus F must be one of two fusion systems, which we denote F1 (of type 3) and *
 *F2

(of type 4).  The restriction of Out F(R4) to Q  is generated by ,|Q (and eleme*
 *nts of

OutT*(Q )). Hence if we let X0   OutF (Q ) be the subgroup generated by Out T*(*
 *Q )

and restrictions of F-automorphisms of R4, then

         OQ (X0) = <(1 2)(3 4), (1 3)(2 4), (1 2)(7 8), (3>1~8)(4=2C7)22o  3
                      OQ(ca1)    OQ(cb1)     OQ(cc)       OQ(,|Q)

(i.e., the group of even permutations which fix 5 and 6, and permute the three *
 *subsets

{1, 2}, {3, 4}, {7, 8}). Since OQ (Out F(Q )) is generated by OQ (X0) and the r*
 *estrictions

of elements in OutF (R2), Proposition 7.5 implies
                          (
                            <OQ (X0), (1 3 2)(6 7>8)~=GL3(2)if F = F1
         OQ (Out F(Q )) =
                            <OQ (X0), (1 3 2), (6>7~8)=A7  if F = F2.


When F = F1, the isomorphism with GL3(2) is obtained by giving to 8_the structu*
 *re of

an F2-vector space by associating to n 2 8_the last three digits in the binary *
 *expansion

of n + 3. Thus 1 corresponds to (1, 0, 0), 2 to (1, 0, 1), 5 to (0, 0, 0), etc.


Step 2:   Now assume that H1 and H2 are F-essential.  Then Out F(Hi) ~=  3 for

i = 1, 2, since Out(Hi)=O2(Out (Hi)) ~= 3 by Lemma 7.7. As seen in Case 1,

              OQ (Out T*(Q )) = <(1 2)(3 4), (1 3)(2 4), (1>2)(7~8)=D8.


  Fix ffi2 AutF (Hi) of order 3. By Lemma 7.7, OQ (ff1|Q ) normalizes OQ (Aut H*
 *1(Q )) =

<OQ (cc)> = <(1 2)(7 8)>, and it is normalized by OQ (ca1) = (1 2)(3 4). Thus O*
 *Q (ff1|Q ) 2

<(3 4 x)> for some x 2 {5, 6}. Similarly, OQ (ff2|Q ) normalizes <OQ (a1c)> = <*
 *(3 4)(7 8)>

and is normalized by (1 2)(7 8), and hence OQ (ff2|Q ) 2 <(1 2 y)> for some y 2*
 * {5, 6}. If

x 6= y, then this implies that OQ (Out F(Q )) contains all even permutations wh*
 *ich leave

{7, 8} invariant, thus has order a multiple of 24, which is impossible since Ou*
 *tT*(Q ) 2

Syl2(Out F(Q )) has order 8. Hence x = y, and OQ (Out F(Q )) contains all even *
 *permu-

tations which leave the sets {1, 2, 3, 4, x} and {7, 8} invariant.

  In particular, since OQ (j(2)12|Q ) = (1 3 2) (where j(2)122 Out (R2)), this *
 *implies that

j(2)12|Q  2 Out F(Q ), and hence (by the extension axiom) that j(2)122 Out F(R2*
 *).  So

F has type 4 in the notation of Proposition 7.5.  Thus OQ (Out F(Q )) also cont*
 *ains

OQ (j(2)3|Q ) = (6 7 8); and we conclude that x = 5 and OQ (Out F(Q )) is the g*
 *roup of all

even permutations which leave the sets {1, 2, 3, 4, 5} and {6, 7, 8} invariant.

  Since the kernel of the restriction map from Out (Hi) to Out (Q ) has order 2*
 * for

each i (Lemma 7.7), an element of order 3 in Out(Hi) is completely determined b*
 *y its

restriction to Out(Q ). Hence there is only one fusion system satisfying these *
 *conditions,

and we denote it by F3.
54                     BOB OLIVER AND JOANA VENTURA

Step 3:   It remains to identify the three distinct fusion systems we have foun*
 *d. As

described above, they are distinguished by OutF (Q ). Recall (Lemma 7.1) that Q*
 * is the

only subgroup of T *of order 27 with quotient group C62. Hence to determine Out*
 *F (Q )

for any fusion system F over T *, it suffices to find any subgroup C62in NF (z)*
 *=<z> for

some involution z, and determine its automorphism group.

  The centralizer in Co3 of a Sylow central involution is isomorphic to 2.Sp6(2*
 *) (cf.

[Fi, Lemma 4.4]), and Sp6(2) contains a maximal subgroup C62oGL3(2) (the stabil*
 *izer

subgroup of an isotropic plane). Hence FS(Co3) ~=F1 for S 2 Syl2(Co3).

  The centralizer in Sol(3) of any involution is the fusion system of Spin7(3) *
 *[LO ,

Theorem 2.1], and  7(3) contains a maximal subgroup C62oA7 (the elements which

leave an orthonormal basis invariant up to sign and permutation). Hence FT*(Sol*
 *(3)) ~=

F2.

  Finally, the group Aut(P Sp6(3)) has Sylow subgroup isomorphic to T *, and ha*
 *s invo-

lution centralizer (SL2(3)xC2Sp4(3))o C2 (the subgroup of elements which leave *
 *invari-

ant an orthogonal decomposition F23x F43of the vector space). Since P Sp4(3) co*
 *ntains

a subgroup C42oA5, we have FS(Aut (P Sp6(3))) ' F3 for S 2 Syl2(Aut (P Sp6(3))).


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  LAGA, Institut Galil'ee, Av. J-B Cl'ement, 93430 Villetaneuse, France

  E-mail address: bobol@math.univ-paris13.fr


  Departamento de Matem'atica, Instituto Superior T'ecnico, Av. Rovisco Pais, 1*
 *049-
001 Lisboa, Portugal

  E-mail address: jventura@math.ist.utl.pt