A Group-Theoretic Approach to a Family of 2-Local ﬁnite Groups Constructed by Levi and Oliver

Home , Category of groups, Functor, Image (mathematics)

ANNALS OF MATHEMATICS

A group-theoretic approach to a family of 2-local ﬁnite groups constructed by Levi and Oliver

By Michael Aschbacher and Andrew Chermak

SECOND SERIES, VOL. 171, NO. 2 March, 2010

anmaah Annals of Mathematics, 171 (2010), 881–978

A group-theoretic approach to a family of 2-local ﬁnite groups constructed by Levi and Oliver

By MICHAEL ASCHBACHER and ANDREW CHERMAK

Abstract

We extend the notion of a p-local finite group (defined in [BLO03]) to the notion of a p-local group. We define morphisms of p-local groups, obtaining thereby a category, and we introduce the notion of a representation of a p-local group via signalizer functors associated with groups. We construct a chain G D .Ᏻ0 Ᏻ1 / of 2-local finite groups, via a representation of a chain G ! ! D .G0 G1 / of groups, such that Ᏻ0 is the 2-local finite group of the third ! ! Conway sporadic group Co3, and for n > 0, Ᏻn is one of the 2-local finite groups constructed by Levi and Oliver in [LO02]. We show that the direct limit Ᏻ of G exists in the category of 2-local groups, and that it is the 2-local group of the union of the chain G. The 2-completed classifying space of Ᏻ is shown to be the classifying space B DI.4/ of the exotic 2-compact group of Dwyer and Wilkerson [DW93]. Introduction In[BLO03], Broto, Levi, and Oliver introduced the notion of a p-local finite group Ᏻ, consisting of a finite p-group S and a pair of categories Ᏺ and ᏸ (the fusion system and the centric linking system) whose objects are subgroups of S, and which satisfy axioms which encode much of the structure that one expects from a finite group having S as a Sylow p-subgroup. If indeed G is a finite group with Sylow p-subgroup S, then there is a canonical construction which associates to G a p-local finite group Ᏻ ᏳS .G/, such that the p-completed nerve of ᏸ is D homotopically equivalent to the p-completed classifying space of G.A p-local finite group Ᏻ is said to be exotic if Ᏻ is not equal to ᏳS .G/ for any finite group G with Sylow group S. From the work of various authors (cf.[BLO03, 9]), it has begun to appear that for p odd, exotic p-local finite groups are plentiful. On the other hand, exotic

The work of the ﬁrst author was partially supported by NSF-0203417.

881 882 MICHAEL ASCHBACHER and ANDREW CHERMAK

2-local finite groups are – as things stand at this date – quite exceptional. In fact, the known examples of exotic 2-local finite groups fall into a single family ᏳSol.q/, q an odd prime power, constructed by Ran Levi and Bob Oliver[LO02]. With hindsight, the work of Ron Solomon[Sol74] in the early 1970’s may be thought of as a proof that ᏳSol.q/ ᏳS .G/ for any finite simple group G with Sylow group S. 6D Solomon considered finite simple groups G having a Sylow 2-subgroup isomorphic to that of Co3 (the smallest of the three sporadic groups discovered by John Conway), and he showed that any such G is isomorphic to Co3. While proving this, he was also led to consider the situation in which G has a single conjugacy class G z of involutions, and CG.z/ has a subgroup H with the following properties: n H Spin .q/; q r ; q 3 or 5 mod 8; and CG.z/ O.CG.z//H: Š 7 D Á D Here Spin7.q/ is a perfect central extension of the simple orthogonal group 7.q/ by a group of order 2, and for any group X, O.X/ denotes the largest normal subgroup of X all of whose elements are of odd order. Solomon showed that there is no finite simple group G which satisfies the above conditions – but he was not able to do this by means of “2-local analysis” (i.e. the study of the normalizers of 2-subgroups of G). Indeed a potential counterexample possessed a rich and internally consistent 2-local structure. It was only after turning from 2-local subgroups to local subgroups for the prime r that a contradiction was reached. One of the achievements of[LO02] is to suggest that the single “sporadic” object Co3 in the category of groups is a member of an infinite family of exceptional objects in the category of 2-local groups. But in addition,[LO02] establishes a special relationship between the ᏳSol.q/’s and the exotic 2-adic finite loop space DI.4/ of Dwyer and Wilkerson[DW93]. Namely, in[LO02] it is shown that the classifying space B DI.4/ is homotopy equivalent to the 2-completion of the nerve n of a union of subcategories of the linking systems ᏸSol.q /, with the union taken for any fixed q as n goes to infinity. (This result was prefigured in, and motivated by, work of David Benson[Ben94]. Benson showed, first, that the 2-cohomology ring H .B DI.4/ 2/ is finitely generated over H .Co3 2/, and second, that the 2- I I cohomology of the space of fixed points in B DI.4/ of an unstable Adams operation q, would be that of the “Solomon groups”, if such groups existed.) Moreover [BLO05] introduces the notion of a “p-local compact group”, and Theorem 9.8 in[BLO05] shows that each p-compact group supports the structure of a p-local compact group. As a special case, DI.4/ supports such a structure. We give here an alternate, constructive proof of this fact. Our paper is built around an alternate construction (Theorem A) of the 2-local finite groups 2k 1 Ᏻ Ᏻ ᏳSol.r C / .S ; Ᏺ ; ᏸ / k D k;r D D k k k 2-LOCALFINITEGROUPS 883 where r is a prime congruent to 3 or 5 mod 8. The construction is based on a notion of the “representation” of a p-local finite group Ᏻ .S; Ᏺ; ᏸ/ as the p-local group D of a not necessarily finite group G, by means of a “signalizer functor” . This means, first of all, that S is a Sylow p-subgroup of G (in a sense which we shall make precise), and that the fusion system Ᏺ may be identified with the fusion system ᏲS .G/ consisting of all the maps between subgroups of S that are induced by conjugation by elements of G. Second, it means that whenever P is a subgroup of S which contains every p-element of its G-centralizer (i.e. whenever P is centric in Ᏺ), there is a direct-product factorization

CG.P / Z.P / .P / D where the operator is inclusion-reversing and conjugation-equivariant. Then gives rise to a centric linking system ᏸ associated with Ᏺ, with the property that

Aut .P / NG.P /=.P / ᏸÂ D for any Ᏺ-centric subgroup P of S. One says that Ᏻ is represented in G via if the p-local groups Ᏻ and .S; Ᏺ; ᏸ / are isomorphic. The notions of p-local finite group and of representation via a signalizer functor can be generalized to obtain a representation of the 2-local compact group of DI.4/, by allowing S to be an infinite 2-group. We also introduce a notion of morphism, to obtain a category of p-local groups, having p-local finite groups as a full subcategory. As an application we show in Theorem B that the 2-local finite group Ᏻ0 associated with Co3 is a “subgroup” of each Ᏻk. In the final section of this paper we introduce a notion of direct limit of a directed system of embeddings of p-local groups, and in this way obtain (Theo- rem C) a 2-local compact group ᏳG which is the direct limit of a directed system G .Ᏻk Ᏻk 1/k 0 of embeddings of 2-local finite groups. The identification of D ! C ᏳG with the 2-local compact group of DI.4/ (Theorem D) is a corollary of results in[LO02], obtained by setting up a homotopy equivalence between the nerve of c our direct limit and the nerve of a category ᏸSol.p1/ constructed in[LO02]. The 2-completion of the latter category is shown in[LO02] to be homotopy equivalent to B DI.4/. In view of the length of this paper, the reader may find the following outline helpful. The first three sections are concerned with general principles and support- ing results. Then in Section 4, which provides information on certain spin groups, the argument actually begins to take shape. Let p be an odd prime. For reasons which will not be immediately apparent, it will be necessary to take p to be congruent to 3 or 5 mod 8. Let F be an algebraic x closure of the field of p elements. There is a subfield F of F, obtained as the union n x of the tower of subfields of F of order p2 , n 0. Take H to be the group Spin .F/ x x 7 x 884 MICHAEL ASCHBACHER and ANDREW CHERMAK

– the universal covering group of the simple orthogonal group .F/. Let be an 7 x endomorphism of H such that CH . / Spin7.p/, and set x x Š [ 2n H CH . /: D x n 0 Then H is a group of F-rational points of H. One finds that all fours groups in x H containing Z.H / are conjugate, and that if U is such a fours group then the 0 identity component B of the group B NH .U / is a commuting product of three D copies of SL2.F/. In Section 5 we show that there is an automorphism y of B0 of order 3, which transitively permutes the three SL2.F/ components of B, and which when chosen carefully, interacts in a special way (to be described shortly) with the normalizer in H of a maximal torus T of B. It is at this point, in choosing an appropriate automorphism y, that we require that p be congruent to 3 or 5 mod 8. Once y has been fixed in the appropriate way, we form a group K B; y which is D h i isomorphic to a split extension of B0 by the symmetric group of degree 3. We then form the amalgam Ꮽ .H > B < K/, and its associated free amalgamated D product G H B K: D This is the group which informs and guides our investigation. We need the following notion of “Sylow 2-subgroup”: A subgroup S of G is a Sylow 2-subgroup of G if every element of S has order a power of 2 (i.e. S is a 2-group), S is maximal with respect to inclusion among the 2-subgroups of G, and every finite 2-subgroup of G is conjugate to a subgroup of S. It turns out that the normalizer in H of a maximal torus T of B0 contains a Sylow 2-subgroup S of G. Moreover, if T is chosen to be y-invariant then S is a Sylow 2-subgroup of each of the groups H , B, and K. The special way in which y interacts with S may be summarized as follows: for the Sylow 2-subgroup S S T of T , we have 1 D \

. /NG.S / NH .S / NB .S / NK .S /; and 1 D 1 1 1

. / AutG.S / NG.S /=CG.S / GL.3; 2/ C2: 1 WD 1 1 Š The effect of . / is that S=S may be identified with a Sylow 2-subgroup of 1 AutG.S /, and it is this property which, as is made clear in[LO02], turns out to be the key1 to fulfilling the axioms for “saturation” (defined in 1.5, below). One feature of our treatment is the use of amalgams (cf. 3) to keep track of the various fusion systems which can be constructed from H, B, and K, and to distinguish the system with property . /. We prove in Theorem 5.2 that the amalgam Ꮽ with property . / is unique. Then we carry out the remainder of our analysis in the universal completion G of Ꮽ, using the “standard tree” of G 2-LOCALFINITEGROUPS 885 as a source of geometric intuition, and as the basis for geometric arguments. The formalization by means of the amalgam Ꮽ and its free amalgamated product G provides, at the very least, a useful system of bookkeeping. For example, the language of amalgams provides a conceptual framework within which one can rigorously consider the question of which of the fusion systems constructed from H , B, and K is the “right” system. We mention that amalgams have also been used in recent work of G. Robinson[Rob07], and of Ian Leary and Radu Stancu [LS] as a tool for studying abstract fusion systems. Setting Z Z.H / one has Z 2, and CG.Z/ is in fact a rather complicated D j j D subgroup of G, properly containing H. Our proof that the fusion system ᏲS .G/ is saturated is modeled on the proof of saturation in[LO02] for the fusion systems ᏲSol.q/ defined over finite 2-groups. Thus, the main step is to establish that H controls CG.Z/-fusion in S. That is, the fusion system ᏲS .H / is equal to the a priori larger system ᏲS .CG.Z//. The proof of saturation in G is accompanied by the construction of a linking system by means of a signalizer functor. These steps require information on fusion among the centric subgroups of S, obtained in Sections 6 through 8. After this, in order to prepare the way for the construction of morphisms, we determine in complete detail the radical centric subgroups of S and of S , where is an automorphism of G which fixes S and which induces a Frobenius endomorphism of H . Here a subgroup P of S is defined to be radical if Inn.P / O2.AutG.P //. D One of the radical centric subgroups of S is an elementary abelian group A of order 16 which has the property, as in . /, that NG.A/ NH .A/ NK .A/: D NB .A/ Here one can do better than to determine AutG.A/ in analogy with . /. Indeed, there is a surjective homomorphism

A NG.A/ L; W ! with CG.A/ A ker.A/, where L is a maximal subgroup of the sporadic group D Co3, isomorphic to a nonsplit extension of A by Aut.A/. We then deﬁne a normal subset X of G by [ g X ker.A/ : D g G 2 For any centric subgroup P of S we deﬁne a subset .P / of CG.P / by

.P / CX.P /O.CG.P //: D It turns out that .P / is a subgroup of CG.P / and that is a signalizer functor (cf. Theorem 8.8 below). 886 MICHAEL ASCHBACHER and ANDREW CHERMAK

We next consider the groups G of fixed points of automorphisms of G, such that fixes both H and K, and such that the restriction of to H is a 2n Frobenius map with H Spin .Fq/, q p . The groups G , for n 0, provide Š 7 D representations of the 2-local finite groups of[LO02]. In particular, for each such

(chosen so that S is a Sylow 2- subgroup of G ), the fusion system ᏲS .G / is saturated, and the signalizer functor given by

.P / CX .P /O.CG .P //; D for centric subgroups P of S , defines a centric linking system ᏸ .S / associated with ᏲS .G /. This completes our outline of the proof of Theorem A. One aim of this paper is thus to suggest the possibility that many p-local finite groups may best be studied via a representation in terms of free amagalmated products and signalizer functors. For example, to study the fusion system Ᏺ on a p-group S generated by systems ᏲS .Gi / for some family Ᏻ .Gi i I/ of finite groups with Sylow D j 2 group S, perhaps one should study the various amalgams Ꮽ obtained from Ᏻ, and the corresponding free amalgamated products G G.Ꮽ/. If Ꮽ is well chosen, D then S is Sylow in G and Ᏺ ᏲS .G/ is saturated. Then one can consider suitable D overamalgams Ꮾ of Ꮽ, and the kernels of surjections from subgroups G.Ꮾ/ of G onto suitable finite groups and use these kernels to construct a signalizer functor and the corresponding p-local finite group from Ᏺ. If Ꮽ is the amalgam of some family of subgroups generating a finite group G, then the kernel of the surjection g c y G G will be .P / P Ᏺ ; g G , whence Ᏺ ᏲS .G/ (cf. Example 2.13, ! y h j 2 2 i Š y below). But in other cases one may hope for exotic p-local finite groups, such as ᏸsol.q/. Now here are the main theorems.

THEOREM A. Let p be a prime, p 3 or 5 mod 8, let F be an algebraic Á x closure of the ﬁeld Fp of p elements, and let F be the union of the subﬁelds of F 2n x of order qn p , n 0. Then there is a group G G.p/, an automorphism 0 D D of G, a Sylow 2-subgroup S of G, and a 0-invariant normal subset X of G such 2n that, for any power of 0 of the form 0 , we have the following. (1) G H B K is the free amalgamated product of an amalgam D Ꮽ .H B K/; D ! where H is a group of F-rational points in Spin .F/, B is the normalizer in H 7 x of a fours group U of H containing Z.H /, and K is a group which contains B as a subgroup of index 3 where K has the property that AutK .U / GL.2; 2/. Š (2) 0 leaves invariant each of the subgroups H , K, and B of G; the restriction of to H is the restriction of a Frobenius automorphism of Spin .F/, and 0 7 x CH . 0/ Spin .p/. Š 7 2-LOCALFINITEGROUPS 887

(3) The group S CS ./ is a ﬁnite Sylow 2-subgroup of the group G CG./, D D and there exists a unique choice of the amalgam Ꮽ such that, for all , the

fusion system Ᏺ ᏲS .G / is isomorphic to the fusion system ᏲSol.q/ of n D [LO02], .q p2 /. D (4) For any Ᏺ -centric subgroup P of S , the set

.P / CX G .P /O.CG .P // WD \

is a group, and is a complement to Z.P / in CG .P /. Moreover, defines a 2-local finite group Ᏻ .S ; Ᏺ ; ᏸ / isomorphic to the 2-local finite group D ᏸsol.q/ of [LO02]. (5) The order of a maximal elementary abelian 2-subgroup A of S is 16, and all maximal elementary abelian 2-subgroups of G are conjugate in G. Moreover, CG.A/ A CX.A/, where CX.A/ is a free normal subgroup of CG.A/, D NG.A/=CX.A/ is isomorphic to a nonsplit extension of A by Aut.A/, and X is the union of the conjugates of CX.A/ in G.

THEOREM B. Let p, Ꮽ, G, 0, and X be as in Theorem A. Then there exist subgroups H0, K0, and B0 H0 K0 of H , K , and B , respectively, such D \ 0 0 0 that the following hold.

(1) H0 is isomorphic to a perfect central extension of Sp.6; 2/ by ޚ2, K0 is a 10 3 group of order 2 3 , and B0 is of index 3 in K0. (2) Setting G0 H0;K0 , we have D h i (a) X H0 X K0 1 , and \ D \ D f g (b) G0= X G0 is isomorphic to the colimit of the amalgam ᏹ of maximal h \ i subgroups of Co3 containing a ﬁxed Sylow 2-subgroup of Co3.

(3) Let S00 be a Sylow 2-subgroup of Co3 and S0 a Sylow 2-subgroup of B0. Then there is an isomorphism of 2-local ﬁnite groups

ᏳS .Co3/ ᏳS0 .G0/: 00 Š THEOREM C. For any positive integer i, let Ᏻi be the 2-local ﬁnite group

Ᏻ 2i 1 of Theorem A, and let Ᏻ0 be the 2-local ﬁnite group associated with Co3 0 as in Theorem B. Let Ᏻ be the 2-local group .S; Ᏺ; ᏸ/ associated with G via the fusion system Ᏺ ᏲS .G/ and via the signalizer functor deﬁned by the subset X D of G. Then there exists a directed system

G .ˇi;j Ᏻi Ᏻj /0 i j D W ! Ä Ä of embeddings of 2-local ﬁnite groups, possessing a limit ᏳG which is canonically isomorphic to the 2-local group Ᏻ.

The 2-local group Ᏻ.G/ is a 2-local compact group, as deﬁned in[BLO05]. 888 MICHAEL ASCHBACHER and ANDREW CHERMAK

It will be proved in[COS06] that the exotic fusion systems ᏲSol.q/ of Levi and Oliver, defined over 2-groups Sq, are determined by the isomorphism type of Sq. This implies that for any odd prime power q, and any prime p 3 or 5 mod 8, Á there is a unique such that the Levi-Oliver fusion system ᏲSol.q/ is isomorphic to ᏲS .G /, where G G.p/ is the group in characteristic p constructed here. D This is needed for the proof of the following result. THEOREM D. Let ᏸ ᏸ .G/ be the centric linking system over Ᏺ as WD S; given in Theorem C. Then the 2-completed nerve ᏸ ^ is homotopy equivalent to j j2 B DI.4/. In particular, DI.4/ may be given the structure of the 2-local group Ᏻ, and DI.4/ is then a 2-local compact group. We are grateful to Bob Oliver for many helpful conversations about the 2-local finite groups ᏳSol.q/ which he and Ran Levi constructed, and for his help in under- standing the space B DI.4/ of Dwyer and Wilkerson. The proof of Theorem D was communicated to us by Levi and Oliver. We would also like to thank Ron Solomon and the other members of his seminar at Ohio State, for suggesting improvements to an earlier version of this manuscript. Remarks and questions. (1) One might imagine that the normal subgroup X of G leads to an interesting h i factor group G= X . But the fact is that X G. Moreover, G = X 1 h i h i D h i D for any automorphism of G as in Theorem A, while G0= X G0 is in fact h \ i isomorphic to Co3. These results will appear in[COS06]. (2) To what extent can our method of construction of the Levi-Oliver fusion and linking systems be carried out in a characteristic 0 context? For example, one might consider a subring ᏻ of the field of complex numbers, and ask whether there is a 7-dimensional quadratic space over ᏻ, yielding a group H Spin .ᏻ/, from which to build up a suitable free amalgamated product ᏻ D 7 and linking system as we do here in characteristic p. One requires 1=2 ᏻ 2 in order to have an isomorphism of PSL2.ᏻ/ with a suitable 3-dimensional orthogonal group. The rings

ᏻm ޚŒ!=2; D where ! is a primitive 2m-th root of unity, are possible candidates for this, and there may be others. (3) The sporadic group O0N (or rather, the 2-local ﬁnite group associated with O0N ) can be shown to occur as a subgroup of some of the 2-local ﬁnite groups constructed here. Since O0N and its subgroup J1 are “pariahs”, i.e. are not among the twenty sporadic simple groups which are involved in the Monster, it is of some interest to have a context in which these groups, and Co3 (which is not a pariah) can live together in harmony. This will be the subject of another paper. 2-LOCALFINITEGROUPS 889

1. Fusion systems and Sylow subgroups

We shall need to consider fusion systems both over finite p-groups, and over certain infinite p-groups. In the finite case the definitions are due first of all to Lluis Puig[Pui06], and then to Broto, Levi, and Oliver[BLO03]. The latter three authors also consider a class of infinite p-groups which they call discrete p-toral groups, in[BLO05], and this class includes all of the p-groups that will be studied here. For reasons of exposition, however, we shall present the definitions in a somewhat more general context – but we emphasize that the main concepts, and the proofs of the basic lemmas, come from the above-cited works. We follow the practice, peculiar to finite group theory, of using right-hand notation for conjugation within a group, and for group homomorphisms. But we use left-hand notation for functors, and for auxiliary mappings associated with some of our functors. It may also be worth mentioning that if X is a set admitting action by a group G, and g is an element of G, then the set of fixed points for g g in X is denoted Xg , rather than the topologist’s X . If G is a group, g an element of G, and X a subset or an element of G, we write X g for the image of X under the conjugation automorphism g 1 cg G G; .cg x x g xg for all x G/: W ! W 7! WD 2 We also write cg P Q for the mapping of P into Q given by g-conjugation, W ! whenever P and Q are subgroups of G with P g Q. The transporter of P into Ä Q is the set g NG.P; Q/ g G P Q ; and we define WD f 2 j Ä g

HomG.P; Q/ cg P Q g NG.P; Q/ : WD f W ! j 2 g Denote by Inj.P; Q/ the set of all injective homomorphisms of P into Q. If ˛ P Q is an isomorphism, write ˛ for the isomorphism from Aut.P / to W ! Aut.Q/ deﬁned by ˛ ˇ ˛ 1ˇ˛. W 7! Deﬁnition 1.1. A fusion system Ᏺ over a group S is a category whose objects are the subgroups of S, and whose morphism-sets HomᏲ.P; Q/ satisfy the following two conditions.

(1) HomS .P; Q/ Hom .P; Q/ Inj.P; Q/. Â Ᏺ Â (2) If ˛ Hom .P; Q/ then the isomorphisms ˛ P P ˛ and ˛ 1 P ˛ P 2 Ᏺ W ! W ! are morphisms in Ᏺ. Example 1.2. Let G be a group and S a subgroup of G. For subgroups P and Q of S, set Hom .P; Q/ HomG.P; Q/: Ᏺ D Then Ᏺ is a fusion system over S, denoted ᏲS .G/. 890 MICHAEL ASCHBACHER and ANDREW CHERMAK

Let Ᏺ be a fusion system over S, let P be a subgroup of S, and let ˛ 2 AutᏲ.P /. Set

N˛ x NS .P / .cx/˛ AutS .P / : D f 2 j 2 g Thus, N˛ is the largest subgroup R of NS .P / having the property that, in the group AutᏲ.P /, the conjugation map c˛ carries AutR.P / into AutS .P /.

LEMMA 1.3. Let S be a subgroup of a group G, and let P be a subgroup of S. Set Ᏺ ᏲS .G/, let g NG.P /, and set ˛ cg Aut .P /. D 2 D 2 Ᏺ g g (a) .N˛/ S .CG.P /NS .P //. D \ (b) If S is a p-group, and every p-subgroup of CG.P /NS .P / is conjugate via CG.P / to a subgroup of NS .P /, then there exists ˛ Aut .N˛;NS .P // N 2 Ᏺ extending ˛.

g Proof. Set R N˛. By definition, R consists of those x NG.P / such that g D 2 x S and cx P AutS .P /. But cx P AutS .P / if and only if x CG.P /NS .P /, 2 j 2 j 2 2 and thus (a) holds. g Assume the hypothesis of (b). Then R is a p-subgroup of CG.P /NS .P /, by h (a), so that by the hypothesis of (b) there exists h CG.P / such that R NS .P /. 2 Ä Now ˛ is the restriction to P of ˛ c R NS .P /, and we have (b). N D ghW ! Definition 1.4. Let p be a prime. A group S is a p-group if for every x S, 2 the order of x is a power of p.A p-subgroup S of a group G is a Sylow p-subgroup of G if (1) S is maximal (with respect to inclusion) among all p-subgroups of G, and (2) S contains a conjugate of every finite p-subgroup of G.

The set of all Sylow p-subgroups of G is denoted Sylp.G/. The group generated by the set of normal p-subgroups of G is itself a normal p-subgroup of G, and is denoted Op.G/. Deﬁnition 1.5. Let p be a prime, let S be a p-group, and let Ᏺ be a fusion system over S. A subgroup P of S is fully normalized in Ᏺ if, for every 2 Hom .P; S/, there exists Hom .NS .P /; NS .P // such that maps P Ᏺ 2 Ᏺ to P . We say that Ᏺ is saturated if the following two conditions hold for every subgroup P of S.

(I) There exists Hom .P; S/ such that P is fully normalized in Ᏺ. 2 Ᏺ (II) If P is fully normalized in Ᏺ then:

(A) AutS .P / Syl .Aut .P //, and 2 p Ᏺ (B) each ˛ Aut .P / extends to a member of Hom .N˛;S/. 2 Ᏺ Ᏺ 2-LOCALFINITEGROUPS 891

The preceding definition of saturation is formulated so as to make no mention of NS .P / for P S, and it is equivalent to the standard definition (cf.[BLO03]) j j Ä in the case that S is finite. This follows from[BCG C05, Lemma 2.3], and it is then easy to check that our definition of “fully normalized” is equivalent to the usual one, in a saturated fusion system over a finite p-group.

LEMMA 1.6. Let p be a prime and G a group. Let ᐅ be the set of subgroups Y of G such that the set ᏿.Y / of maximal p-subgroups of Y is nonempty, and such that Y is transitive on ᏿.Y / by conjugation. Assume for each p-subgroup P of G that:

(1) NG.P /, CG.P /P , and CG.P /T are in ᐅ for T ᏿.NG.P //. 2 (2) OutG.P / is ﬁnite. (3) P is Artinian (i.e., any descending chain of subgroups of P stabilizes).

(4) If P 1 then NG.P / is locally ﬁnite. ¤ Then (a) G has a Sylow p-subgroup S.

(b) A subgroup P of S is fully normalized in ᏲS .G/ if and only if NS .P / is a Sylow p-subgroup of NG.P /.

(c) ᏲS .G/ is saturated. Proof. We have G ᐅ, by (1) as applied to P 1. Thus ᏿.G/ Syl .G/, 2 D D p and (a) holds. Let P S and set L NG.P /, K CG.P /, and T NS .P /. Applying Ä D D D (1) to P , we obtain T X for some X Sylp.L/. Let Q S and g G with g Äg 2 Ä 2 Q P . Then NS .Q/ is a p-subgroup of L, and since L ᐅ there exists l L D gl 2 2 with NS .Q/ X. We conclude that P is fully normalized if T X. Further, Ä D as G ᐅ there exists h G with X h S, and so X h Syl .Lh/ and hence P h 2 2 Ä 2 p is fully normalized. This verifies axiom (I) in the definition 1.5 of saturation. Assume that P is fully normalized. Then there exists y G with P hy P n 2 D and with X hy T X. Then X .hy/ X for all n > 0, and it follows from (3) Ä Ä Ä that X X hy, and then that X T . This completes the proof of (b). D D Set L L=PK. We have T Syl .L/ by (b), and L is finite by (2). Let Y x D 2 p x be the pre-image in L of a Sylow p-subgroup of L containing T . By (4) there is a x x finite subgroup U of Y with Y U . As U is a p-group we have U V for some x D x x x D x V Syl .U /. As L ᐅ there exists a L with V a T . Then Y V T Y , 2 p 2 2 Ä j xjDj xjÄj xjÄj xj so that Y T , and we have verified axiom (IIA) in 1.5. x D x Finally, set Ᏺ ᏲS .G/ and let ˛ Aut .P /. As TK ᐅ, by (1), we conclude D 2 Ᏺ 2 from 1.3(b) that ˛ extends to an element of HomᏲ.N˛;S/, verifying axiom (IIB) for saturation, and completing the proof of (c). 892 MICHAEL ASCHBACHER and ANDREW CHERMAK

Notice that if G is a finite group then the hypotheses of 1.6 are satisfied by G. Thus, it is a corollary of 1.6 that for any finite group G and S Syl .G/, the fusion 2 p system ᏲS .G/ is saturated. Let Ᏺi , i 1; 2, be fusion systems over subgroups Si of a group S. We D say that Ᏺ1 is a fusion subsystem of Ᏺ2 (and write Ᏺ1 Ᏺ2) if S1 S2 and Ä Ä Hom .P; Q/ Hom .P; Q/ for all P;Q S1. Ᏺ1 Â Ᏺ2 Ä Given a set F of fusion systems over S, there is a largest fusion system \ ᏲF Ᏺ WD Ᏺ F 2 which is a subsystem of each Ᏺ F. Thus 2 \ Hom .P; Q/ Hom .P; Q/: ᏲF D Ᏺ Ᏺ F Given a set E of fusions systems, each of2 which is defined over a subgroup of S, define E – the fusion system generated by E – to be the fusion system ᏲF, where h i F is the set of all fusion systems over S which contain each member of E. The proof of the following result is straightforward:

LEMMA 1.7. Let S be a group and let .Si i I/ be a collection of subgroups W 2 of S. For each i I , let Ᏺi be a fusion system over Si , and set F .Ᏺi i I/. 2 D j 2 Assume that Si S for at least one index i, and deﬁne a fusion system Ᏻ on S by D taking Hom .P; Q/ to consist of the maps ˛0 : : : ˛r such that, for each 0 j r, Ᏻ Ä Ä there exists i.j / I , Pj Si.j /, and ˛j HomᏲi.j / .Pj ;Si.j //, such that Pj 1 2 Ä 2 C D Pj ˛j , P0 P , and Pr 1 Q. Then F Ᏻ. D C D h i D LEMMA 1.8. Let G be a group, S Syl .G/, and let X be a normal sub- 2 p group of G of index p in S, such that S=X N .S=X/. Then ᏲS .G/ Š G=X D ᏲS .S/; ᏲX .G/ . h i Proof. Set G G=X and Ᏹ ᏲS .S/; ᏲX .G/ . As ᏲS .S/ and ᏲX .G/ D D h i are contained in Ᏺ ᏲS .G/, we have Ᏹ Ᏺ by deﬁnition of Ᏹ, and it remains to D Â establish the opposite inclusion. Let P;Q S and ˛ Hom .P; Q/. If P X Ä 2 Ᏺ — then as S is of order p and is equal to its normalizer in G we get Q X — and ˛ Hom .P; Q/. Similarly if P X but Q X then ˛ ˇ where 2 ᏲS .S/ Ä — D ˇ Hom .P; Pˇ/, Pˇ X, and Pˇ Q is the inclusion map. Thus it 2 Ᏺ Ä W ! remains to show that Hom .P; Q/ Hom .P; Q/ for P;Q X, which follows Ᏺ Â Ᏹ Ä since ᏲX .G/ Ᏹ. Â The next lemma states a weak form of the Alperin-Goldschmidt fusion theorem[Gol70], in the language of fusion systems. This result will be of use in the proof of Theorem B.

LEMMA 1.9. Let G be a ﬁnite group, S Syl .G/, and denote by ᏺ the set 2 p of subgroups N of G having the following properties. 2-LOCALFINITEGROUPS 893

(1) N NG.Op.N //, D (2) S N Syl .N /, and \ 2 p (3) CS .Op.N // Op.N /. Ä Let ᏺ0 be the set of minimal members of ᏺ, with respect to inclusion. Then ᏲS .G/ ᏲS N .N / N ᏺ0 . D h \ j 2 i We end this section with a generalization of a well-known result concerning ﬁnite p-groups.

LEMMA 1.10. Let P be a p-group, set A Aut.P /, and let D Ꮿ .P P0 P1 P 1/ D D k D be a chain of normal subgroups of P . Let ƒ be a subgroup of the group

CA.Ꮿ/ ˛ A ŒPi ; ˛ Pi 1 for all i, 0 i < k ; D f 2 j Ä C Ä g and assume that either ƒ is a torsion group or that P1 is of bounded exponent. Then ƒ is a p-group. Proof. Apply induction on k. The lemma is trivial when k 1, so take k > 1 D and set P P=Pk 1. Then ƒ centralizes the chain Ꮿ .P0 Pk 1 1/. D D D By induction, Autƒ.Ꮿ/ is a p-group, so it remains to show that Cƒ.P / is a p-group. Thus, we may take k 2. Let ˛ ƒ, x P , and set c Œx; ˛. Then ˛n D n 2 2 D c P1 CP .˛/, so that x xc , and hence ˛ x c . If ˛ is of ﬁnite order, 2 Ä D j jh ij D j j or P1 is of bounded exponent, we conclude that ˛ lcm Œx; ˛ x P j j D fj j j 2 g is a power of p.

2. Linking systems, signalizer functors, and p-local groups

Let Ᏺ be a fusion system over a p-group S. A subgroup P of S is Ᏺ-centric if CS .P / Z.P / for every Hom .P; S/, and P is Ᏺ-radical if Op.Aut .P // D 2 Ᏺ Ᏺ Inn.P /. Write Ᏺc for the set of Ᏺ-centric subgroups of S, Ᏺr for the set of Ᏺ- D radical subgroups of S, and Ᏺrc for Ᏺc Ᏺr . \ LEMMA 2.1. Let S be a Sylow p-subgroup of a group G, set Ᏺ ᏲS .G/, D and let P S. Ä c g (a) If P Ᏺ , and g G with P S , then CS g .P / P . 2 2 Ä Ä c (b) If P contains every ﬁnite p-subgroup of CG.P / then P Ᏺ . 2 c (c) If P is ﬁnite and P Ᏺ , then Z.P / contains every p-subgroup of CG.P /. 2 894 MICHAEL ASCHBACHER and ANDREW CHERMAK

Proof. Part (a) is immediate from the definition of Ᏺc. Now suppose that Z.P / contains every finite p-subgroup of CG.P /, and let g NG.P; S/. Then g g 2 c every element of CS .P / is contained in Z.P /, and thus P Ᏺ , proving (b). 2 Finally, assume that P is finite and that P Ᏺc, and let R be a finite p-sub- 2 group of CG.P /. Then RP is a finite p-subgroup of G. Since S is a Sylow p-sub- g c g group of G, there exists g G with .RP / S. As P Ᏺ we have CS .P / g 2 Ä 2 D Z.P /, and so R Z.P /. Thus Z.P / contains every finite p-subgroup of CG.P /, Ä and since every p-group is the union of its finite subgroups, we obtain (c).

LEMMA 2.2. Let S be a Sylow p-subgroup of a group G, and set Ᏺ ᏲS .G/. r D Let P Ᏺ such that P contains every p-element of CG.P /, and let 2 Ꮿ .P P0 P1 P 1/ D D k D be a chain of NG.P /-invariant subgroups of P . Suppose that P1 is of bounded exponent. Then P contains every ﬁnite P -invariant p-subgroup R of CG.Ꮿ/.

Proof. First, let R0 be a p-subgroup of CG.P /P . Then R0P CR0P .P /P E D by the Dedekind Lemma. As P CG.P /P , R0P is a p-group, and then CR0P .P / P by hypothesis. Thus Ä . /R0 R0P CR P .P /P P: Ä Ä 0 Ä Now let R be a p-subgroup of CG.Ꮿ/. Set ƒ C .Ꮿ/. Then ƒ E AutG.P /, D AutG .P / and ƒ is a p-group by 1.10. As P is Ᏺ-radical, it follows that AutR.P / Inn.P /, Ä and thus R CG.P /P . Then R P by . /. Ä Ä A set Ᏺ0 of objects in a fusion system Ᏺ is closed under Ᏺ-conjugation if P Ᏺ0 2 for all P Ᏺ0 and all morphisms Ᏺ defined on P . 2 2 LEMMA 2.3. Let Ᏺ be a fusion system on S. Then Ᏺc, Ᏺr , and Ᏺrc are closed under Ᏺ-conjugation. Proof. Let P S and let Hom .P; S/. Then P Ᏺc if and only if Ä 2 Ᏺ 2 P Ᏺc, by definition. Now let P Ᏺr . The natural map Aut .P / 2 2 W Ᏺ ! AutᏲ.P / is an isomorphism, and since Inn.P / Op.AutᏲ.P //, it follows that r D r Inn.P / Op.Aut .P //. Thus, P Ᏺ if and only if P Ᏺ . D Ᏺ 2 2 Definition 2.4. Let S be a p-group, let Ᏺ be a fusion system over S, and let Ᏹ be a subset of Ᏺc. An Ᏹ-linking system (or linking system on Ᏹ), consists of (1) a category ᏸ with Obj.ᏸ/ Ᏹ and composition (read from left to right), D (2) a functor ᏸ Ᏺ, for which the associated map of objects induces the W ! identity map Obj.ᏸ/ Ᏹ, and ! (3) a collection ı ıP P Aut .P / P Ᏹ of injective group homomor- D f W ! ᏸ j 2 g phisms, such that the following three conditions hold for any P and Q in Ᏹ. 2-LOCALFINITEGROUPS 895

(A) The left action of Z.P / on Morᏸ.P; Q/ given by 1 z ıP .z / W 7! is free (i.e. Morᏸ.P; Q/ is a union of regular orbits for Z.P /), and the map Z.P / ./ is a bijection of Z.P / Mor .P; Q/ with Hom .P; Q/. In 7! n ᏸ Ᏺ particular, is surjective on morphism sets, and .ᏸ/ is a full subcategory of Ᏺ. (B) For all g P , 2 .ıP .g// cg Aut .P /: D 2 Ᏺ (C) For each Mor .P; Q/ and each g P , the following square commutes 2 ᏸ 2 in ᏸ:

P Q ? ! ? ıP .g/? ?ıQ.g . // y y

P Q: ! A centric linking system on Ᏺ is a linking system on Ᏺc.A pre-local group consists of a triple Ᏻ .S; Ᏺ; ᏸ/ where S is a p-group, Ᏺ is a fusion system over S, and D ᏸ .ᏸ; ; ı/ is a linking system on a subset Ᏹ of Ᏺc. If Ᏺ is saturated, and ᏸ D is a centric linking system, then Ᏻ is a p-local group.A p-local ﬁnite group is a p-local group Ᏻ in which S is ﬁnite.

We are following the notational conventions in[BLO03] in writing Morᏸ.P;Q/ (rather than Homᏸ.P; Q/) for the set of morphisms in ᏸ from P to Q, in order to emphasize that in general, ᏸ-morphisms are not mappings. Deﬁnition 2.5. Let G be a group, let S be a Sylow p-subgroup of G, and set Ᏺ ᏲS .G/. Let ᐀.G/ be the set of all subgroups of G. An Ᏺ-signalizer functor D is a mapping Ᏺc ᐀.G/ W ! satisfying the following three conditions:

(1) .P / is a complement to Z.P / in CG.P /. g g (2) .P / .P / for all g NG.P; S/. D 2 (3) .Q/ .P / for all Q with P Q S. Ä Ä Ä Remark. Signalizer functors are bona fide contravariant functors. Namely, in 2.5, view ᐀ ᐀.G/ as a category whose morphism sets are given by D Mor .X; Y / NG.X; Y /; ᐀ D for any subgroups X and Y of G, and where composition is given by multiplication in G. Let ᐀c be the full subcategory of ᐀ whose set of objects is Ᏺc. Condition (2) 896 MICHAEL ASCHBACHER and ANDREW CHERMAK in 2.5 implies that an Ᏺ-signalizer functor is a contravariant functor from ᐀c to ᐀, if we define Mor c .P; Q/ Mor ..P /; .Q// W ᐀ ! ᐀ by .g/ g 1. D Given the setup of 2.5, define ᏸ ᏸ to be the category whose objects are D the Ᏺ-centric subgroups of S, with morphisms

Mor .P; Q/ .P / NG.P; Q/: ᏸ D n The composition of morphisms is defined by .P /g .Q/h .P /gh; D for g NG.P; Q/ and h NG.Q; R/. In fact, this composition is no more than 2 2 ordinary multiplication of subsets of G. To see this, notice that if P g Q, then Ä the signalizer functor axioms (2) and (3) yield .Q/ .P g / .P /g . Thus 1 Ä D .Q/g .P /, and so Ä 1 ..P /g/..Q/h/ .P /.Q/g gh .P /gh: D D Next, define a functor ᏸ Ᏺc D W ! by .P / P and by ..P /g/ cg for g NG.P; Q/. Finally, define a family D c D 2 ı ı ıP P Ᏺ of monomorphisms D D f j 2 g ıP ı P Aut .P / D P; W ! ᏸ c by ıP .g/ .P /g, for P Ᏺ . D 2 LEMMA 2.6. Let S be a Sylow p-subgroup of a group G, set Ᏺ ᏲS .G/, D and let be an Ᏺ-signalizer functor. Then

(a) .ᏸ ; ; ı / is a centric linking system on Ᏺ.

(b) If Ᏺ is saturated then Ᏻ .G/ .S; ᏲS .G/; ᏸ / is a p-local group. S; WD Proof. Let P;Q Ᏺc, let z be a nonidentity element of Z.P /, and let g 2 2 NG.P; Q/. Then .P /g Mor .P; Q/, and 2 ᏸ ıP .z/ .P /g .P /z .P /g .P /zg: D D Here .P /zg .P /g since .P / Z.P / 1. Thus, Z.P / acts freely on ¤ \ D Hom .P; Q/. Similarly, since CG.P / .P / Z.P /, the map ᏸ D Z.P /.P /g cg ..P /g/ 7! D is a bijection from Homᏸ.P; Q/=Z.P / to HomᏲ.P; Q/. Thus, condition (A) in Deﬁnition 2.4 is satisﬁed in our setup. 2-LOCALFINITEGROUPS 897

Now suppose that g P . By construction, we then have 2 .ıP .g// ..P /g/ cg ; D D and so condition (B) is satisfied. Finally, let f .P /x Morᏸ.P; Q/. We then x D 2 x x have .f / cx P Q, and g.f / g . Since also ıQ.g / .Q/g , we D W ! D D obtain x x f ıQ..g.f // .P /x .Q/g .P /xg D D .P /gx .P /g .P /x ıP .g/ f: D D D Thus, condition (C) is satisfied, and (a) is proved. Part (b) follows from (a), by the definition of p-local group. PROPOSITION 2.7. Let S be a Sylow p-subgroup of a group G and set Ᏺ c D ᏲS .G/. Suppose that NG.P / is finite for every P Ᏺ . 2 p (a) There is a unique Ᏺ-signalizer functor given by .P / O .CG.P //. D (b) Set ᏸS .G/ ᏸ , and suppose that Ᏺ is saturated. Then D ᏳS .G/ .S; ᏲS .G/; ᏸS .G// WD is a p-local finite group. Proof. Part (a) follows from 2.1(c), and then (b) follows from (a) and from 2.6(b). The following lemma is intended as a remark, to point out the connection between Definition 2.5 and the usual notion of “balanced signalizer functor” in finite group theory. It will not be used in the sequel. LEMMA 2.8. Let be an Ᏺ-signalizer functor, where Ᏺ is a fusion system over a finite p-group. Then, for any P;Q Ᏺc with P Q, 2 Ä (a) Z.Q/ Z.P /, and Ä (b) .Q/ C .Q/. D .P / Proof. Part (a) is immediate from 2.1(c). By definition,

.Q/ C .Q/ CG.Q/ .Q/ Z.Q/; Ä .P / Ä D so that C .Q/ .Q/ .Z.Q/ .P //. Since Z.Q/ Z.P /, we obtain (b). .P / D \ Ä Deﬁnition 2.9. Let and be fusion systems over the groups S and S, Ᏺ Ᏺz z respectively. A morphism ˛ Ᏺ Ᏺ of fusion systems consists of a functor ˛1 W ! z W Ᏺ Ᏺ, and a homomorphism ˛0 S S of groups, satisfying the following two ! z W ! z conditions.

(MF1) For every subgroup P of S, ˛1.P / ˛0.P /, and D (MF2) for each Hom .P; Q/, we have ˛0 ˛./ ˛0 (in right-hand 2 Ᏺ ı D ı notation). 898 MICHAEL ASCHBACHER and ANDREW CHERMAK

We most often write ˛ for both ˛0 and ˛1. In the case that ˛0 is given by inclusion of S into S, and ˛ is given by inclusion of Hom .P; Q/ into Hom .P; Q/ for z 1 Ᏺ Ᏺ all P;Q Ᏺ, we say that ˛ is an embedding. We reserve the symbolz to denote 2 an embedding of fusion systems. In general, if and are fusion systems over finite p-groups S and S, re- Ᏺ Ᏺz z spectively, then a morphism ˛ Ᏺ Ᏺ of fusion systems need not send Ᏺ-centric W ! z subgroups of S to -centric subgroups of S. For example, let G be a finite group, Ᏺz z take G to be the direct product of G with a nonidentity p-group R, let S be a Sylow z z p-subgroup of G, and take S S G. Then the inclusion map ˛ ᏲS .G/ ᏲS .G/ z D z\ W ! z z carries no centric subgroup to a centric subgroup. Recall that if Ᏺ is a saturated fusion system over a p-group S, then Ᏺrc denotes the set of subgroups of S which are both Ᏺ-centric and Ᏺ-radical. Given a p-local group Ᏻ .S; Ᏺ; ᏸ/, denote by ᏸrc the full subcategory of ᏸ whose objects are D the objects of Ᏺrc. Thus ᏸrc is a linking system on Ᏺrc. By[BCG 05, Th. B], the classifying spaces ᏸ and ᏸrc are homotopy equiv- C j j j j alent in the case that S is finite. This provides some justification for the following definition of morphism of p-local groups. Definition 2.10. Let Ᏻ .S; Ᏺ; ᏸ/ and Ᏻ .S; Ᏺ; ᏸ/ be pre-local groups. A D z D z z z morphism of pre-local groups from Ᏻ to Ᏻ is a pair .˛; ˇ/, where ˛ Ᏺ Ᏺ is a z W ! z morphism of fusion systems, and ˇ ᏸ ᏸ is a functor which, for each pair P;Q W ! z of objects of ᏸ, satisfies the following conditions. (MG1) ˛.P / ˇ.P /. Ä (MG2) For each Mor .P; Q/, the restriction of .ˇ. // to ˛.P / maps ˛.P / 2 ᏸ z into ˛.Q/, and ˛.. // .ˇ. // . D z j˛.P / (MG3) ˇ ıP ı ˛0. ı D ˇ.P / ı We say that the morphism .˛; ˇ/ is an embedding if ˛ is an embedding of fusion systems and

ˇ Morᏸ.P; Q/ Morᏸ.˛.P /; ˛.Q// W ! z is an injection for all P;Q ᏸ. We say that Ᏻ is a pre-subgroup of Ᏻ if there is an 2 z embedding .; ˇ/ of Ᏻ into Ᏻ, and in this case one may say simply that ˇ Ᏻ Ᏻ z W ! z is an embedding. If and are p-local groups, then a morphism of p-local groups Ᏻ Ᏻz from to is a morphism of pre-local groups Ᏻ Ᏻz .˛; ˇ/ .S; Ᏺ; ᏸrc/ .S; Ᏺ; ᏸrc/: W ! z z z Such a morphism is an embedding of p-local groups if it is an embedding of pre- local groups, and if ˛ is given by inclusion, we say that is a subgroup of . Ᏻ Ᏻz The next two results provide tools for carrying out the construction of morphisms, and particularly of embeddings, of p-local groups. 2-LOCALFINITEGROUPS 899

PROPOSITION 2.11. Let G1 be a subgroup of a group G2, and assume that there are Sylow p-subgroups Si of Gi with S1 G1 S2. Assume that for D \ each i, the fusion system Ᏺi ᏲSi .Gi / is saturated, and that we are given an WD rc rc Ᏺi -signalizer functor i . In addition, assume given a mapping ˇ Ᏺ Ᏺ such W 1 ! 2 that the following conditions hold for every P Ᏺrc. 2 1 (1) P ˇ.P /. Ä g g (2) For each g NG .P; S1/ we have ˇ.P / ˇ.P / . 2 1 D (3) For each Q Ᏺrc with P Q we have ˇ.P / ˇ.Q/. 2 1 Ä Ä (4) 2.ˇ.P // G1 1.P /. \ D Let Ᏺ1 Ᏺ2 be the inclusion functor, and write Ᏻi .Si ; Ᏺi ; ᏸi / for the p- W ! D local group which is canonically associated with Ᏺi and i (cf. 2.6). Then for any P;Q Ᏺrc there is a mapping 2 1 . / ˇP;Q Mor .P; Q/ Mor .P; Q/ W ᏸ1 ! ᏸ2 given by 1.P /g 2.ˇ.P //g; and .; ˇ/ is an embedding of Ᏻ1 into Ᏻ2. That is, 7! Ᏻ1 is a p-local subgroup of Ᏻ2. rc g rc Proof. Let P;Q Ᏺ and let g NG .P; Q/. Then P Ᏺ by 2.3. 2 1 2 1 2 1 Since P g Q, (3) yields ˇ.P g / ˇ.Q/. Then ˇ.P /g ˇ.Q/ by (2), and so Ä Ä Ä 1 g NG2 .ˇ.P /; ˇ.Q//. We have 1.P /g 1.P /h if and only if hg 1.P /, 2 1 D 1 2 while by (4), hg 1.P / if and only if hg 2.ˇ.P // G, which holds if 2 2 \ and only if 2.ˇ.P //g 2.ˇ.P //h. This shows that the mappings ˇP;Q in . / D are well-deﬁned injections. Visibly, ˇ preserves composition, so that ˇ is a functor from ᏸ1 to ᏸ2. Axiom (MG1) in 2.10 is an immediate consequence of (1). Next,

˛.1.1.P /g// ˛.cg / cg P 2.2.ˇ.P /g/ P 2.ˇ.1.P /g// P ; D D j D j D j so that (MG2) holds. Finally, for any g P , 2 ˇ.ı1;P .g// ˇ.1.P /g/ 2.ˇ.P //g ı .g/ ı .˛.g//; D D D 1;ˇ.P / D 1;ˇ.P / and so (MG3) holds.

PROPOSITION 2.12. Let G1 be a subgroup of a group G2, assume that there are Sylow p-subgroups Si of Gi with S1 G1 S2, and let i be a signalizer D \ functor on the fusion system Ᏺi ᏲSi .Gi /. Assume also that each Ᏺi is saturated. rc WD For P Ᏺ , denote by Ꮾ.P / the set of NG .P /-invariant p-subgroups of G2, 2 1 1 and set ˇ.P / Ꮾ.P / . Assume that the following two conditions hold for each D h i P Ᏺrc. 2 1 rc (1 ) ˇ.P / Ᏺ . .In particular, ˇ.P / S2:/ 0 2 2 Ä (2 ) 1.P / 2.ˇ.P // G1. 0 D \ 900 MICHAEL ASCHBACHER and ANDREW CHERMAK

Then ˇ satisfies conditions (1) through (4) of 2.11, and defines an embedding .; ˇ/ Ᏻ1 Ᏻ2 of p-local groups. W ! rc rc rc Proof. By (10), ˇ is indeed a map from Ᏺ1 to Ᏺ2 . Let P Ᏺ1 . As P Ꮾ.P /, g g 2 2 condition (1) of 2.11 holds. We have Ꮾ.P / Ꮾ.P / for any g NG .P; S1/, D 2 1 so that condition (2) holds. rc x Let Q Ᏺ1 with P Q. For any x NG1 .Q/ we have P S1. Then x rc 2 x Ä x 2 NG .Q/ Ä P Ᏺ , and so ˇ.P / ˇ.P / S2 by (2). Thus ˇ.P / 1 is a p-group, 2 1 D Ä h i invariant under NG1 .Q/, and hence contained in ˇ.Q/. Thus condition (3) holds. Condition (4) is equivalent to (20), and so the proof is complete. Example 2.13. Let G be a group and K E G such that G G=K is finite. Let x WD S be a p-subgroup of G such that S K 1, S Syl .KS/, and S Syl .G/. Set p x p x c \ D 2 2 Ᏺ ᏲS .G/, and for P Ᏺ set .P / Op .CG.P //. Let .P / be the preimage WD 2 N D 0 x in CG.P / of .PN /. Set Ᏺ ᏲS .G/. Then is an Ᏺ-signalizer functor, and Ᏻ ᏳS; .G/ is x D x x N x N WD x N x the natural p-local group ᏳS .G/. Let ˛0 S S be the restriction to S of the x x W ! x quotient map G G. For P S define ˛1 on HomG.P; S/ by ˛1 cg cg . ! x Ä W 7! N Observe that for P S, ˛0 NG.P; S/ NG.P; S/ is a surjection. Namely, Ä W ! x x x if g G with P g S then, as S Syl .KS/, there is k K with P gk S, and 2 x N Ä x 2 p 2 Ä we have g gk. Similarly C .P/ C .P /. Also, if g; h N .P; S/ with N G x G G N D g x h D 1 2 1 ˛1.cg / ˛1.ch/ then P S P with gh centralizing P , so gh NG.P / D 1 Ä x 2 with ŒP; gh P K 1. Thus ˛1 HomᏲ.P; S/ HomᏲ.P; S/ is a bijection. Ä \ D W ! x x x Therefore .˛ ; ˛ / is an isomorphism of with , and since is saturated, so is 0 1 Ᏺ Ᏺx Ᏺx Ᏺ. It is now easy to check that is an Ᏺ-signalizer functor, and then Ᏻ Ᏻ .G/ x WD S; is a p-local finite group by 2.7(b). Define ˇ ᏸ ᏸ by ˇ.P / P (on objects), and by ˇ .P /g .P/g W ! N D x W 7! N x N (on morphisms). One may now check that .˛; ˇ/ Ᏻ Ᏻ is an isomorphism of W ! N p-local finite groups. The hypothesis that S Syl .KS/ was used only to verify that the maps 2 p ˛ N .P; S/ N .P; S/ .P/ C .P / P 0 G G are surjective, and that N G for each rcW ! x x x x Ä 2 Ᏺ . Thus, that hypothesis may be replaced by the hypothesis that ˛0 NG.P; S/ rc W ! NG.P; S/ is surjective and .P/ 1 for each P Ᏺ . x x x N x D x 2 x 3. Amalgams In this section, an amalgam of groups will always mean a pair

˛1 ˛2 Ꮽ .A1 A1;2 A2/ D ! of injective group homomorphisms. A morphism from the amalgam Ꮽ to an amal- ˇ1 ˇ2 gam Ꮾ .B1 B1;2 B2/ is a triple . J ¿ J 1; 2 / of injective D ! D j 6D Â f g group homomorphisms J AJ BJ , such that ˛i i 1;2ˇi for i 1; 2. W ! D D 2-LOCALFINITEGROUPS 901

For example, if G is a group with subgroups G1 and G2, and G1;2 is a subgroup of G1 G2, then there is a subgroup amalgam Ᏻ given by the inclusion maps \ ˛i G1;2 Gi .A completion of an amalgam Ꮽ is an isomorphism Ꮽ Ᏻ of W ! W ! Ꮽ with a subgroup amalgam Ᏻ in a group G, such that G G1;G2 . One often D h i abuses the terminology and says simply that G is a completion of Ꮽ. Let

Ꮽ .G1 B G2/ D ! be an amalgam and let G G1 B G2 be the associated free amalgamated product. D Then G is a completion of Ꮽ, and indeed the universal completion of Ꮽ. We identify Ꮽ with the subgroup amalgam of G which is the image of Ꮽ under this completion, and in particular regard G1, G2 and B as subgroups of G with G1 \ G2 B. D For any subgroup X of G, denote by X G the set of right cosets of X in G. n Set i Gi G. Then .Ꮽ/ is the graph whose vertex set is the disjoint union D `n D V ./ 1 2, and whose set of edges is the set E./ of 2-subsets G1x; G2x D f g with x G. We call .Ꮽ/ the standard tree associated with Ꮽ and with G, and 2 D we refer to [Se] for the fact that really is a tree. Observe that G is represented as a group of automorphisms of via right multiplication, and that the kernel of this representation is the largest normal subgroup of G which is contained in B. Evidently, G acts transitively on E./, while 1 and 2 are the (distinct) orbits for G on V ./. It is also evident that G is locally transitive on ; that is the stabilizer Gı of any vertex ı acts transitively on the set .ı/ defined by .ı/ ı; E./ : D f 2 j f g 2 g For any subgroup or element X of G, write X for the subgraph of induced on the set of vertices which are fixed by X. For any connected graph , and vertices ˛ and ˇ of , the length of the shortest geodesic path from ˛ to ˇ in is denoted d.˛; ˇ/. Then .; d/ is a discrete metric space, and automorphisms of are isometries. An isometry of a tree is said to be hyperbolic if it fixes no vertices or edges. The following two results, noticed first by J. Tits[Tit70], are elementary.

LEMMA 3.1. Let h be an automorphism of a tree , and denote by ƒ.h/ the intersection of all the h-invariant subtrees of . Suppose that there exists an edge ; ı of such that f g (1) d. ; h/ d.ı; ıh/ 0 and D ¤ (2) ; ı is not ﬁxed by h. f g Then h is hyperbolic, and the set of all edges ; ı which satisfy condition .1/ is f g the edge set of ƒ.h/. 902 MICHAEL ASCHBACHER and ANDREW CHERMAK

LEMMA 3.2. Let be a tree, let g be a hyperbolic isometry of , and deﬁne ƒ ƒ.g/ as in 3.1. Set d min d.ı; ıg / ı V ./ . Then: D D f j 2 g (a) ƒ is isomorphic to the graph ޚ whose vertex set is the set ޚ of integers, and N whose edges are the pairs n; n 1 for n ޚ. f C g 2 (b) There is an isomorphism ƒ ޚ such that 1g n n d for all n ޚ. W ! N W 7! C 2 (c) For any vertex of , the geodesic in from to g has length d 2e, C where e is the minimal distance from to a vertex of ƒ.

LEMMA 3.3. Let be a tree, and let x; y Aut./, ı x, y, and 2 2 2 .˛0; : : : ; ˛d / the geodesic from to ı. Suppose x does not ﬁx ˛ ˛d 1 and y D does not ﬁx ˛1. Then xy is hyperbolic.

Proof. Observe ﬁrst of all that .˛0y; : : : ; ˛ y/ is the geodesic from ˛0y d D to ıy, so that the path .˛ y; : : : ; ˛0y; ˛1; : : : ; ˛ / contains a geodesic g from D d d ıy to ı. Indeed as ˛1 ˛1y, g is of length 2d. Then g0 .˛d 1y; : : : ; ˛0y; ¤ D 1D ˛1; : : : ; ˛ / is the geodesic from ˛y to ı, and since ˇ ˛x ˛, it follows that d D 6D g ˇ is the geodesic of length 2d from ˛y to ˇ. But ıy ıxy and ˇxy ˛y, and 0 D D so d.ı; ıxy/ d.ˇ; ˇxy/. The lemma now follows from 3.1. D We will work under the following hypothesis for the remainder of this section.

HYPOTHESIS 3.4. G G1 B G2 is the free amalgamated product associated D with an amalgam Ꮽ .G1 B G2/; D ! and G1, G2, and B are regarded as subgroups of G in the canonical way. Let be the standard tree associated with Ꮽ and G, and let i be the subset Gi G of the n vertex set of . Assume (1) There is a subgroup S of G such that S is a (possibly inﬁnite) Sylow p-subgroup of each of the groups G1, G2, and B. (2) NG .S/ B Gi , for i 1 and 2. i Ä ¤ D The vertex Gi of will most often be denoted i . LEMMA 3.5. Assume Hypothesis 3.4, and assume also that g g B . / S g G1 G2;S B S : f j 2 [ Ä g D Then:

(a) S 1; 2 , and D f g (b) S Syl .G/. 2 p Proof. It follows from . / and from[Asc86, 5.21] that NG .S/ is transitive i on S . i /. Since NG .S/ B, by 3.4, we obtain (a). i Ä Let S be a p-subgroup of G containing S and let x S . Then x is finite, 2 j j so that 3.2 implies that x ¿. Choose ı x and S with d d.ı; / 6D 2 2 WD 2-LOCALFINITEGROUPS 903 minimal. Suppose that d > 0 and let .ı; ı0; : : : ; 0; / be the geodesic from ı to in . Then x Gı , and there exists y S G . As xy is finite, this contradicts … 0 2 0 j j 3.3, and so we conclude that d 0. Then (a) yields S Gi for some i. Since D Ä S is a maximal p-subgroup of Gi we then have S S, and thus S is a maximal D p-subgroup of G. Let P be a finite p-subgroup of G, and let P0 be a subgroup of P which fixes a vertex of , and which is maximal for this condition. Then NP .P0/ acts on the tree 0 P , and no element of NP .P0/ P0 fixes a vertex of 0. Now, NP .P0/ P0 D 0 consists of hyperbolic isometries on 0, by 3.2, and hence NP .P0/ P0 P . D D Since G is edge-transitive on , it follows that P is conjugate to a subgroup of G1 or G2. Then P is conjugate to a subgroup of S, by 3.4(1), and S is a Sylow p-subgroup of G.

LEMMA 3.6. Assume Hypothesis 3.4, let P be a ﬁnite subgroup of S, and let g NG.P; S/. Set F0 P . Then there exists a positive integer n, elements 2 D g1; : : : ; gn of G1, elements h1; : : : ; hn of G2, and subgroups Ei and Fi of S, 1 Ä i n, such that the following conditions hold. Ä (a) gi B for 1 < i n, and hi B for 1 i < n. … Ä … Ä gi hi (b) Ei Fi 1 and Fi Ei for all i with 1 i n. D D Ä Ä (c) g g1h1 : : : gnhn. D Moreover, the minimal length of g as a word in the generating set G1 G2 is 2n 2 [ if g1; hn B, 2n 1 if exactly one of g1 and hn is in B, and 2n if neither g1 nor 2 hn is in B.

Proof. Since G1 and G2 generate G, we may choose elements gi G1 and 2 hi G2, satisfying the conditions in (a) and (c). Set ˛0 2, ˇ0 1hn, and for 2 D D 1 i n set wi gn i 1hn i 1 : : : gnhn, ˛i 2wi , and ˇi 1hn i wi . Then Ä Ä D C C D D q .˛0; ˇ0; : : : ; ˇn 1/ is a path in with ˇi ˇi 1 and ˛i ˛i 1 for i < n 1. D 6D C 6D C Thus q is a geodesic from ˛0 to ˇn 1, and if g1 B then also the path q˛n is a … geodesic. In particular if n > 1 or g1 B then d.a0; an/ > 0, and wn B. … … Take (b) as the deﬁnition of the groups Ei and Fi for i > 0. We now show that these groups are contained in B. Let x F0, set y0 x, and for 1 i n 2 D Ä Ä deﬁne xi and yi recursively, by

gi hi xi yi 1 and yi xi : D D Suppose that for some j , either xj or yj is not in B, and let j be the smallest such gj index. Suppose that xj B. Then yj 1 B, and so xj yj 1 G1 B. Then … 2 D 2 g 1 1 1 x h g : : : h xj hj : : : gnhn D n n j 904 MICHAEL ASCHBACHER and ANDREW CHERMAK is an alternating product of elements of G2 and G1, in which none of the factors lies in B except possibly for the ﬁrst and the last. It follows from paragraph one g g that x B, whereas x S B. A similar argument shows yj B. Therefore … 2 Ä 2 xi and yi are in B for all i, and so each of the groups Ei and Fi is contained in B. Since S is a Sylow p-subgroup of B, we may adjust our choices of the elements gi and hi , via right multiplication by elements of B, to ensure that Ei and Fi are in S for all i. It remains to prove the ﬁnal statement in the lemma. This follows since, by paragraph one, the length `.g/ of g as a word in the generating set G1 G2 for G [ is equal to the shortest distance in the tree .Ꮽ/ from a vertex in 1g; 2g to a f g vertex in 1; 2 . f g We have the following immediate consequence of 3.6.

COROLLARY 3.7. Assume Hypothesis 3.4. Then

ᏲS .G/ ᏲS .G1/; ᏲS .G2/ : D h i For any subgroup X of G, and any elementary abelian p-subgroup A of X, denote by Ᏹn.X; A/ the set of elementary abelian p-subgroups of X which have n order p and which contain A. Write Ᏹn.X/ for Ᏹn.X; 1/. In the remainder of this section we assume the following hypothesis.

HYPOTHESIS 3.8. Hypothesis 3.4 holds, and so do the following conditions.

(1) There is a normal subgroup Z of G1 of order p, and Z is the unique subgroup of order p in Z.S/.

(2) There exists U Ᏹ2.G2;Z/ with U E G2, and G2 acts transitively on Ᏹ1.U /. 2 (3) B NG .U / NG .Z/. D 1 D 2 (4) For each X H;K;B , X is transitive on its set of maximal p-subgroups. 2 f g LEMMA 3.9. Let P be a subgroup of S and let X be a subgroup of Z.P / of order p. g (a) Let g G2 with P S. Then one of the following holds. 2 Ä g (i) g B, and neither P nor P is contained in CG.U /. 2 (ii) P CG.U /. Ä g g (b) If X Z and X U then there exists g G2 with X Z and with P S. ¤ Ä 2 D Ä g Proof. Let g be as in (a), and set Q P . Since U E G2, we have P CG.U / D Ä if and only if Q CG.U /. Since conclusion (ii) of (a) does not hold, neither P nor Ä Q is contained in CG.U /. Set G2 G2=CG .U /. Then P and Q are nontrivial x D 2 x x p-subgroups of S. Hence by 3.8(2), G is isomorphic to a subgroup of GL .p/ x x2 2 containing SL2.p/, and B NG .Z/ has index p 1 in G2. Thus P Q S, D 2 C x D x D x 2-LOCALFINITEGROUPS 905

g and B NG .S/. Since P Q, we then have g B. Thus g B and (a) is x D x2 x D N 2 x 2 proved. g Suppose that Z X U . By 3.8(2) there exists g G2 with X Z. ¤ Äg 2 D Since X Z.P /, we have P CG .Z/ B. By 3.8(4) there exists h B with Ä Ä 2 Ä 2 P gh S. Replacing g with gh, we obtain (b). Ä g For ı and g G with ı i g for some i, Z will denote Z if i 1, 2 2 D ı D and U g if i 2. This notation is well deﬁned as a consequence of 3.8(3). D LEMMA 3.10. Let † be a subtree of and let †. Set Y Z ı † . 2 Dh ı j 2 i Then CG .Y / ﬁxes † vertex-wise.

Proof. Let †, set X CG .Y /, and assume that X G†. Among all 2 D — pairs .ı; x/ with ı † and x X with ıx ı, choose .ı; x/ so that d d.ı; ıx/ 2 2 ¤ WD is minimal. Let ˛ †.ı/ be of distance d 1 from . Then X ﬁxes ˛ and 2 centralizes Y , so X centralizes Z . Thus X CG .Z / G by 3.8(3), and ı Ä ˛ ı Ä ı contrary to our choice of ı.

LEMMA 3.11. Let ı and be distinct vertices in 1 with Z Z . Then ı D d.ı; / 6. Moreover, the following hold. (a) Let ˛; ˇ 1 with d.˛; ˇ/ 2, and let X be a subgroup of G ﬁxing ˇ and 2 D centralizing Z˛ and Zˇ . Then X ﬁxes ˛.

(b) Let ˛0; ˛4 1 with d.˛0; ˛4/ 4, and such that Z˛ centralizes Z˛ . Then 2 D 4 0 Za4 fixes a0. Proof. Suppose d d.ı; / < 6. Then d 2 or 4. If d 2, and .ı; ˇ; / WD D D is the geodesic from ı to , then Z Z Z is of order p, which is not the case. ˇ D ı Thus d 4. Write .ı; ˇ; ı ; ˇ ; / for the geodesic from ı to . Then D 0 0 Zˇ Zı Zı Z Zı Zˇ : D 0 D 0 D 0 By edge-transitivity, we may take ı and ˇ . Then U Z Z , and 0 1 0 2 ˇ 0 ˇ D gD D g D by local transitivity there exists g G1 with ˇ ˇ . Then U U , so g B, 2 D 0 D 2 and ˇ ˇ . Then d < 4, and we have a contradiction. D 0 Assume the hypothesis of (a). Without loss, ˇ 1 and 2 .˛/ .ˇ/. D f g D \ As X fixes ˇ and centralizes Z˛ and Z , we obtain X CH .Z˛Z / CH .U / ˇ Ä ˇ D Ä B G˛, establishing (a). Ä Now assume the hypothesis of (b), and take X to be Z˛4 . Let .˛0; : : : ; ˛4/ be the geodesic from ˛0 to ˛4. Then X Za Z˛ Za CG.Z˛ /, and so X Ä 3 D 2 4 Ä 2 fixes ˛2 by (a). Then, since dist.˛0; ˛2/ 2 and X centralizes Z˛ for i 0; 2, X D i D fixes ˛0 by another application of (a).

HYPOTHESIS 3.12. Hypothesis 3.8 holds, and every subgroup of G1 of order p is conjugate in G1 to a subgroup of U . 906 MICHAEL ASCHBACHER and ANDREW CHERMAK

LEMMA 3.13. Assume Hypothesis 3.12. Let P be a subgroup of S and let X be a subgroup of Z.P / of order p. g g (a) If X U then there exists g G1 with X U and with P S. — 2 Ä Ä (b) If X Z then there exists g G with X g Z, .XZ/g U , .P U /g S, ¤ 2 g D D Ä and g g1g2 where gi Gi and P 1 S. D 2 Ä g Proof. Suppose that X U . By 3.12 there exists g G1 with X U . Set g — g 2 Ä Y X . Then YZ U , so that P CG1 .U / B, and by 3.8(4) there exists D ghD Ä Ä h B such that P CS .Y /. Replacing g with gh, we obtain (a). 2 Ä Now assume that X Z. If X U we appeal to 3.9(b), with PU in the role ¤ Ä g g of P , in order to obtain g2 G2 with X 2 Z and .P U / 2 S. With g1 1, (b) 2 D Ä D holds in this case. So assume that X U . If U centralizes X we apply (a) to PU , g — g g obtaining g1 G1 such that X 1 U and .P U / 1 S. Then .XZ/ 1 U , and by 2 Äg g Ä g g D 3.9(b) there exists g2 G2 with X 1 2 Z and .P U / 1 2 S. Thus (b) also holds 2 D Ä in this case, and we are reduced to the case where U does not centralize X. Since P S, 3.8(3) implies that ŒU; X ŒU; S Z. Then XZ E PU . By 3.12 there Ä g1 D g1D g1 exists g1 G1 with X U . Then .XZ/ U , so that .P U / NG1 .U / B. 2 Ä D g Ä D By 3.8(4) we may assume that g1 was chosen so that .P U / 1 S. Then (a) applies, Ä and completes the proof of (b). The next result amounts to a re-working of[LO02, Lemma 1.4] in our tree- theoretic setup. The formulation given here is different in several respects from the one in[LO02], but the main idea of the proof has not been altered. We remark that we shall only use part (c) of 3.14, and this will occur only once, in the proof of 9.2.

PROPOSITION 3.14. Assume Hypothesis 3.12. Set D NG.Z/ and assume D that ᏲS .D/ ᏲS .G1/. Denote by the set of all pairs .P; g/ such that P is a ¤ ﬁnite subgroup of S, g ND.P; S/, and cg HomG .P; S/. Set 2 … 1 ᏼ P .P; g/ for some g D ; D f j 2 2 g and let P ᏼ. Choose .P; g/ so that the length `.g/ of g, as a word in the set 2 2 G1 G2 of generators of G, is minimal. Then ŒP; U 1, .P U; g/ , and upon [ D 2 replacing P with a suitable subgroup of CS .P /P containing P , we have

(a) `.g/ 5, and g g1g2g3g4g5 where gi G2 for i odd, and gi G1 for i D D 2 2 even. g :::g (b) The elements g1 through g5 in (a) may be chosen so that U Z.P 1 i /, Ä and P g1:::gi S for all i, 1 i 5. Ä Ä Ä g g g (c) There exists E Ᏹ3.Z.P /; U / such that U E , CB .E/ B, and CB .E / 2 Ä Ä Bg . Ä 2-LOCALFINITEGROUPS 907

g Proof. Set Q P . Since g D, also g ND.PZ; QZ/, so that we may D 2 2 assume Z P . Ä By 3.6 we have g g1 : : : gn, where each gi is in G1 G2, and where g :::g D [ P 1 i S for all i, 1 i n. Moreover, the sequence .g1; : : : ; gn/ may be Ä Ä Ä chosen so that `.g/ n. D gi gi Set P0 P , Z0 Z, and for 1 i n set Pi Pi 1 and Zi Zi 1. If Zi D D Ä Ä D D D Z for some i with 0 < i < n, then the minimality of n implies that, for x g1 : : : gi D and y gi 1 : : : gn, we have cx HomG1 .P; Pi / and cy HomG1 .Pi ;Q/. But D C 2 2 in that case we get cg cxcy HomG .P; Q/, contrary to hypothesis. Thus: D 2 1

(1) Zi Z if and only if i 0 or n. D D

By 3.13(b) there exist elements v1 through vn 1 of G1G2, such that

vi v v (2) Z Z, .ZZi / i U , and .Pi U/ i S. i D D Ä

Set v0 vn 1, and for each i, 0 i n, choose ri G1 and si G2 with D D 1 Ä Ä vi 2 2 vi ri si . Set ki vi 1gi vi for i 1, and set Pi0 Pi for i 0. Then Pi0 S D D ki D Ä for all i, by (2), and .Pi0 1/ Pi0 for i 1. Notice that D . / g1 : : : gn k1 : : : kn: D

Suppose that, for all i 1, we have cki HomG1 .Pi0 1;Pi0/. Choose ti G1 1 2 1 2 so that ki ti CG.Pi0 1/, and set t t1 : : : tn. Then gt CG.P / by . /, so 2 D 2 that cg HomG1 .P; Q/, contrary to hypothesis. Thus, cki HomG1 .Pi0 1;Pi0/ 2 1 … for some i 1. Since ki .ri 1si 1/ gi ri si is of length 5, it follows from the D minimality of n that n 5. Ä Set ˛ ˛0 1, and deﬁne vertices ˛1 through ˛n by ˛i ˛i 1gi . Let D D D † be the subtree of generated by ˛0; : : : ; ˛n , and let †i be the subtree of † f g generated by ˛0 and ˛i . Thus † is the union of its subtrees †i , and †i is the geodesic from ˛ to ˛i in . Minimality of n implies that g1 G2 G1, as otherwise we may replace g 2 .P; g/ with .P 1 ; g2 : : : gn/. Then gi is in G2 G1 for i odd, and in G1 G2 for i even. Observe that ( i 1 if i is odd, and . / d.˛; ˛i / C D i if i is even:

Notice that Zi Z˛ . Then 3.11 implies that Z Z for any ı V .†/ with D i ı ¤ 2 d.˛; ı/ 5. Since n 5, it now follows from . / that n 5. Ä Ä D The reader may ﬁnd it convenient to have a “picture” of †, at this point, as a visual reference for the remainder of the argument. 908 MICHAEL ASCHBACHER and ANDREW CHERMAK

˛2

j ˛1 ı j j ˛4 ˛ ˇ ˛5 ı ı ı ıııı j ı j ı j ˛3

Since Pi S G˛ , Pi fixes every vertex of †i , and then 3.10 implies that Ä Ä 0 ŒZ ;Pi 1 for every ı †i . Set ˇ 2. Then from paragraph one of the proof ı D 2 D of 3.6, for i odd, †i is the tree induced on the geodesic .˛; ˇ; ˛gi ; ˇgi 1gi ;:::; ˛g1 : : : gi /. In particular, ˇ is a vertex of †i for i odd. Since Z U , it follows ˇ D that ŒU; Pi 1 for i odd. But for i odd we also have gi G2 NG.U /, and then Dgi 2 Ä since Pi Pi 1, we conclude that: D (3) ŒU; Pi 1 for all i, 0 i 5. D Ä Ä From the description of †5 above, ˇg is the vertex of †5 at distance 5 from ˛, and ˛g ˛1g .ˇg/. Set ˇ ˇg, let ˛ be the vertex in ˇ at distance 4 6D 2 0 D 0 0 from ˛, and let ˛00 be the vertex of †5 at distance 2 from ˛. By (3), U centralizes g g g Z˛ , so as Zˇ ZZ˛ Z1 , we get Z1 CG.U / CG.Z˛ /. Then Z1 0 0 D 0 g Ä Ä g 00 fixes ˛00, by 3.11(b). Also ŒZ1 ; Z 1, so that 3.11(a) implies Z1 fixes ˛. Since g g g D g g U .ZZ1/ ZZ , we conclude that U G˛ G . That is, U B, and so D D 1 Ä \ ˇ Ä QU g is a p-subgroup of B. By 3.8(4) there exists h B with .QU g /h S, and 2 Ä we may replace .P; g/ with .P U; gh/, without increasing the length n. Thus, we may assume henceforth that U P . By symmetry between .P; g/ and .Q; g 1/), Ä we may assume also that U Q. Then since g1 NG.U /), also U P1. Since Ä 2 Ä g5 NG.U / we then obtain U P4. 2 Ä As g2; g4 G1 and U P1 P4, we have Z P2 P3. Since g3 NG.U /, 2 Ä \ Ä \ 2 we have U P2 if and only if U P3. Suppose that U P2. Then Ä Ä — g3 g3 Z .U P2/ U P3 Z; D \ D \ D contrary to g3 G2 G1. Therefore U Pi for all i. Then by (3): 2 Ä (4) U Z.Pi / for all i, 0 i 5. Ä Ä Ä 2-LOCALFINITEGROUPS 909

g :::g Set U0 U U 1, and for 1 i 5, let Ui U 1 i . For 0 i 5 set D D Ä Ä D Ä Ä Ei Ui 1Ui . As g1 G2, we have U U1 E2, and so D 2 D Ä g4 g4 g3g4 g3g4 Z Z U U E E4: D Ä D Ä 2 D g 1 1 g 1 Also g5 G2 G1, so that Z Z 5 E E4. Then U ZZ 5 E4. 2 6D Ä 5 D D Ä Since g5 G2, also U E5. 2 Ä g g Set F CB .E0/. Then F CBg .E5/, and B is the stabilizer in G of the D g Dg g g edge ˛5; ˇg of †5. Next U E0 E5, and U Zˇg , so that F centralizes f gg Ä D D Zˇg . Then F fixes the vertex ˛0 of †5, by 3.11(a). Denote by ˇ00 the vertex of †5 at distance 3 from ˛, and hence adjacent to ˛0. From an earlier remark, we have ˇ ˇg4g5, and so 00 D g4g5 g3g4g5 g3g4g5 Zˇ U U E E5: 00 D D Ä 2 D Thus F g centralizes Z , and so F g fixes every vertex in .ˇ / by 3.11(a). In ˇ 00 00 particular, F g fixes ˛ . Since ŒU; F g 1, F g fixes every vertex in .ˇ/ by 00 D 3.11(a). Thus, F g fixes ˛ and ˇ, and so F g B. This yields the first part of (c), Ä g with E0 in the role of E. Since Zˇ Zˇ E5, 3.11(a) yields CB .E5/ B , and 00 0 Ä Ä thus all parts of 3.14 have been established.

4. Spin7.F/ Let p be an odd prime, let F be an algebraic closure of the field of p elements, x let V be a vector space over F (of finite dimension d), and let f be a symmetric, z x nondegenerate bilinear form on V . The form f is essentially unique, as V has an z z orthonormal basis with respect to f . The isometry group O.V ; f / will be denoted z also O.V/ (and O .F/). The identity component of O.V/ is denoted .V/, and z d x z z has index 2 in O.V/. Indeed, we have O.V/ .V/ , where is a reflection z z D z h i on V . In the case that d is odd, we have O.V/ .V/ I . The universal z z D z f˙ g covering group of .V/ is denoted Spin.V/ (or Spin .F/). z z d x There is a rational representation Spin.V/ .V/, with kernel contained W z ! z in Z.Spin.V //. From [C], one has ker./ 2, and ker./ Z.Spin.V // if d is z j j D D z odd.

For any subset or element D of Spin.V/, we write CV .D/ and ŒV ; D for z z z CV .D/ and ŒV ; D, respectively. z z Let T be a maximal torus of Spin.V/. By a weight of T on V we mean a z z z z homomorphism T F such that the space V v V va .a/v for all W z ! x z D f 2 z j D a T is nonzero. The set of such weights is denoted ƒ.T/. 2 zg z A hyperbolic line in V is a nondegenerate subspace ` of V of dimension 2. z z Such a subspace has exactly two 1-dimensional singular subspaces (or points), and 910 MICHAEL ASCHBACHER and ANDREW CHERMAK from this one may easily deduce that .`/ F and that O.`/ .`/ t , where Š x D h i t is an involution which interchanges the singular points of `. The following result is well known, and its proof is elementary. LEMMA 4.1. Let .V ; f / be a nondegenerate orthogonal space over F of z x dimension d, and let T be a maximal torus of .V/. Then there exists a set z z `.T/ `1; : : : ; ` of T -invariant, pairwise orthogonal hyperbolic lines in V , z D f kg z z such that the following hold. (a) d 2k or d 2k 1. D D C (b) ŒV; T `1 `k, and either CV .T / 0 or CV .T/ is a nonsingular z z D C C z D z z 1-space, orthogonal to ŒV; T . z z 1 (c) Each ì is a sum of two weight spaces V and V 1 , where . These z z ¤ weight spaces are the singular points of ì . The set `.T/ is uniquely determined by the conditions (a) and (c). Conversely, for z any maximal set of pairwise orthogonal, hyperbolic lines in V , there is a unique ᐁ z maximal torus T in .V/ with `.T/ ᐁ. z z z D From now on, let V be a vector space of dimension 7 over F, and set H z x z D Spin.V/. Write Z for the kernel of . Then Z z where z is of order 2. z D h i It is well known that an involution t in an orthogonal group (over a field of characteristic different from 2) lifts to an involution in the corresponding spin group if and only if the dimension of the commutator space of t is divisible by 4. This implies the following result. LEMMA 4.2. Let x H with .x/ 2. Then x 2 if and only if 2 z j j D j j D dim.ŒV ; x/ 4. z D Let T be a maximal torus of H . By 4.1, the commutator space ŒV; T is the z z z z orthogonal direct sum of three hyperbolic lines `1, `2, and `3, where each ì is 1 a sum of two weight spaces for T , with weights i and . For i 1; 2; 3, fix z i D a basis x2i 1; x2i for ì of singular vectors, with f .x2i 1; x2i / 1. Then 4.1 f g D yields

.4:2:1/ ŒV; T `1 `2 `3: z z D C C Let x7 CV .T/, with f .x7; x7/ 1. Then 2 z z D

CV .T/ ŒV; T ? Fx7: z z D z z D x Identify V with F.7/, via the ordered basis .x ; : : : ; x /. z x 1 7 The semidirect product V H is an algebraic group in which T is a maximal z z z torus, and in which V is the unipotent radical. Let be a Frobenius endomorphism z of V H which induces the pth-power map on T . Then ﬁxes `.T/ pointwise, and z z z z ﬁxes the vectors x1 through x7. 2-LOCALFINITEGROUPS 911

Denote also by the pth-power automorphism of F, and set x [ 2k F CF. /: D x k 0 Evidently, F is a subfield of F. Denote by V the F-span of x1; : : : ; x7 in V , and x f g z by H the group of F-rational points in H , with respect to the matrix representation z given by the chosen basis for V . The restriction of f to V V defines an orthogonal z form on V , and .H / is contained in O.V /. Set T T H, and set E x D z \ D f 2 T x2 1 . j D g Since O.V/ .V/ t for any reflection t O.V/, it follows that .V/ is the z D z h i 2 z z group SO.V/ of determinant 1 isometries. Any element x of O.ŒV; T / extends z z z to an element of .V/, since we are free to adjust the action of x on CV .T/ by z z z 1. In particular, there exists an element w0 of H such that w0 acts on ŒV; T by ˙ z z z the permutation .x1 x2/.x3 x4/.x5 x6/ of the basis vectors; and then x7w0 x7. D Evidently w0 commutes with , so w0 H. By 4.1, w0 NH .T /, and one can 2 2 check that w0 acts on .T / by inversion. Since every element of F is a square, it follows that w0 acts on T by inversion. Similarly we choose elements w1, w2, w3, w, and of NH .T / so that:

.w/ .x1 x3 x5/.x2 x4 x6/; D ./ .x1 x3/.x2 x4/; and D .wi / .x2i 1 x2i /; 1 i 3: D Ä Ä Thus, we may take w0 w1w2w3. D n (4.2.2) Fix n 0 and set q p2 . Denote by ! both the inner automorphism D of H induced by w0, and the identity map on F. For any k 0, let be the z x k automorphism of H deﬁned by z 8 k < .!/2 if p 3 mod 4, and Á k k D : 2 if p 1 mod 4: Á We now ﬁx n, and set

n: D n Notice that since !2 1, we in fact have 2 unless n 0 and p 3 mod 4. D D D Á The restriction of to H will again be denoted . For any subgroup D of H , write D for CD./, and write F for CF./.

LEMMA 4.3. Set W z; w1; w2; w3; w; . Then W H , and the follow- D h i Ä ing hold.

(a) T CH .T /, and w0 acts on T as inversion. D 912 MICHAEL ASCHBACHER and ANDREW CHERMAK

(b) NH .T / WT . D (c) AutH .T / Sym.4/ C2. Š 2 (d) Set E x T x 1 . Then E is an elementary abelian subgroup of T D f 2 j D g of rank 3, CH .E/ T w0 , NH .E/ TW , and NH .E/ NH .T /. D h i z D z D Proof. Set ᏸ `1; `2; `3 , and denote by T the pointwise stabilizer of ᏸ D f g z in H . Thus T T , and .T / is contained in the direct product of the orthogonal z Ä z z groups O.ì /, so that T T w1; w2; w3 . z D zh i By 4.1, CH .T/ T and NH .T/ permutes ᏸ. Since AutT .ì / contains its z z Ä z z z z centralizer in GL.ì /, we have T CH .T/. Similarly CH .T / T . z D z z D Evidently, each of the elements wi , w, and commutes with both and w0, and so W NH .T /. From the definitions of these elements, we obtain Ä .w0/ Z..W //, . w1; w2; w3 / is elementary abelian of order 8, . w; / 2 h i h i Š Sym.3/, and w; acts naturally on . w1; w2; w3 / and on ᏸ. We conclude that h i f g WT contains H T , WT NH .T /, that W T=T Sym.4/ C2, and that \ D Š w0 T=T Z.W T=T /. As w0 inverts T , it inverts T . Thus, parts (a) through (c) h i D z hold. Notice that CH .ŒV; T / Z since .H / contains no reflections. As ! inverts D 2 T and induces a power map on T , T contains the group E t T t 1 . z D f 2 j D g From 4.2, .E/ is a fours group and E is elementary abelian of order 8. The lines ` are the fixed point spaces for the three involutions in .E/ on ŒV ; E, i z so that NH .E/ NH .T/, and hence NH .E/ NH .T /. Since w0 inverts T , z D z z D w0 CH .E/. 2 Since CH .E/ T , we have CH .E/ T w1; w2; w3 . By 4.2, for i; j; k Ä z D h i f g D 1; 2; 3 and x E such that ŒV ; x `j ` , each of wi , wi x, wj w , and wj w x f g 2 z D C k k k is of order 4. Then x does not centralize wi or wj w , and CH .E/ T w0 , k D h i completing the proof of (d).

We may choose z1 T so that z1 acts as the scalar 1 on `1 `2, and as 1 2 z C on `3. Then z1 centralizes x7. Set U z; z1 , B NH .U /, B B H, and D h i z D z D z \ denote the identity component of B by B0. Set B0 B0 H . z z D z \ LEMMA 4.4. The following hold.

(a) V is the orthogonal direct sum of ŒV ; U and CV .U /, of dimensions 4 and 3, z z z respectively, over F . zN (b) B is the stabilizer in H of ŒV ; U and of CV .U /. z z z z 0 (c) B CH .U / L1L2L3, where Li SL2.F/ and where Li Lj SL2.F/ z D z D z z z z Š x z z Š x SL .F/ for all distinct i and j . Moreover, the indexing may be chosen so that 2 x (i) L3 centralizes ŒV ; U and L3 .CV .U //. z z z D z (ii) L1L2 centralizes CV .U / and L1L2 .ŒV ; U /. z z z z z D z 2-LOCALFINITEGROUPS 913

(iii) The maximal singular subspaces of Œ V ; U spanned by x1; x4 and z f g x2; x3 over F are natural SL2.F/-modules for L1, and the maximal f g x x z singular subspaces spanned by x1; x3 and x2; x4 are natural SL2.F/- f g f g x modules for L . z2 0 (d) U Z.B /, and notation may be chosen so that z1 L1. When zi is the D z 2 z involution in L , then zi z z1z2 z3: D D 0 0 (e) B B w1 B w2 , where both w1 and w2 interchange L1 and L2 by z D z h i D z h i z z conjugation.

Proof. As ŒV ; U ŒV ; z1 and CV .U / CV .z1/, part (a) is immediate from z D z z D z our choice of z . The stabilizer in H of ŒV ; U normalizes the unique subgroup U 1 z z of H containing Z which acts as I on ŒV ; U and as I on ŒV ; U . Similarly, z z z ? the stabilizer in H of CV .U / normalizes U , establishing (b). z z Set K C .C .U //0 and L C .ŒV ; U /0. From the Steinberg rela- z H V z3 H z D z z D z 0 tions, K L1 L2, and Li SL2.F/ for i 1; 2; 3. Thus B is a commuting z D z z z Š D 0 z product of these three copies of SL2.F/, and U Z.B /. Here ŒV ; U is a natural x Ä z z 4.F/-module for L1L2, and is therefore a direct sum of two natural SL2.F/- x z z 0 x modules for each of L1 and L2. Observe that T B since T is connected. z z 0 z Ä z z Then T is a maximal torus of B , and hence T T1T2T3, where Ti T Li . z z z D z z z z WD z \ z Since ŒL1; L2L3 1, the irreducible L1T -submodules of ŒV ; U are weight z z z D z z z spaces for T T . Since these irreducible L T -submodules are also maximal sin- z2 z3 z1 z gular subspaces of ŒV ; U , the only possibilities are that the two irreducible L T - z z1 z submodules of ŒV ; U are z

x1; x3 ; x2; x4 or x1; x4 ; x2; x3 : fh i h ig fh i h ig We may therefore choose the indexing so that (c)(iii) holds. 0 To complete the proof of (c), it remains to show that B CH .U /. This z D z will follow from (e), and since (d) is immediate from the parts of (c) which have already been established, we now need only prove (e). The group O.ŒV ; U / is generated by .ŒV ; U / together with a reflection z z interchanging .L1/ and .L2/. Similarly, O.CV .U // is generated by .CV .U // z z z z together with a reflection on C .U /. Since .H/ contains no reflections on V , we V z z 0 z have B B 2, and then (e) follows from the definitions of w1 and w2. j zW z j D Notice that each of the groups Li is -invariant. Set Li Li H, 1 i 3. z D z \ Ä Ä The following result should then be evident:

LEMMA 4.5. All parts of 4.4 hold, with B, B0, F, and L in place of B, B0, i z z F, and L . x zi 914 MICHAEL ASCHBACHER and ANDREW CHERMAK

Given a basis v1;:::; vm of an F-space V, and elements di of F, 1 i m, Ä Ä we write @.d1; d2; : : : ; dm/ for the diagonal map vi di vi for each i. 7!

LEMMA 4.6. Let V be a 2-dimensional F-space with basis Ꮾ v1; v2 , set D f g L SL.V/, and let T be the maximal torus @.a; a 1/ a F of L determined D f j 2 g by Ꮾ. Set X L L L, set ŒŒX X= . I; I; I/ , and write ŒŒa; b; c for D D h i the image of .a; b; c/ X under the canonical surjection X ŒŒX. Finally, let 2 ! 1; ı1; 2; and ı2 be maps from Ꮾ into V which send the ordered basis .v1; v2/ to the pairs .x1; x4/, .x3; x2/, .x1; x3/, and .x4; x2/, respectively.

(a) There are isomorphisms ˛i L Li , i 1; 2; 3, such that W ! D (i) ˛1; 1 and ˛1; ı1 are quasi-equivalences of the representation of L on V with the representations of L1 on x1; x4 and x3; x2 , respectively. h i h i (ii) ˛2; 2 and ˛2; ı2 are quasi-equivalences of the representation of L on V with the representations of L2 on x1; x3 and x4; x2 , respectively. h i h i (iii) The map ˛3 is the 3-dimensional orthogonal representation of L in which 1 2 2 @.c; c / acts as @.c ;1;c / with respect to the ordered basis .x5;x7;x6/ of CV .U /. 0 (b) The map ˛1 ˛2 ˛3 X B given by W ! .a; b; c/ .a˛1/.b˛2/.c˛3/ 7! induces an isomorphism of ŒŒX with B0. 1 (c) .T Li /˛ is the set of diagonal maps in L. For each i, let ˇi F T Li be \ i W ! \ the composition of @ with ˛i . Set Y F F F, and ŒY Y= . 1; 1; 1/ , D D h i with Œa; b; c the image of .a; b; c/ Y in ŒY . Then the map ˇ1 ˇ2 ˇ2 2 W Y T induces an isomorphism Œa; b; c .aˇ1/.bˇ2/.cˇ3/ of ŒY with T . ! 7! Proof. This is straightforward, given 4.4.

From now on we use 4.6 to identify B0 with the set of equivalence classes

ŒŒa; b; c ŒŒ a; b; c; a; b; c SL2.F/; D 2 and identify T with the set of equivalence classes

Œa; b; c Œ a; b; c a; b; c F : D 2 LEMMA 4.7. (a) The element Œa; b; c of T acts diagonally as

1 1 1 1 2 2 @.ab; a b ; ab ; a b; c ; c ; 1/

with respect to the ordered basis .x1; x2; x3; x4; x5; x6; x7/ of V . 2-LOCALFINITEGROUPS 915

(b) The action of W on T is given as follows. 1 1 w1 Œa; b; c Œb ; a ; c; W 7! w2 Œa; b; c Œb; a; c; W 7! 1 w3 Œa; b; c Œa; b; c ; W 7! w Œa2; b2; c2 Œabc2; a 1b 1c2; ab 1; W 7! Œa; b; c Œa; b 1; c: W 7! Proof. Again, this is straightforward, given 4.6. Denote by S the set of elements of T whose order is a power of 2. Set 1 WS w1; w2; w3; , and set S S WS . Also, for any k 1 set D h i D 1 k T t T t 2 1 : k D f 2 j D g Thus Tk is a subgroup of S , and T1 is the group E appearing in 4.2. We shall henceforth take1 p to be congruent to 3 or 5 mod 8. One reason for this choice is that it allows us to keep track of the structure of Sylow 2-subgroups of the groups H and H , as in the following two lemmas.

LEMMA 4.8. Let k be a nonnegative integer, and set k. Then CS . / k 2 D 1 Tk 2 is homocyclic abelian of rank 3 and exponent 2 C . D C Proof. For any integer m, and any k 1, k k 1 k 1 m2 1 .m2 1/.m2 1/: D C A straightforward induction argument then yields the following fact. . / For any integer m with m 5 mod 8, and for any nonnegative integer k, we Á have k m2 1 2k 2 mod 2k 3: Á C C C k 2 k 3 Set D d F d 2 C 1 , set D f F f 2 C 1 , and ﬁx D f 2 j D g 0 D f 2 j D g f D0 D. Then D0 D Df . Set Qk Œa; b; c T a; b; c D and 2 D [ 2k 2 D f 22k j 2 g Rk Tk 2. That is Rk x T x C 1 . As Œa; b; c 1 in T if and only 2Dk C2k 2k D f 2 j D g S D if a b c 1, it follows that Rk Qk QkŒf; f; f . Thus Qk has D D D ˙ D k 2 index 2 in Rk. Let A be a homocyclic abelian group of rank 3 and exponent 2 C . Since Œa; b; c Œ a; b; c, there is an exact sequence D 1 C2 A Q 1; ! ! ! k ! and thus A 2 Q R . Since R is abelian of rank 3, exponent 2k 2, and j j D j kj D j kj k C order A , it follows from the fundamental theorem of ﬁnite abelian groups that j j R A. k Š 916 MICHAEL ASCHBACHER and ANDREW CHERMAK

Suppose that p 3 mod 8 and that k 0. For Œa; b; c T , Á D 2 Œa; b; c Œa p; b p; c p; D p 1 p 1 p 1 so that Œa; b; c T if and only if a b c 1. As p 3 mod 8, 4 2 C D C D C D˙ Á is the largest power of 2 dividing p 1, and it follows from the preceding paragraph C that CS . / Rk in this case. On the other hand, suppose that p 5 mod 8 or 1 D Á that k > 0. Then 2k 2k 2k Œa; b; c Œap ; bp ; cp : D k Notice that if p 3 mod 8 then p 5 mod 8, while for k > 0 we have p2 2k Á Á D . p/ . Now . / shows that, in any case, we have CS . / Rk. This yields 1 D the lemma.

LEMMA 4.9. The following hold.

(a) S CS ./WS , and S is a Sylow 2-subgroup of H . D 1 (b) S is a Sylow 2-subgroup of every subgroup X of H which contains S.

(c) T2 is the unique homocyclic abelian subgroup of S of rank 3 and exponent 4. Moreover, we have T CH .T2/, and T2 T CH .T2/. D Ä D (d) S B is the set of maximal 2-subgroups of B containing a subgroup isomorphic to T2.

Proof. For any subgroup P of S, denote by Ꮽ.P / the set of homocyclic abelian subgroups of P of rank 3 and exponent 4. Let and Rk be defined as in the preceding lemma, set Q WS R , and set Q0 w0 R . D k D h i k Suppose first that that there exists A Ꮽ.Q/ with A T2. Then A T . 2 ¤ — Suppose that A Q0 has rank 3 and exponent 4. Since w0 inverts R , we then \ k have A Q0 R , and A contains the unique elementary abelian subgroup E of \ Ä k R of order 8. By 4.3, Q=Q0 acts faithfully on E, so that A Q0, and then A T . k Ä Ä This is a contradiction, and so we conclude that A Q0 has exponent less than 4 or \ has rank less than 3. Since Q=Q0 is dihedral of order 8, it follows that AQ0=Q0 is cyclic of order 4. Then A Q0 is homocyclic of rank 2 and exponent 4. Again, \ A Q0 R , and now A E 4. The faithful action of Q=Q0 on E implies \ Ä k j \ j D that C .A E/ has exponent 2. Since AQ0=Q0 centralizes A E, we again Q=Q0 \ \ have a contradiction, and thus A R . That is, Ꮽ.Q/ T2 . Ä k D f g Let P be a Sylow 2-subgroup of (the finite group) H containing Q. By the preceding paragraph, T2 E NP .Q/. It follows from 4.1 that NP .Q/ preserves the set `1; `2; `2 of hyperbolic lines, and hence NP .Q/ normalizes T . Then f g NP .Q/ TW by 4.3. Since Q is a Sylow 2-subgroup of .W T / , we conclude that Ä 2k NP .Q/ Q. Then P Q, and so Q Syl .H /. We recall that D D 2 2 D k D 2-LOCALFINITEGROUPS 917

2k or .w0/ for some k, and that [ H CH .. /: D k k 0

Then S is the union of its subgroups S H , and so Ꮽ.S/ T2 . \ k D f g By Zorn’s Lemma, there is a maximal 2-subgroup S of H containing S. We have [ X X D k k 0 for any subgroup X of H. Taking X S , we conclude that S S . Taking X D D to be an arbitrary subgroup of H containing S, we note that every ﬁnite subgroup of X is contained in X k for some k, so that every ﬁnite 2-subgroup of X is X-conjugate into S. Thus, S is a Sylow 2-subgroup of X, and we have (a) and (b). Notice that for some k. Then (c) follows as T2 CH . 0/, Ꮽ.S/ D k Ä D T2 , and NH .T2/ NH .T /. f g D By (c) and 4.3.b, NB .T2/ TWS TS. Let S2 be a subgroup of B iso- D D morphic to T2, and X a maximal 2-subgroup of B containing S2. By (b), S is b b Sylow in B, and so S S for some b B, and by (c), S T2. Thus we may 2 Ä 2 2 D take T2 S2. Similarly for each k, X k is contained in a conjugate of S, and so D T by (c), X NB .T2/ TS. Hence X is a maximal 2-subgroup of TS, X S , k Ä D 2 establishing (d).

5. The amalgam Ꮽ, and an amalgam for Co3 We now undertake the construction of the amalgam which provides the focus for this work. (See the beginning of Section 3 for a discussion of amalgams.) We continue the setup and notation of the preceding section. In particular, we have p 3 or 5 (mod 8). Let i be a square root of 1 in F, and let be the element Á 0 w2Œ1; 1; i of B. Then B B , by 4.4(e). By deﬁnition, w2 interchanges the D h i two singular points of `2, centralizes `1 and `3, and acts as 1 on CV .T /. Then acts as I on CV .U / `3 CV .T /, and acts as a transvection on ŒV; U . In D C particular, commutes with .L3/, hence also with L3 (since L3 is perfect), and is an involution by 4.2. Further, acts as w2 on ŒV; T , and then 4.4(c)(iii) yields

ŒŒ˛; ˇ; ŒŒˇ; ˛; ; W 7! for all ŒŒ˛; ˇ; B0. 2 0 Deﬁne y0 to be the automorphism of B given by

(5.0) y0 ŒŒ˛; ˇ; ŒŒ ; ˛; ˇ: W 7! 918 MICHAEL ASCHBACHER and ANDREW CHERMAK

Then y0 3, and y0; acts faithfully as the symmetric group Sym.3/ on the j j D h i set ᏸ L1;L2;L3 . In the semidirect product D f g 0 K B y0; ; D h i we may then identify B with the subgroup B0 of K, and form the amalgam h i Ꮽ1 .H B K/: D Ä For any Aut.B/, denote by the composition of with the inclusion map of 2 B into K, and form the amalgam

Ꮽ .H B K/: D ! The corresponding free amalgamated product will be denoted G. Subject to the usual identiﬁcations, H and K are subgroups of G , with H K B. Here it is \ D important to note that the inclusion map of B into K, within G, is obtained by “twisting” via the “ordinary” inclusion map occurring in Ꮽ1.

LEMMA 5.1. For X H;K , write AX for Aut .B/. Set ˆ Aut.F/, 2 f g Aut.X/ D and regard ˆ as the group of ﬁeld automorphisms of SL2.F/. Deﬁne a representation of ˆ on B0 by ŒŒ˛; ˇ; ŒŒ˛; ˇ; for ˆ: W 7! 2 Then ˆ commutes with on B0, and the representation of ˆ on B0 extends thereby to a representation on B. Moreover:

(a) Inn.B/ AH AK . Ä \ (b) For each X H;K we have Aut.B/ AX ˆ AH AK ˆ, and AX ˆ 1. 2 f g D D \ D (c) For ; Aut.B/, we have Ꮽ Ꮽ if and only if AH AK AH AK . 2 Š D Proof. Identify ˆ with a subgroup of Aut.B0/ via the prescribed representation. Evidently Œˆ; 1, so we may even regard ˆ as a subgroup of Aut.B/. As D B H K, (a) holds. As is well known (cf.[Ste]), Aut.SL2.F// Inn.SL2.F//ˆ. D \ 0 D 0 Then, since B is the central product of three copies of SL2.F/, Aut.B / is the 3 split extension of Aut.SL2.F// by Sym.3/, where Sym.3/ permutes the three components of B0 faithfully. 0 Recall that B B , where centralizes L3 and interchanges L1 and L2. D h i For X H;K we have AX Inn.B/ˆX , where ˆX ˆ and ˆX is diago- 2 f g D 3 3 Š nally embedded in the subgroup ˆ of Aut.SL2.F// , and centralizes . Similarly, Aut.B/ Inn.B/.ˆX ˆ/, where ˆ centralizes L1L2 and acts faithfully as ˆ on D L3. Now (b) follows, and Aut.B/ AH AK ˆ. By[Gol80, Lemma 2.7], Ꮽ Ꮽ D Š if and only if AH AK AH AK , and now (c) follows from (b). D We may now state the ﬁrst main result of this section. 2-LOCALFINITEGROUPS 919

THEOREM 5.2. Let S be the Sylow 2-subgroup of T , and for any Aut.B/ 1 2 set N NH .S /; NK .S / , where H and K are regarded as subgroups of G D h 1 1 i in the canonical way. Set ᏺ AutN .S /, deﬁne ˆ as in 5.1, and set D 1 ƒ ˆ ᏺ GL.3; 2/ C2 : D f 2 j Š g Then the following hold. (a) ƒ 1. j j D (b) For ƒ, 2 ( CN .S / if e 2, and 1 CN .Te/ D CN .S / w0 if e 1. 1 h i D (c) AH AK . D (d) The map Ꮽ is a bijection of ˆ with the set of isomorphism classes of 7! amalgams Ꮽ with Aut.B/. 2 Remark. It can be shown, by means of a lengthy computation based on 4.7(b), that the unique in the set ƒ of Theorem 5.2 is not an algebraic endomorphism of B.

0 Let !3 be the automorphism of B induced by conjugation by w3. By 4.7(b), 0 w3 L3 and w3 inverts T L3. Denote by 3 the automorphism of B which 2 th \ induces the p power Frobenius map on L3 and which centralizes L1 and L2. 0 Then let 0 be the automorphism of B given by ( 3!3 if p 3 mod 8, 0 Á D 3 if p 5 mod 8: Á Thus, ( ŒŒ˛; ˇ; w3 if p 3 mod 8, 0 ŒŒ˛; ˇ; N Á W 7! ŒŒ˛; ˇ; if p 5 mod 8, N Á th where is the element of SL2.ކ/ whose entries are the p powers of the corre- N sponding entries of . For any e ގ, set 2 2e e : D 0 2e Notice that T is invariant under e, that ŒL1L2; e 1, and that e for e 1. D D 3 Recall from 4.8 that we have deﬁned subgroups Te of T0 by 2e Te t T t 1 ; e 1; D f 2 j D g and that Te is homocyclic abelian of rank 3. e 3 LEMMA 5.3. Let e be a nonnegative integer, and let c F with c2 C 1. 2 D Then 920 MICHAEL ASCHBACHER and ANDREW CHERMAK

2e 2 (a) Œa; b; ce Œa; b; c if c C 1, and otherwise Œa; b; ce Œa; b; c. D D D (b) e centralizes a subgroup of index 2 in Te 2 containing Te 1, and ŒTe 2; e Z. C C C D Proof. Suppose ﬁrst that e 0 and that p 3 mod 8. Then c8 1, and D Á D p 5 Œa; b; ce Œa; b; c Œa; b; c : D D Since c5 c if c 8, and c5 c if c4 1, (a) holds in this case. On the other D j j D D D hand, suppose that either e > 0 or p 5 mod 8. We saw in the proof of 4.8 that Á for any integer m with m 5 mod 8, Á e m2 1 2e 2 mod 2e 3: Á C C C Taking m p if p 5 mod 8, and taking m p if p 3 mod 8, we then D Á D Á have e m2 1 2e 2 Œa; b; ce Œa; b; c Œa; b; c C : D D C Thus (a) holds in every case. Part (b) follows from (a) and the fact that Z 2 2 e 3 D Œ1; 1; 1 and Te 2 Œc ; 1; 1; Œ1; c ; 1; Œc; c; c , where c 2 C . h i C D h i j j D PROPOSITION 5.4. There is a uniquely determined sequence .ye e 0/ of 0 j automorphisms of B , with y0 as in 5.0, and having the following two properties.

e 1 (a) ye ye 1; ye 1 for e > 0. 2 f g (b) The group ᏺe of automorphisms of Te 1 induced by the action of W; ye C h i is isomorphic to GL.3; 2/ C2 for e > 0, and is isomorphic to GL.3; 2/ for e 0. D The proof of 5.4 will be based on the following result.

LEMMA 5.5. Let N be the Steinberg module for GL.3; 2/ over the ﬁeld ކ2 of two elements, and let X be an extension of N by GL.3; 2/. Then the following hold: (a) X splits over N .

(b) Let D be a complement to N in a Sylow 2-subgroup of X. Then CN .D/ g D h i is of order 2. (c) Let D be as in (b), and denote by ᏼ the set of subgroups P of X such that g D P Sym.4/. Then ᏼ P1;P2;Q1;Q2 , where Qi P for i 1; 2, Ä Š D f g D i D Pi ;Qi is a complement to N in X, and P1;Q2 P2;Q1 X. h i h iDh iD Proof. Let R Syl .X/. Then N is a free ކ2R=N -module, so that CN .R/ 2 2 D g is of order 2, and for each overgroup Y of R in X we have H i .Y=N; N / 0 h i D for i 1; 2. In particular, (a) and (b) hold. D Choose R so that R DN and let P and Q be maximal subgroups of X such D that P=N and Q=N are the maximal parabolic subgroups of X=N over R=N . It 2-LOCALFINITEGROUPS 921 follows from the preceding paragraph that for each Y X;P;Q;R , Y splits over 2 f g N and is transitive on its complements to N . Thus the set ᏼY of complements to N in Y containing D is nonempty for Y X;P;Q , and by a standard argument 2 f g (cf. 5.2.1 in[Asc86]), NY .D/ is transitive on ᏼY . Then as NX .D/ D g , g D h i ᏼY Y1;Y2 with Y Y2. Since CN .Y / 0 for Y P;Q , each Pi and each D f g 1 D D 2 f g Qi is contained in a unique complement to N in X. In particular Xi GL.3; 2/ Š and the indexing may be chosen so that Xi is generated by Pi and Qi PQ. Then 2 Mi Pi ;Q3 i is not a complement to N in X. Since X is irreducible on N , D h i Mi N . This completes the proof. D We may now prove Proposition 5.4. Since T1 is elementary abelian of order 8, we have Aut.T1/ GL.3; 2/. From 4.3(c), W induces the stabilizer in GL.T1/ of Š Z, and so the image of W in GL.T1/ is maximal. The closure of Z under the action of y0 is the fours group U , and thus ᏺ0 GL.T1/. We may therefore h i D assume that e 1, and that for all indices e with 0 < e < e: 0 0 e 1 . / There exists a unique y y ; y 0 such that the group ᏺ of au- e0 e0 1 e 1 e0 2 f 0 g tomorphisms of Te 1 induced by the action of W; ye is isomorphic to 0C h 0 i GL.3; 2/ C2. Set R Te 1, x ye 1, and denote by ᏺ the image of W; x in Aut.R/. D C D h i Applying . / to e e 1, we have ᏺ=C .Te/ GL.3; 2/ if e 1, and GL.3; 2/ 0 D ᏺ Š D C2 if e > 1. Let d C .Te/. Writing 2 ᏺ d Œa; b; c Œa; b; c; W ! z z z for Œa; b; c R, we obtain a2 "a2, b2 "b2, and c2 "c2, for some " 1 . 2 D z D z D z 2 f˙ g Thus either u u; u for all u a; b; c , or u iu; iu for all u a; b; c , z 2 f g 2 f g z 2 f g 2 f g where i is a square root of 1. Therefore as T1 Œ 1; 1; 1; Œ1; 1; 1; Œi; i; i ; D h i d acts trivially on R=T1. Then d 1 for some Homޚ.R=Te;T1/, and D C d d 2 the mapping C .Te/ Homޚ.R=Te;T1/ ᏺ ! given by d d is an ᏺ-homomorphism. Observe that R=Te and T1 are isomor- 7! 2e 1 phic as modules for W; x via the map ' rTe r C . Thus, there is an ᏺ-equi- h i 1 W 7! variant monomorphism d ' from C .Te/ into M Endޚ.T1/. Since ᏺ 7! d ᏺ WD acts as GL.3; 2/ on T1, M may be identified with the ᏺ-module of 3 3 matrices over the field F2 of two elements, and we regard Cᏺ.Te/ as an ᏺ-submodule of M . Now M is a vector space of dimension 9 over F2, and the subspace M0 of trace-zero matrices is an 8-dimensional ᏺ-submodule of M . Indeed ᏺ=C .M / ᏺ Š GL.3; 2/, and M0 is the Steinberg module for ᏺ=Cᏺ.M /. The element w0 of W inverts R, and hence w0 CM .ᏺ/, and M M0 CM .ᏺ/. Further, ᏺ is an h i D D ˚ 922 MICHAEL ASCHBACHER and ANDREW CHERMAK extension of ᏺM ᏺ M by L3.2/, and as CM .ᏺ/ w0 ᏺM and ᏺ is D \ D h i Ä irreducible on M0, ᏺM M or CM .ᏺ/. As WS =.WS T/ Z2 D8 (cf. 4.3), D \ Š ᏺ=M0 or ᏺ is isomorphic to Z2 L3.2/ in the respective case. Further, in the latter case, we obtain (a) and (b) of 5.4 by setting ye 1 ye. Thus we may assume that C D ᏺM M . D Set X Œᏺ; ᏺ. From the previous paragraph, M0 X M and X=M0 D D \ Š GL.3; 2/. Denote by D the image of WS in ᏺ, and set D D X. Then D \ D D Z.ᏺ/, and D is dihedral of order 8. Denote by P and Q the images D in ᏺ of W and WS ; y , respectively, and set P P X and Q Q X. By h i D \ D \ 5.5(b), CM0 .D/ g is of order 2. By definition ᏺ is generated by P and Q, D h i g and so X is generated by P and Q. By 5.5(c), P;Q is a complement to M0 in h i X. By construction, e 1 centralizes WS , and by 5.3(b), e 1 induces a nontrivial automorphism of R centralizing a subgroup of index 2 containing Te. Then, since

CM0 .WS / g , it follows that the action of g on R is the same as that of e 1. D h gi Setting ye x , we obtain (a) and (b) of Proposition 5.4. D 0 Let e e 0 be the sequence of automorphisms of B deﬁned by 0 1 0 , f j g D B and for e > 0 by the recursive formula

( e 1 if ye ye 1 and D e e 1 D e 1e 1 if ye ye 1 D where ye is as in Proposition 5.4. For k 0, take as deﬁned just prior to 4.3. k

LEMMA 5.6. Each e extends to an automorphism of B which commutes with the element of B B0, and with for each k. Further, for each e 0, k

e 1 CB . e 1/ e CB . e 1/ : C j D j 2e 0 Proof. Recall that e , where 0 is the automorphism of B given by D 0 0 ŒŒ˛; ˇ; ŒŒ˛; ˇ; ; W 7! 0 for the automorphism of SL2.F/ such that 7! 0 0 ŒŒ˛; ˇ; ŒŒ˛ ; ˇ ; : W 7! 0 0 0

It follows that e commutes with k for all e and k, and also with the automorphism 0 ŒŒ˛; ˇ; ŒŒˇ; ˛; of B . In constructing the amalgam Ꮽ1, we identiﬁed W 7! 0 B with the semidirect product B , so e may now be regarded as an automor- h i phism of B. Since e is the product of some elements of 1; 0; : : : ; e , we may f g also regard e as an automorphism of B, commuting with k. The proof is then completed by the observation that e centralizes CB . e 1/. 2-LOCALFINITEGROUPS 923

Notice that 4.8 yields [ B CB . e/: D e 0 By the preceding lemma, we may then deﬁne an automorphism of B by taking e for each e. Set jCB . e/D Ꮽ Ꮽ .H B K/ D D ! (where denotes inclusion) and form the corresponding free amalgamated product G G . D We may now complete the proof of Theorem 5.2. We have

ˆ Aut.F/ lim Aut.F 2e /; D D p and there is an isomorphism

ޚ2e Aut.F 2e / ! p given by sending a residue class Œk to the pk-th power map, 0 k < 2e. The Ä sequence of inverses of these isomorphisms then defines an isomorphism of Aut.F/ with the ring ޚ.2/ of 2-adic integers. Let Aut.F/, and denote by e the restriction of to the subfield Fpe of 22e e F of order p . For any e 0, there is then a unique integer ke, 0 ke < 2 , such ke Ä that e is given by the p -th power map on F2e . Define a sequence ."e e 0/ j of elements of 0; 1 by taking "0 k0, and for e > 0 by f g D ( ) 0 if ke ke 1 D "e e 1 : D 1 if ke ke 1 2 D C We may represent the action of on B0 as in 5.1; namely acts on SL.2; F/ in the natural way, and on B0 by ŒŒa; b; c ŒŒa; b; c: W ! Observe that the restriction e of to CB0 . e 1/ is given by C "0 "1 "e e : : : ; D z0 1 e where 0 0!3 if p 3 mod 8, and where 0 0 if p 5 mod 8. z D Á z D Á Recall from 5.1 that we have identified ˆ with a subgroup of Aut.B/, and that parts (b) and (c) of 5.1 show that for any Aut.B/ there exists ˆ such 0 2 2 that Ꮽ Ꮽ . In particular if ˛ with ˛ Inn.B/ then Ꮽ Ꮽ , and Š 0 0 D 2 Š 0 hence there is an induced isomorphism G G of universal completions. Š 0 Take !3 if both p 3 mod 8 and 1 0, and take otherwise. By D Á D D our construction of we have ˆ, and then since !3 B we obtain G G . 2 2 Š Adopt the notation of 5.2. In particular N NH .S /; NK .S / , N W; y T , D h 1 1 i D h i 924 MICHAEL ASCHBACHER and ANDREW CHERMAK

e and ᏺ AutN .S /. Since e centralizes Te 1 by 5.3(b), we have y y , 0 Te 1 0 D 1 C j C D where e is as deﬁned prior to 5.6. By induction on e, we then get

( ye 1 if ye ye 1 ) y D ; 0 Te 1 e 1 j C D ye 1 if ye ye 1 ¤ and so y ye. Then 5.4 shows that is in the set ƒ deﬁned in 5.2. 0 Te 1 j C D Now let be an arbitrary element of ƒ. Set !3 if both p 3 mod 8 and D Á 1 "0 1, and otherwise set . Then G G and N N. Set x y0, D D Š Š D regard x as an automorphism of S , and denote by xe the restriction of x to Te 1, 1 "e C e e 0. Then x0 y0 and, by induction on e, xe 1 xe . As ƒ, the D C D 2 uniqueness of the sequence in 5.4 implies that "e 0 if and only if ye ye 1, 1 D D and hence that x y. Since y Œa; b; c Œc ; b; a, also , establishing D W 7! D 5.2(a). Now 5.2(b) follows from the action of ᏺ on T in 5.4. 1 If AH AK then AH AK ˆ 1, by 5.1(b), and then 5.1(c) implies that there ¤ \ ¤ exists ˆ with Ꮽ Ꮽ . This is contrary to 5.2(a), so that AH AK 0 2 f g Š 0 D and 5.2(c) holds. Now 5.2(d) follows from (c) and from 5.1(c), and this completes the proof of 5.2. Regarding H and K as subgroups of G G in the canonical way, we have D B H K. From 5.6, n commutes with e for each e, so commutes D \ D 0 with . Since commutes with y0 and with as automorphisms of B , and since 0 y acts on B as y0 , commutes with y. Then since K is the semidirect product 0 of B with y; , it follows that induces an automorphism K of K, commuting h i with y; . The universal property of the free amalgamated product now implies h i that induces an automorphism of G, whose restriction to K is K . We record this result for future reference.

LEMMA 5.7. For each positive integer n, n H extends uniquely to an automorphism of G such that Œy; 1. j D We next show that the third Conway simple group Co3 is the completion of a subamalgam of Ꮽ, and that this subamalgam generates a fusion system which is isomorphic to that of Co3. These will be key ingredients in our proof of Theorem B. THEOREM 5.8. Let G be the simple group Co , let S be a Sylow 2-subgroup x0 3 0 of G0, set Z0 Z.S0/, and let U0 be the unique normal fours group in S0. Set x D H0 CG .Z0/; K0 NG .U0/; B0 H0 K0; D x0 D x0 D \ and let Ꮽ0 .H0 B0 K0/ be the amalgam of inclusion maps among these D ! groups, within G0. Set and set 0 B , where is an automorphism of x D D j 0 B which satisﬁes the conditions of Theorem 5.2. Let Ꮽ .H B K /. 0 D ! Then the following hold: 2-LOCALFINITEGROUPS 925

(a) There is a morphism ' Ꮽ0 Ꮽ of amalgams, displaying Ꮽ0 as a subamal- W ! 0 gam of Ꮽ0 .

(b) Let G0 be the subgroup of G generated by the images of H0 and K0 under the morphism ' of part (a), and let Ᏺ0 be the fusion system ᏲS .H0/; ᏲS .K0/ h 0 0 i contained in ᏲS .G0/. Then Ᏺ0 ᏲS .G0/ ᏲS .G0/. D D 0 x Proof. We refer to[Fin73] for the structure of the maximal subgroups of G0. Thus, H0 is isomorphic to the covering group of Sp6.2/, which is the perfect central extension of Sp .2/ by a group of order 2. Since Sp .2/ C2 is a reﬂection 6 6 group (namely, the Weyl group of type E7), we have Sp .2/ O7.ޒ/, and then, 6 Ä by taking the standard ޚ-form of O7.ޒ/ and reducing mod p, one obtains Sp6.2/ as a subgroup of 7.p/. Identifying 7.p/ with H , we then have an inclusion of H0 in H . In particular, Z0 Z. D 2 Let ƒ be GL2.3/ S3 and ƒ ƒ=Z.ƒ/. Set B1 O .B0/ and D O2.B0/. o N D 10 D D Sylow 2-subgroups of H0 and H are of order 2 , so conjugating in H , we may take S S0 Syl .H0/, and U U0. Next, B0 is the preimage in H0 of the D 2 2 D solvable maximal parabolic subgroup of Sp6.2/, so that B0=O2.B0/ is isomorphic to Sym.3/ Sym.3/, and B1 is isomorphic to a subgroup of index 3 in O2;3.ƒ/, N contained in the ƒ-orbit of length 4 on such subgroups. In particular N (1) D is a commuting product of three quaternion groups Qi , 1 i 3, with the Ä Ä property that ᏽ Q1;Q2;Q3 is the set of normal subgroups of B1 of order D f g 8. Moreover CD.Qi / Qj Q for any ordering .i; j; k/ of .1; 2; 3/. D k The preimage B of B in ƒ is the 2-covering group of B , and so Aut.B / y1 1 1 1 acts on B1. By (1), Aut.B1/ permutes ᏽ, and hence ᏽ Qi 1 i 3 , where y y D f y j Ä Ä g Qi ŒQ1; B1 and where Qi is the preimage in ƒ of Qi . Therefore Aut.B1/ y D z y z D Autƒ.B1/. x Referring once more to[Fin73], we ﬁnd that K0 B0 3. Since CK .U / j W j D 0 Ä B0 we have B1 CB .U / E K0, and so B1 E K0 and K0=CB .U / Sym.3/. Ä 0 0 Š From the preceding paragraph, AutK0 .B1/ Autƒ.B1/. Then, since the Ä WD x Sylow 2-subgroup S0 of B0 is Sylow in K0 and contains the kernel U of the map from Nƒ.B1/ onto , we may regard K0 as a subgroup of ƒ. By (1), ᏽ is x x K0-invariant. As K0=CK0 .U / Sym.3/, it follows that K0 is transitive on ᏽ. Š 2 3 Here ƒ=D Sym.3/ Sym.3/, and K0=D is a subgroup of ƒ=D of order 2 3 . x Š o x Since K0=CK .U / Sym.3/, it follows that K0=B1 Z2 Sym.3/, and then that 0 Š Š K is determined up to conjugacy in ƒ. The same argument shows that K is in 0 x this conjugacy class, and so K0 K and we may choose K0 K . Š D Observe that Nƒ.B1/ B0 3 and that B0 is equal to its normalizer in j x W j D Nƒ.B1/. Then x

(2) NAut.B1/.B0/ Autƒ.B0/ Inn.B0/. D x D 926 MICHAEL ASCHBACHER and ANDREW CHERMAK

By (2) we have Aut.B0/ Inn.B0/C0, where C0 C .B1/. Then D D Aut.B0/ ŒC0;B0 CB .B1/ U . Let X0 Syl .B0/ and let X0 X Syl .K0/. Then Ä 0 D 2 3 Ä 2 2 CD.X0/ U , CD.X/ 1, and as we saw above K0=B1 Z2 Sym.3/. It follows D D Š that there is a Sylow 2-subgroup R0 of NB .X0/, of the form U s r , where 0 h i h i r; s is a Sylow 2-subgroup of NK .X/, U s is a dihedral group of order 8, and h i 0 h i r; s is a fours group. h i We have ŒC0;R0 CB0 .B1/ U , and so R0 is C0-invariant. Since r Z.R0/ Ä D 2 we then have ŒC0; r U Z.R0/ Z. For any C0, s is an involution in sU , Ä \ D 2 so also ŒC0; s Z. Therefore C0 Hom.B0=B1;Z/ B0=B1, and so C0 is a Ä Š Š fours group. Since Z2 AutU .B0/ C0, we conclude that Aut.B0/ Inn.B0/ Š Ä j W j D C0 AutU .B0/ 2. Thus j W j D (3) Aut.B0/ Inn.B0/ Inn.B0/0, where 0 C0 AutU .B0/. D [ 2 It follows from (3) and[Gol70, Lemma 2.7] that there are, up to isomorphism, ˛ at most two amalgams .H0 B0 K0/ with B0˛ B0, and that any such ! D amalgam is isomorphic to Ꮽ0 or to Ꮽ0;0 , where

0 Ꮽ0; .H0 B K0/: 0 D !

Thus, either Ꮽ0 or Ꮽ0;0 is a subamalgam of the amalgam Ꮽ . Referring again to[Fin73], there is a subgroup M of G , containing S , such x0 0 that M is a nonsplit extension of E16 by GL4.2/. Set A O2.M /, and denote D by M0 the stabilizer in M of the unique S0-invariant hyperplane E0 of A. Then ŒO2.M0/; M0 is homocyclic abelian of exponent 4 and rank 3, and hence E0 E D by 4.9(c). Since ŒO2.M0/; E0 1 we then have O2.M0/ T2 w0 , by 4.3. D D h i Moreover

M0 CM .Z/; NM .U / M0 H0;M0 K0 : D h 0 0 i D h \ \ i Let ˛ S0 NK .T2/ be the embedding of S0 S in NK .T2/. Then S0˛ W ! D 0 D S0 K0 K , and we have the two amalgams Ä D ᏭM .M0 H0 S M0 K0/ 0 D \ ! \ and ˛ .NH .T2/ S0 NK .T2//: ! On the other hand, the reader will recall from the proof of 5.4 that if ᏭM0 is “twisted” by 0, to obtain an amalgam

0 .M0 H0 S M0 K0/; \ ! \ then M0 H0; x 0 induces on T2 the full automorphism group of T2, of order 9 h \ i 2 GL3.2/ . We therefore conclude that, of the two amalgams Ꮽ0 and Ꮽ0; , only j j 0 the ﬁrst is a subamalgam of Ꮽ . This completes the proof of (a). 2-LOCALFINITEGROUPS 927

G0 Set ᐆ0 z and denote by Ᏹ the set of elementary abelian subgroups F D # of S0 such that F ᐆ0. Denote by ᏺ the set of subgroups N of G0 such that Â x N NG .O2.N //, S0 N Syl2.N /, and CS0 .O2.N // O2.N /. It is a property D x0 \ 2 Ä of Co3 that xy ᐆ0 for any two distinct commuting elements x and y of ᐆ0 (cf. 2 [Fin73]), from which it follows that for any N ᏺ there exists F Ᏹ with F E N . 2 2 By Lemmas 5.8 and 5.9 in[Fin73], all members of Ᏹ of any given order are fused in G0, each member of Ᏹ is normal in a Sylow 2-subgroup of G0, and if F Ᏹ x x 2 with F 8 then NG .F / is contained in the normalizer of some F Ᏹ with j j D x0 2 F 16. Then, for any N ᏺ, we have S0 N , and N is contained in one of j j D 2 Ä the groups H0, K0, or M . Then 1.11 yields

.4/ ᏲS .Co3/ ᏲS .H0/; ᏲS .K0/; ᏲS .M / : 0 D h 0 0 0 i Now .M H0/=A and .M K0/=A are distinct maximal parabolic subgroups of \ \ M=A GL4.2/, and so by 1.9: Š .5/ ᏲS .M / ᏲS .M H0/; ᏲS .M K0/ : 0 D h 0 \ 0 \ i From (4) and (5) we have ᏲS .G0/ ᏲS .H0/; ᏲS .K0/ , and it follows that 0 x D h 0 0 i ᏲS .G0/ ᏲS .G0/. Since G0 is a homomorphic image of G0, by (a), the reverse 0 x Â 0 x inclusion of fusion systems is obvious, and we therefore have (b).

Some well-known properties of Co3 (some of which were mentioned in the proof of 5.8(b)), which depend only on fusion, now yield corresponding properties of the subgroup G0 of G.

COROLLARY 5.9. Identify H0 and K0 with subgroups of G, via the morphism ' of 5.8(a), and set G0 H0;K0 . Then D h i (a) G0 has two classes of elements of order 2. (b) If t and t are distinct, commuting elements of zG0 , then tt zG0 . 0 0 2 G (c) Let F be an elementary abelian 2-subgroup of G0. Then F z 0 is the set of \ nonidentity elements of a subgroup of F .

(d) For any X G0, and any subgroup F of X, denote by Ᏹ.X; F / the set of Ä z all subgroups P of X such that F P and P # zG0 . Write Ᏹ.X/ for Ä Â z Ᏹ.X; 1/. Then Z;U;E;E w0 is a set of representatives for the orbits of z f h ig G on .G /, and for the orbits of H on .H ;Z/. 0 Ᏹz 0 0 Ᏹz 0

6. Discrete p-toral groups

The notion of a discrete p-toral group, and the results in this section on such groups, come from[BLO05], particularly Sections 1 and 7 of that paper. As [BLO05] is unpublished at this time, we reproduce some of its deﬁnitions and 928 MICHAEL ASCHBACHER and ANDREW CHERMAK results here, and supply sketches of proofs in special cases, for the sake of com- pleteness.

Definition 6.1. Let p be a prime and denote by Z=p1 the group of all complex roots of unity whose order is a power of p.A discrete p-toral group is a p-group P with a normal subgroup P0 of finite index, such that P0 is the direct product of a finite number of copies of Z=p1. Write Ᏸp for the class of discrete p-toral groups.

We record some facts about Ᏸp from[BLO05]:

LEMMA 6.2. Let P Ᏸp. Then 2 (1) P has unique subgroup P 0 which is minimal subject to the condition that P P 0 be finite. (Call P 0 the identity component of P .) j W j 0 (2) P is the direct product of a finite number r of copies of Z=p1. (Write rk.P / for r and call rk.P / the rank of P .) (3) P 0 has no proper subgroups of finite index. (4) P is locally finite and Artinian.

(5) Subgroups and homomorphic images of P are in Ᏸp. (6) Torsion subgroups of Out.P / are finite. (7) Each injective homomorphism from P into P is an isomorphism. (8) If R P then R0 P 0. Ä Ä Proof. As P Ᏸp, P has a normal subgroup P0 of finite index which is the 2 direct product of r copies of Z=p1 for some 0 r Z. As Z=p1 has no proper 0 Ä 2 subgroups of finite index, it follows that P P0, and (1)–(3) hold. Parts (4), (5), D and (6) are 1.2, 1.3, and 1.5(a) in[BLO05], respectively. Part (7) follows as P is Artinian, and (8) follows from (3).

LEMMA 6.3. Let F be the field of Section 4, V a finite-dimensional vector space over F, and G GL.V /. Then Ä (1) G is locally finite.

(2) All 2-subgroups of G are in Ᏸ2. (3) Syl .G/ ¿, Syl .G/ is the set of maximal 2-subgroups of G, and G is 2 6D 2 transitive on Syl2.G/.

(4) Let S Syl .G/ and P S. Then P is fully normalized in ᏲS .G/ if and only 2 2 Ä if NS .P / Syl .NG.P //. 2 2 (5) ᏲS .G/ is saturated. 2-LOCALFINITEGROUPS 929

Proof. The proof of this lemma comes from[BLO05, 7, particularly Lemma 7.8]. The proof is a bit easier in our special case, and we supply a sketch. As GL.V / is the union of the finite groups GL.V / , Aut.F/, (1) holds. 2 Then (2) follows from (1) and from[Weh73, 2.6]. By[Weh73, 9.10], G is transitive on its maximal 2-subgroups, and such subgroups exist, and so (3) holds. Observe that G satisfies the hypotheses of Lemma 1.6: Condition (1) of 1.6 follows from (3) applied to subgroups of NG.P /. Condition (2) of 1.6 is satisfied by (1) and 6.2(6). Condition (3) holds by 6.2(4), and (4) holds by (1). Now 1.6 implies (4) and (5). Remark 6.4. Let H;K;B;S be the groups defined in Sections 4 and 5. Each of these groups has a faithful finite-dimensional representation over F, and so we can apply Lemma 6.3 to these groups. By 4.9(b), S is a Sylow 2-subgroup of each of these groups. By 6.3(2), S and each of its subgroups is a discrete 2-toral group. By 6.3(5), ᏲS .X/ is saturated for each X H;K;B and by 6.3(3), X is 2 f g transitive on Syl2.X/ where Syl2.X/ is the set of maximal 2-subgroups of X. Let G be the group constructed in Section 5. It will be shown, in Theorem C, that there is a 2-local group Ᏻ .S; ᏲS .G/; ᏸS .G//. Since S is a discrete 2-toral D group, Ᏻ is then a 2-local compact group, as defined in[BLO05].

7. Local subgroups and fusion in the free amalgamated product

Let Ꮽ be the amalgam Ꮽ constructed in Section 5, and let G be the associated free amalgamated product, G H B K. We shall view Ꮽ as being given by the D inclusion maps of H, K and B into G, so that Ꮽ .H B K/: D Ä Viewed in this way, the key point in the construction of Ꮽ is that the element y of 1 K acts on T as y0. That is y Œa; b; c Œc 1; a; b; W 7! for all Œa; b; c T . 2 Recall that we have an automorphism n of H, with H Spin7.Fq/, 2n D Š q p , and by 4.8, S is a Sylow 2-subgroup of H . By 5.7, induces an D automorphism of Ꮽ which induces an automorphism of G. Form the semidirect products H , B , and K , and the amalgam h i h i h i Ꮽ .H B K /; D h i h i ! h i in which the arrows are inclusion maps. Denote the free amalgamated product of Ꮽ by G. The inclusion Ꮽ Ꮽ induces an isomorphism of G with the semidirect y ! y product G , and we identify these groups via that isomorphism. h i The following result is trivially veriﬁed. 930 MICHAEL ASCHBACHER and ANDREW CHERMAK

LEMMA 7.1. Let and be the standard trees associated with the amalgams y Ꮽ and Ꮽ, respectively. Then there is an isomorphism given by ! y Xg X g for X H;B;K and g G. 7! h i 2 f g 2 If and are identiﬁed via this isomorphism, then the action of on is given y by .Xg/ Xg for X H;B;K and g G. D 2 f g 2

For any X G , we write X or C .X/ for the subgraph of induced Ä h i on the set of ﬁxed points of X on . If X ¿ then X is a subtree of . For ¤ any graph and vertex ı of , we write .ı/ for the set of vertices of such that ; ı is an edge of . If .ı/ 1 then ı is a boundary vertex of , and f g j j Ä otherwise ı is an interior vertex of . For any subtree of , let be the graph obtained by deleting the boundary z vertices from . Thus either is a tree or has at most one edge, in which case z is empty. z Set G1 H and G2 K, and denote by i the set of vertices of given by D D the cosets of Gi in G. For any vertex of , write Z. / for the largest normal 2-subgroup of G . Deﬁne i to be the vertex of given by the coset Gi .

LEMMA 7.2. Let be a vertex of .

(a) If 1 then Z. / Z.G / is of order 2. 2 D (b) If 2 then . / 3, and Z. / is a fours group, whose nonidentity cyclic 2 j j D subgroups are the groups Z.ı/, ı . /. 2 (c) If 2 then CG .Z. // is the pointwise stabilizer in G of . /. 2 Proof. The stabilizer of any vertex in i is conjugate in G to Gi , and the stabilizer of any edge is conjugate to B. All parts of the lemma follow trivially from these observations.

LEMMA 7.3. Let X G and let 2 X . Then: Ä h i 2 \ (a) is an interior vertex of X if and only if X centralizes Z. /.

(b) Either of the following conditions implies that the inclusion maps from NH .X/ and NK .X/ into NG.X/ induce an isomorphism of NH .X/ NK .X/ NB .X/ with NG.X/. (i) X B0, and X H B X K B X B . Ä \ D \ D 0 (ii) X B0 and X H B0 X K B0 X B . Ä h i h i \ h i D h i \ h i D Moreover, NG.X/ acts edge-transitively on X in case (b)(i), and edge-transitively on in case (b)(ii). zX 2-LOCALFINITEGROUPS 931

Proof. Set X . Then is an interior vertex of if and only if X fixes D 0 at least two distinct vertices ˛ and ˇ in X . Since Z. / Z.G / Z˛Z , we D D ˇ obtain (a). Set N NG .X/; NG .X/ , and assume that either (i) or (ii) holds. Take D h 1 2 i ƒ in case (i), and ƒ in case (ii). Then NG.X/ acts on ƒ, and N NG.X/. D D y 0 Ä By hypothesis, X CB .U / B , so that X fixes . 2/ pointwise, and Ä h i D h i hence 2 ƒ. In (i), a standard argument (cf.[Asc86, 5.21]) shows that NG .X/ 2 i acts transitively on ƒ. i / for i 1 and 2. Assume that we are in case (ii) and that h 1 D h . i ; 3 i / is an edge in ƒ for some h Gi . Then X .B / , and so h 1 h 1 2 0 h i Ä h i X B . Then X CB .U / B by (a). The hypotheses of (ii) Ä h i 0 Ä h i D h i then yield h B NGi .X/, so that NGi .X/ acts transitively on ƒ. i /. 2 h i h i h i We now claim that N is transitive on the set of edges of ƒ. As ƒ is connected, it suffices to show for each ƒ that N is transitive on ƒ./. Pick i and g with 2 i g and set D d./ min d.; j / j 1; 2 : D f j D g Choose to be a counterexample with d d./ minimal. By the preceding D paragraph, d > 0. Thus there exists ˛ ƒ./ with d.˛/ < d, and N˛ is transitive 2 on ƒ.˛/. Then there is ˇ ƒ.˛/ with d.ˇ/ < d, contrary to the choice of . This 2 completes the proof of the claim. As the stabilizer NB .X/ of an edge of ƒ is contained in N , we now obtain N NG.X/. Now[Ser80, Th. 6, p. 32] yields the conclusions concerning edge- D transitivity, and the identification of NH .X/ NK .X/ with N . NB .X/ LEMMA 7.4. We have the following.

(a) CH ./ H . z D (b) The inclusion maps from H and K into G induce an isomorphism of G with H B K , and G acts edge-transitively on the tree . z (c) Deﬁne the subgroups G0, H0, K0, and B0 of G as in 5.8. Then the inclusion maps from H0 and K0 into G0 induce an isomorphism of G0 with H0 B K0, 0 the universal completion of the amalgam Ꮽ0 of subgroups of Co3. Proof. Let h H such that h B0. By Lang’s Theorem there exists b B0 h b2 1 2 2 z such that . Then hb CH ./ H as n. Thus (a) holds and b H . D 2 z D D 0 2 Here b B0 since H B0 B0, and so H B0 B . Since K C ./B0, z K 2 K \0 DB0 \ D D we also have B . Now by 7.3(b), G H ;K and G is edge- \ D D h i transitive on the tree . Since H and K ﬁx adjacent vertices in , the lemma z z now follows from[Ser80, Th. 6, p. 32]. The same theorem implies (c).

From now on, G0 H0 B K0 is the subgroup of G deﬁned in 7.4(c), such D 0 that G0 is the universal completion of an amalgam of subgroups of Co3. 932 MICHAEL ASCHBACHER and ANDREW CHERMAK

LEMMA 7.5. Let D G;G ;G0 , set Di D Gi , i 1; 2, and set 2 f g D \ D R S D. Then: D \ (a) Hypotheses 3.4 and 3.8 hold, with D, Di and D B in the roles of G, Gi and \ B, and with R in the role of S. (b) R is a Sylow 2-subgroup of D.

Proof. When R is finite so is Di , and by construction R is a Sylow 2-subgroup of Di and of D B. If R is infinite then R S, and by Remark 6.4, R is a Sylow \ D 2-subgroup of Di and of D B. In each case Z and U are characteristic subgroups \ of R, so that ND .R/ D1 D2. A free amalgamated product decomposition for i Ä \ G is given by 7.4(b), and for G and G0 by the definition of these groups. Thus, Hypothesis 3.4 holds. The verification of the first three parts of Hypothesis 3.8 is immediate in each case. Part (4) of Hypothesis 3.8 holds by Sylow’s Theorem when D is finite, and by Remark 6.4 when D G. Thus (a) is established. By D 3.8(4), RDi B RDi B . Part (b) follows from (a), 3.5(c), and this observation. \ D \ Finally, when D G or G , Hypothesis 3.12 follows from part (a) of Lemma 7.6 D below. For any subgroup X of G, and any elementary abelian 2-subgroup F of X, denote by Ᏹn.X; F / the set of elementary abelian 2-subgroups of X containing F , n of order 2 . Write Ᏹn.X/ for Ᏹn.X; 1/. Recall from the preceding sections that

Z U E A Ᏹ4.G / Ä Ä Ä 2 is a chain of elementary abelian 2-groups, where Z Z.H / z , U Z.B0/ 2 D D h i D D z; z1 , E e T e 1 , and A E w0 . h i D f 2 j D g D h i LEMMA 7.6. The following hold.

H H (a) Ᏹ2.H; Z/ U , and Ᏹ2.H ;Z/ U . D D G G (b) Ᏹ1.G/ Z , and Ᏹ1.G / Z . D D H Proof. By 4.2 there is a unique class z z of noncentral involutions in H . Then 1 z 0 since CH .z1/ B is connected, it follows from Lang’s Theorem that H has a z D z z unique class of noncentral involutions. As H H , (a) follows. # z D Since K is transitive on U , it follows from (a) that all involutions in S (resp. S ), are fused in KH (resp K H ). By 7.5(b), S and S are Sylow in G and G , respectively, so (b) holds. B0 H G LEMMA 7.7. We have E Ᏹ3.B; U /, E Ᏹ3.H; Z/, and E Ᏹ3.G/. D D D 0 Proof. Let F Ᏹ3.B; U / and f F U . Then CB .U / B L1L2L3, 2 2 2 2 2 2D D # so that f f1f2f3 with fi Li , and 1 f f f f . Since U is the D 2 D D 1 2 3 2-LOCALFINITEGROUPS 933 set of involutions in Li Lj for i j , it follows that fi is an element of order 4 ¤ 0 in Li . Since Li is transitive on its elements of order 4, involutions in B U are 0 B0 0 conjugate in B . Thus E Ᏹ3.B ;U/. The lemma now follows from 7.6. D LEMMA 7.8. Set B./ CL ./ 1 i 3 . Then D h i j Ä Ä i (a) There exists an involution v in S B0 B./ such that B0 B./ v , and \ D h i v induces a diagonal automorphism on each CLi ./.

B./ B B./ (b) E Ᏹ3.B./; U / and Ᏹ3.B ;U/ E .E / , where E U v . D D [ 0 0 D h i H H (c) Ᏹ3.H ;Z/ is the disjoint union of E and .E0/ .

(d) EE0 is a Sylow 2-subgroup of CH .E0/, and is elementary abelian of order 16.

(e) For each F E;E , we have AutH .F / C .Z/ and AutG .F / 2 f 0g D Aut.F / D Aut.F /. Proof. Recall from Section 4 that we may regard T as a set of equivalence classes Œa1; a2; a3. Let a be a 2-element in F with a a, and set f Œa; a; a. 0 D D Then f .S B / B./, and since w0 inverts S , the element v f w0 2 1 \ 1 WD is an involution in .S B0/ B./. Recall from 4.4 that B0 is J = i , where J \ z z h i z is the direct product of three copies of SL .F/ and i is an involution diagonally 2 x embedded in Z.J/. Thus J is simply connected, so that B./ J = i , and B./ z z D z h i is of index i 2 in B0. Then B0 B./ v , completing the proof of (a). j j D D h i Let X B0; H , and set 2 f z zg † .F; / F EX ; X ; ŒF; 1 : D f j 2 2 D g Set X X †0 .E; / † and †1 F E .F; / † : D f 2 j 2 g D f 2 j 2 g There is a natural bijection ˇ between the set of NX .E/-orbits on †0 and the set g of CX ./-orbits on †1. Explicitly, if i i I is a set of representatives for f1 j 2 g g the orbits of NX .E/ on †0, then E i i I is a set of representatives for the f j 2 g orbits of CX ./ on †1. By 4.3(d), NH .E/ TW , so that NX .E/ T .W X/. z D z D z \ Let †0. Then 2 X CX .E/ T w0 T T w0: 2 h i \ Â zh i D z [ z When we apply Lang’s Theorem to the connected algebraic group T , we ﬁnd that z T is transitive on T and T w . Since W centralizes both and w , we conclude z z z 0 0 that T and T w0 are the orbits for NX .E/ on †0, with representatives ; w0 . z z f g Applying Lang’s Theorem to the connected group B0, we obtain an element g X g z 2 such that .w0/ . D 934 MICHAEL ASCHBACHER and ANDREW CHERMAK

When we apply the bijection ˇ, CX ./ has two orbits on †1, with represen- g tatives E and E . By 7.7, †1 Ᏹ3.CX . ; Z.X/ /. Thus, CX ./ has two orbits D h g i 0 on Ᏹ3.X; Z.X//, with representatives E and E . In the case that X B we 0 D z have CX ./ B by 6.4(a), and then since E is in the normal subgroup B./ of D CX ./, but E0 is not, it follows that E and E0 are representatives for the two orbits 0 0 0 of B on Ᏹ3.B ;U/. Since ŒE; E0 1, and B B./E0 by (a), these are also D D g the orbits for B./, establishing (b). In particular, E0 is fused to E in B , and so E is not fused to E in H . In the case that X H we get CX ./ H by 0 D z D 7.4(a), and this yields (c). 2n Recall that n for some n 0. Set q p . Let ı 1 with q ı D D D3 ˙ Á mod 8. Then T is homocyclic abelian of rank 3 and order .q ı/ , and CH .E/ D T w0 by 4.3(d). On the other hand, we have seen that .E; w0/ †0. As h i 2 3 w0 inverts T , CT .w0/ is homocyclic abelian of rank 3 and order .q ı/ . In z C particular, E is a Sylow 2-subgroup of CT .w0/. Therefore a Sylow 2-subgroup of CH .w0/ CH .E/ is of order at most 16. Since ŒE; E 1, (d) follows. \ 0 D From 4.3, AutTW .E/ C .Z/, and hence AutTW .E/ AutH .E/. D Aut.E/ D Similarly, since W Hw , we have Ä z 0

AutH .E/ CAut.E/.Z/: zw0 D 0 g Conjugating by the element g of B with .w0/ , we obtain z D g AutH .E / CAut.E g/.Z/: z D Since ŒE; y ŒE ; y U , (e) holds. D 0 D LEMMA 7.9. The following hold.

(a) Ᏹ5.G/ ¿. D X 0 B0 K (b) Ᏹ4.X/ A for X G;H , and Ᏹ4.B / A A . D 2 f g D D (c) AutH .A/ C .Z/, and A CH .A/. D Aut.A/ D 0 Proof. Let Y be H or B , and let A0 be an elementary abelian subgroup of Y of maximal order. Then Z.Y / A , and after conjugation in Y we may assume, Ä 0 by 7.7, that E A . Then A T w0 , by 4.3(d), so that A 16. Since H Ä 0 0 Ä h i j 0j D contains a Sylow 2-subgroup of G, by 7.5(b), we obtain (a). Every element of T is a square, so since w0 acts on T as inversion, all elements in T w0 are fused by T . Thus, A0 is fused to A via T , and this yields (b).

By 4.3(d), CH .A/ CH .E/ CH .w0/ CT w0 .w0/ A. Since w0 Z.W / D \ D h i D 24 and W NH .T / NH .E/, we have W NH .A/. Set T1 t T t 1 . Ä Ä Ä D f 2 j D g Then ŒT1; w0 E, and so T1 NH .A/ and T1 induces on A the subgroup X.E/ D Ä of Aut.A/ consisting of all transvections with axis E. 2-LOCALFINITEGROUPS 935

Let F be a hyperplane of A containing Z. Then F is conjugate to E in H by 7.7. It follows that X.F / AutH .A/ for all such F . Since Ä C .Z/ X.F / F a hyperplane of A over Z ; Aut.A/ D h j i we obtain (c).

LEMMA 7.10. Set M NG.A/, and for any subgroup X of G set MX D D M X. Then the following hold. \ (a) The inclusion maps from MH and MK into M induce an isomorphism M ! MH M MK , and M is edge-transitive on the tree A. B z (b) M is contained in the subgroup G0 of G deﬁned in 5.8(b). In particular M G . Ä (c) There is a surjective homomorphism A M M0, where M0 is a nonsplit W ! extension of an elementary abelian group of order 16 by GL.4; 2/, and such that ker.A/ MH ker.A/ MK 1. \ D \ D (d) For any A satisfying the conditions in (c), we have CG.A/ ker.A/ A, and D ker.A/ acts freely on .

Proof. The induced isomorphism of M with MH M MK is immediate from B 7.3(b) and 7.9(b). The edge-transitivity of M on is given by the ﬁnal statement zA in 7.3, so that (a) holds. By 5.8(a), there are maximal subgroups H0 and K0 of the group Co3 such that H0 may be regarded as a subgroup of H , and K0 as a subgroup of K , in such a way that the resulting amalgam Ꮽ0 .H0 B0 K0/ of subgroups D Ä of G is isomorphic to the corresponding amalgam of subgroups of Co3. Set M0 NCo .A/. Then M0=A GL4.2/, and M0 does not split over A, as one D 3 Š ﬁnds from the list of maximal subgroups of Co3 in[Fin73]. Moreover, as seen in the proof of 5.8, we have M0 M0 H0;M0 K0 , as subgroups of Co3. As D h \ \ i subgroups of G, we have M0 H MH and M0 K MK , so it follows from \ D \ D (a) that M G G . Moreover, (a) implies that M0 is a homomorphic image Ä 0 Ä of M , via a homomorphism A whose restriction to each of MH and MK is the “identity” map. In particular, the restriction of A to MH MK is faithful, and this [ yields (b) and (c). From (c), it is immediate that CG.A/ ker.A/ A. Since M is edge- D transitive on A, and since ker.A/ intersects both H and K trivially, it follows that ker.A/ acts freely on A. That is, every nonidentity element of ker.A/ induces a hyperbolic isometry of A, and hence also a hyperbolic isometry of , by 3.3. Thus, (d) holds.

Set R0 S ; and R1 NS .R0 w0 /. Then R0 has index 8 in R1. Fix a D 1 D 1 h i set X of coset representatives for R0 in R1, and recall that q0 denotes the exponent 936 MICHAEL ASCHBACHER and ANDREW CHERMAK of R0. Then X xe e E D f j 2 g q0 where xe e. Set D xe Ae A ; e E; D 2 and Ᏹ Ᏹ4.R0 w0 /: D h i All elements of R0w0 are fused by R1, so R1 is transitive on Ᏹ. Since NR .A/ 1 Ä R0, R0 has R1 R0 8 orbits on Ᏹ and Ae e E is a set of representatives j W j D f j 2 g for those orbits. Every subgroup of T is -invariant, so each xe is -invariant. Since 2 h i centralizes xe and does not centralize xe, we obtain . / xe e and .e/xe for all e E: D D 2 LEMMA 7.11. For each e E: 2 (a) AutH .Ae/ C . z; e /, D Aut.Ae/ h i (b) AutK .Ae/ N .U / C .e/, and D Aut.Ae/ \ Aut.Ae/ (c) AutG .Ae/ C .e/. D Aut.Ae/ Proof. For any e E, set e e, regarded as an automorphism of G. By 2 D 7.10, M NG.A/ G and AutG.A/ Aut.A/, so that WD Ä D AutG .A/ C .e/ C .e/: e D AutG .A/ D Aut.A/ xe Then as Ae A , conjugating this equality by xe and appealing to . /, we con- D clude that (c) holds. Next, AutH .A/ C .Z/ by 7.9(c), so that D Aut.A/ AutH .A/ C . z; e /: e D Aut.A/ h i Since xe centralizes z; e , conjugation by xe yields (a). h i As AutG .A/ AutG.A/ Aut.A/, as y K, and as NH .U / K, we D D 2 Ä conclude that AutK .A/ N .U /. Then D Aut.A/ AutK .A/ N .U / C .e/; e D Aut.A/ \ Aut.A/ and conjugation by xe yields (b). LEMMA 7.12. Let u U Z and let e E U . Then 2 2 (a) A; Az;Au is a set of representatives for the orbits of H on Ᏹ4.H /, and f g H fuses Au and Ae.

(b) A; Az;Ae is a set of representatives for the orbits of K on Ᏹ4.K ;U/, and f g K fuses Az and Au.

Proof. Since S Syl .G /, each A Ᏹ4.G / is fused under G into S , 2 2 2 and then by 7.6, we may take U A S . Then by parts (b) and (d) of 7.8, we Ä Ä may take E A. Since CS .E/ R0 w0 , we are reduced to the problem of Ä D h i fusion via H , K , and G on Ᏹ. Notice that W has the three orbits 1 , z , and E Z on E, and that S y # f g f g h i has the three orbits 1 , U and E U on E. Hence there are three orbits for S W f g on Ᏹ, with representatives A, Az, and Au, and three orbits for S y on Ᏹ, with h i # representatives A, Az, and Ae. Since S ; W; y is transitive on A f A , h i f f j 2 g it is now enough to show that there is no further fusion among these groups. As AutG .A/ Aut.A/, A is not fused to Az or to Au in G , by 7.11(c). Since Az D is not fused to Au in H by 7.11(a), and Az is not fused to Ae in K , by 7.11(b), the lemma is proved.

LEMMA 7.13. Set N NG.T2/, and for any subgroup X of G set NX D D N X. Let A be deﬁned as in 7.10, and set D C .T2/. Then the following \ D ker.A/ hold.

(a) The inclusion maps from NH and NK into N induce an isomorphism of N with NH N NK . In particular, N is generated by NH and NK . B (b) N is edge-transitive on the tree S . 1 (c) N NG.T / NG.T w0 /. D D h i (d) CG.T2/ CG.S / O.T /D S , and O.T /D is N -invariant. D 1 D 1 (e) AutG.S / GL.3; 2/ Z2. 1 Š (f) Ᏺ .NH / Ᏺ .CN .Z//, and Ᏺ .NH;˛/ Ᏺ .CN .Z//. CS .E/ D CS .E/ CS .E/ D CS .E/ Proof. By 4.9(b) S is a Sylow 2-subgroup of B, and by 4.9(c) T2 is weakly closed in S with respect to G. Therefore T B T g T g B; g H K : 2 D f 2 j 2 Ä 2 [ g Now (a) and (b) follow from 7.3(b). Set T T w0 . Since NH .T2/ NH .E/ TW by 4.3(d), we have NH D h i Ä D D NH .T / NH .T /. Then NB NB .T / NB .T /, and NK NB y NK .T / D D D D h iD D NK .T /. It follows now from (a) that N NG.T / and N NG.T /. Since the Ä Ä reverse inclusions are obvious, we obtain (c). Since W; y NG.A/ NG.T2/, D is W; y -invariant, and evidently so h i Ä \ h ix is O.T /. Let x CG.S /. Then (c) implies that w0 T T , and since T is 2 1 xt 2 transitive on T w0, there exists t T with w w0. Thus xt CG.A/, so it 2 0 D 2 follows from 7.10(d) that xt da for some d ker.A/ and some a A. Since D 2 2 T2 NG.A/, we have ŒT2; d ker.A/, and since ker.A/ acts freely on we Ä Ä have T ker.A/ 1. Let s T2. Then \ D 2 1 Œs; xt Œs; da Œs; aŒs; da; D D D 938 MICHAEL ASCHBACHER and ANDREW CHERMAK

a where Œs; a T2 and where Œs; d ker.A/. It follows that ŒT2; a ŒT2; d 1, 2 2 1 D D and so a E and d D. Now x dat DT , and thus CG.S / DT . By 2 2 D 2 1 Ä 5.1 and (a), ŒD; S 1, so that CG.S / DT . Since D T 1 we have 1 D 1 D \ D S O.T /D 1, and thus O.T /D is a complement to S in CG.S /. This 1 \ D 1 1 completes the proof of (d). Part (e) follows from (a) and Theorem 5.2. Let P S and let g NG.P; S/ CN .Z/. Then g nd for some n NH Ä 2 \ D 2 and d D, by (d). But CS .E/ S w0 , and w0 A CG.D/, so CS .E/ 2 D 1h i 2 Ä centralizes D. Thus d centralizes P , so cg cn on P , establishing (f). D 8. Centric subgroups and signalizer functors

We continue the hypotheses and the notation of Sections 4, 5, and 7. Thus Ꮽ is the amalgam .H B K /, and Ꮽ0 is the amalgam .H0 B0 K0/ ! ! given by 5.8. As in 7.4, we regard Ꮽ0 as a subamalgam of Ꮽ , H0 and K0 as subgroups of G ; and we set G0 H0;K0 and S0 S G0. D h i D \ There is a fair amount of notation which we now need to establish, and which will remain ﬁxed in the remainder. First, we set

Ᏺ ᏲS .G/; Ᏺ ᏲS .G /; Ᏺ0 ᏲS .G0/: D D D 0 Let D be one of the groups G, G , or G0, and let Ᏸ be the fusion system Ᏸ ᏲS D.D/. For any subgroup Y of G such that S D is a Sylow 2-subgroup D \ \ of Y D, we write YD for Y D, and ᏰY for ᏲS .YD/. \ \ D For any subgroup P of SD, set D ZP z Z.P / : D h \ i # Thus ZP 1.Z.P // if D G0, by 7.6(b), and in any case we have ZP D D ¤ D z Z.P /, by 5.9(b). Although the deﬁnition of ZP depends on D, the reader \ may think of D as being ﬁxed, so there need be no cause for confusion. Denote by 0 the smallest G0-invariant subtree of which contains the edge H;K . Recall that i i G denotes the subset Gi G of vertices of , where f g D D n D G1 H and G2 K. Write also for the standard tree for D. That is, is D D the smallest D-invariant subtree of containing the edge H;K . Thus D is , f g , or 0, for D equal to G, G , or G0, respectively, and 0. Ã Ã LEMMA 8.1. Let Y B;K;H , and let P be a 2-subgroup of Y . Then 2 f g NY .P; S/ ¿. ¤ Proof. This follows from Remark 6.4.

LEMMA 8.2. Let P be a subgroup of SD, and let Y G;H;K . Then 2 f g c (a) P Ᏸ if and only if Z.P / contains every ﬁnite 2-subgroup of CY .P /. 2 Y D rc (b) If P Ᏸ then P contains every ﬁnite NY .P /-invariant 2-subgroup of YD. 2 Y D 2-LOCALFINITEGROUPS 939

Proof. We first prove (a). By 2.1 and 7.5(b), we may assume D G, P c D is infinite, P Ᏺ , and there exists a 2-element x of CY .P / with x P . Set 2 Y … P P; x . D h i Since C .P / is x-invariant, and x is finite, it follows from 3.2 that x fixes j j a vertex ı of C .P /, and we may take ı Y if Y H;K . Now 8.1 implies that D 2 f g P is contained in a conjugate of S in G , and then x P since P Ᏺc . Thus ı 2 2 Y (a) is established. rc Now suppose that P ᏰY , set N NYD .P /, and let R be a finite, N -invariant 2 D E 2-subgroup of YD. Set R0 NR.P /. Then R0 N , and so AutR0P .P / D rc Ä O2.AutY .P / Inn.P / as P Ᏸ . Then R0P CR P .P /P , so that R0 D D 2 Y D 0 Ä CR P .P /P P by (a). Thus R0 R P . But the p-group P induces a finite 0 Ä D \ p-group P of automorphisms on R, and C .P/ NR.P /=R0 R0=R0. z NR.R0/=R0 z Ä D It follows that R P , proving (b). Ä c For any P Ᏸ we have CS .P / Z.P /, and thus Z ZP . The following 2 D Ä Ä lemma derives most of the remaining information that we shall need, concerning Ᏸ-centric subgroups of SD, including everything that is needed for the construction of signalizer functors.

LEMMA 8.3. Let P SD. Ä c c (a) Suppose that ZP 2 and that either P Ᏸ or P ᏲH . Then ND.P; S/ j jc D 2 2c H, and if P Ᏺ then NG.P; S/ H and P Ᏺ . Â 2 H Â 2 c h (b) Suppose that P Ᏸ and ZP 4. Then CD.P / H, ZP U , and h 2 j j D Ä c D ND.P / K for some h HD. If also ZP U and P Ᏺ , then NG.P / Ä 2 D 2 K K and P Ᏺc. Ä 2 c HD (c) Suppose that P Ᏸ and ZP 8. Then ZP E . If also Ᏹ4.P / ¿ 2 j j D 2 D then P S D and P is not Ᏸ-radical. D 1 \ c (d) Suppose that Ᏹ4.P / ¿. Then P Ᏺ and O.CG.P // O.CD.P // 1. ¤ 2 D D D Proof. Set † C .P / and † † . Then 1; 2 is an edge of † , D 0 D \ f g 0 since P SD BD. Let ᏼ be the set of paths . 1; ˛; ˇ/ in † such that ˇ 1, Ä Ä D ¤ and ᏼ0 the paths in ᏼ contained in †0. If ᏼ then Z˛ ZZˇ CH .P / by 2 c D Ä 3.11(a), so 8.2 implies that Z˛ ZP if either P Ᏺ or ˇ † . In particular, Ä 2 H 2 0 ZP > 2 and ZP Z˛ if ZP 4. j j D j j D Now assume the hypothesis of (a). Then ᏼ0 ¿ by the preceding paragraph, D g and thus 1 is the unique vertex in C1 .P /. The same is then true for P , for any c g ND.P; S/, and thus ND.P; S/ H. Similarly NG.P; S/ H if P Ᏺ . 2 Â Â 2 H Since CG.P / NG.P; S/, (a) is proved. Â h Suppose next that ZP 4. Since Z ZP , we then have ZP U for j j D Ä D h some h HD, by 7.6(a) and 5.9. We conclude from paragraph one that ˛ for 2 D 2 each ᏼ , and hence that ˛ is the unique vertex in 2 which is in the interior 2 0 940 MICHAEL ASCHBACHER and ANDREW CHERMAK

h h of †0, so that ND.P / K . Now CD.P / CKh .Z/ CK .Z/ H, and the Ä Ä D Ä c first part of (b) is established. Now suppose that ZP U and that P ᏲK . Then c D 2 CH .P / NH .U / K, so that P Ᏺ by 8.2. Then from paragraph one, ˛ 2 Ä Ä 2 H D for each ᏼ, and so 2 is the unique vertex in 2 which is in the interior of †. 2 Thus NG.P / K, and we have (b). Ä Suppose next that ZP 8. If ZP is not conjugate to E in HD, then D G j j D D by 7.7 and 5.9, and ZP is conjugate in HD to the group E0 defined in 7.8(a). But in that case we conclude from 7.8(d) that P does not contain every 2-element of h CD.P /, contrary to 8.2(a). Thus, ZP E for some h HD. Set R CS .E/ D 2 D D and R0 SD T . Then R R0 w0 is a Sylow 2-subgroup of CHD .E/, by 4.3(d), D \ D h i h and we may choose h so that P R . Suppose further that Ᏹ4.P / ¿. Since Ä h D R R0 consists entirely of involutions, we then have P R0 . Since P contains h Ä all 2-elements in CH .P / it follows that P R . Then P R0 by 4.9(c). Since D D 0 D w0 inverts R0, O2.AutD.P // Inn.P /, and therefore (c) holds. ¤ We now remove the hypothesis that P is Ᏸ-centric, and assume that Ᏹ4.P / ¿. G ¤ Let F Ᏹ4.P /. Then F A by 7.9(b), so that F contains every element of 2 2 CG.F / of finite order by 7.10(d). The same is then true of P , and so O.CD.P // 1, D and P Ᏸc by 2.1. That is, (d) holds. 2 c COROLLARY 8.4. Let P Ᏸ , and assume that ZP 4. Then CD.P / 2 j j Ä D CH .P / Z.P / O.CD.P //. D D 2 Proof. By 8.3, CD.P / H, while O 0 .CD.P // Z.P / by 8.2(a). Let X Ä D be a finite subgroup of CD.P / containing Z.P /. The Schur-Zassenhaus Theorem then yields O2.X/ O.X/. Since H is the union of an ascending chain of finite D subgroups, the result follows. Recall from 7.10 that there is a surjective homomorphism

A NG.A/ M0; W ! where M0 is a nonsplit extension of A by GL.4; 2/, and that M0 may be viewed as a subgroup of G0 Co3. From 7.4(c), G0 is the free amalgamated product H0 B x WD 0 K0. The universal property of G0 with respect to Ꮽ0 yields a homomorphism G G0 G0. Then M M0, and we may choose A to be M . For any A0 A , W ! x D g j 2 choose g G with A A and let A NG.A / M0 be the homomorphism 2 0 D 0 W 0 ! given by c 1 A. Then ker.A / does not depend on the choice of the conjugating g 0 element g. Set [ g X ker.A/ ; D g G 2 c and for any P Ᏺ deﬁne a subset .P / of CG.P / by 2 .P / CX.P /O.CG.P //: D 2-LOCALFINITEGROUPS 941

Thus, .P / is a union of cosets of the largest normal subgroup of odd order in c CG.P /. For P Ᏺ , set X X G , and 2 D \ .P / CX .P /O.CG.P // : D c For P Ᏺ set X0 X G0 and 2 0 D \ 0.P / CX .P /: D 0 Write D for 0, , when D G0, G , or G, respectively. D Recall from 7.2 that, for any vertex of , the largest normal 2-subgroup of G is denoted Z. /. G LEMMA 8.5. Let x X, and let A A with x CG.A /. Denote by ƒ.x/ 2 0 2 2 0 the intersection of all the x-invariant subtrees of , set E.x/ Z. / ƒ.x/ , D h j 2 i and denote by Gƒ.x/ the vertex-wise stabilizer of ƒ.x/ in G. Then the following hold.

(a) x ¿, and x induces a hyperbolic isometry on . D (b) E.x/ A , and A E.x/ 2. Ä 0 j 0 W j Ä (c) Let ; ı be an edge of ƒ.x/. Then Gƒ.x/ CG Gı .E.x//. f g D \ (d) If E.x/ A , and H;K is an edge of ƒ.x/, then G is a B-conjugate of ¤ 0 f g ƒ.x/ T w0 . h i G Proof. By the definition of X we have x ker.A / for some A A . 2 2 By 7.10(d), x fixes no vertices (and inverts no edges) of . That is, x induces a hyperbolic isometry of , in the sense of Section 3, and we have (a). Then 3.2 shows that ƒ.x/ is a linear subtree of , on which x acts as a translation. Since ƒ.x/ is contained in every x-invariant subtree of , and since x centralizes A0, we have ƒ.x/ A . Then 7.2 implies that A centralizes E.x/. Since A contains Â 0 0 0 every 2-element in CG.A /, by 7.10(d), we then have E.x/ A . 0 Ä 0 Let .ı0; ı1; ı2/ be a geodesic in ƒ.x/ with ı1 1. Then ı2 ı0h for h 2 D some h in G G , and so also Z.ı0/ Z.ı2/. Since B NH .U /, we have ı1 ı0 D D Z.ı0/ Z.ı2/, and since Z.ı0/Z.ı2/ E.x/ we conclude that E.x/ 8. This ¤ Ä j j yields (b). As G centralizes Z.˛/ for each ˛ 1 ƒ.x/, we get G ƒ.x/ 2 \ ƒ.x/ Ä J CG .E.x//. Conversely J G by 3.10, and this proves (c). WD ;ı Ä ƒ.x/ Suppose that E.x/ A and that H;K is an edge of ƒ.x/. Since Ᏹ4.B; U / ¤ 0 f g AB by 7.9(b), we may assume that A A. By 7.9(c), all hyperplanes of A D 0 D containing U are fused in NB .A/, so we may assume also that E.x/ E. Then D (c) yields G CB .E/, and now (d) follows from 4.3(d). ƒ.x/ D Our aim is to show that D is a signalizer functor on Ᏸ, as defined in 2.5. The key to this is the next result.

LEMMA 8.6. The following hold. 942 MICHAEL ASCHBACHER and ANDREW CHERMAK

(a) CX.A/ ker.A/. D

(b) CX.S / CX.T2/ Cker.A/.T2/O.T /. 1 D Â Proof. Let x CX.A/, and suppose that x ker.A/. Then x ker.A / 2 … 2 0 for some A Ᏹ4.G/ A . Let ƒ.x/ and E.x/ be as in 8.5. Then 8.5 yields 0 2 f g ƒ.x/ A A , E.x/ A A , and E.x/ 8. Since NG.A/ is edge-transitive Â \ 0 Ä \ 0 j j D on A, we may assume that ƒ.x/ contains the edge H;K . By 8.5(d) and as T is z G f g transitive on A CB .E/, we may then assume also that G T w0 . Thus \ ƒ.x/ D h i T w0 is x-invariant, and then so is T . We then have h i .1/ x NG.T / CG.A/ NG.T / ker.A/A N .T /A: 2 \ D \ D ker.A/ Since A E w0 , we have NT .A/ T2. Then D h i D .2/ N .T / N .T2/ C .T2/; ker.A/ Ä ker.A/ D ker.A/ since ker.A/ is invariant under T2 and intersects T2 trivially. Since all involutions in T w0 E are fused by T , there exists t T such that t t h i 2 .A0/ A. Then x ker.A/. By (1) and (2), x ga for some g Cker.A/.T2/ D 2 t D t2 t t and a A. By 7.13(d) we have g Cker.A/.T2/O.T /, so that x a g ky 2 2 t 1 t D D for some k ker.A/ and y O.T /. Thus ya k x ker.A/, and then since t 2 2 D 2 t ya is of ﬁnite order and ker.A/ is torsion-free, we have ya 1. Therefore D y a 1, and x g ker.A/, contrary to our choice of x. This contradiction D D D 2 proves (a). G Let x CX.S /. Then there exists A0 A with x Cker.A /.S /. As in 2 1 2 2 0 1 the proof of (a), x induces a hyperbolic isometry of , and ƒ.x/ is contained in . Setting F Z ı ƒ.x/ , we ﬁnd that F C .S /, and since S A0 ı G 1 \ D h j 2 g i Ä 1 NG.S / is edge-transitive on S , F CH .S / for some g NG.S /. Since 1 1 Ä 1 2 1 g all involutions in CH .S / are contained in E, we conclude that F F E. 1 D D Since ƒ.x/ we get ŒE; A 1, and then E A since A O20 .C .A //. A0 0 0 0 G 0 Â D Ä g D Again by the edge-transitivity of NG.S / on S , we have A0 CH .E/ T , gt 1 1 Ä D for some g NG.S /. Then A0 A for some t T , and so 2 1 D 2 1 .gt/ Cker.A /.S / Cker.A/.S / Cker.A/.S /O.T /; 0 1 D 1 Ä 1 by 7.13(d). This completes the proof of (b).

S G0 PROPOSITION 8.7. X0 ker.A/ . D G Proof. Let x X0. By deﬁnition, there exists A Ꮽ with x ker.A /. 2 0 2 2 0 Since x G , 8.5(a) implies that the axis ƒ.x/ is contained in G . Set 0 A0 0 2 \K E Z.ƒ.x//. Then 8.5(b) yields E A . We have Z Z.S0/, and Z 0 U , 0 D 0 Ä 0 D h i D so that U G0. Since G0 H0 B K0, G0 is edge-transitive on 0, and so Ä D 0 Z. / G0 for any 0. Thus, E G0. Ä 2 0 Ä 2-LOCALFINITEGROUPS 943

Denote by Ᏹ the set of elementary abelian 2-subgroups F of G0 such that F ZG0 F , and by Ᏹ the set of all F Ᏹ with F 2n. By construction, D h \ i n 2 j j D S0 S B0 is a Sylow 2-subgroup of G0, and then 5.9 implies that for any n, all WD \ members of Ᏹ are fused in G0. We have A Ᏹ by 7.10, and evidently E Ᏹ . If n 2 4 0 2 E A we conclude that A AG0 , and there is nothing more to show. Thus, we 0 D 0 0 2 may assume henceforth that E0 is a proper subgroup of A0, and then 8.5(b) yields E 8. Moreover, E is conjugate to E in G0, since G0 is transitive on Ᏹ . j 0j D 0 3 Since G0 is edge-transitive on 0 we may assume that H;K is an edge of f g ƒ.x/. Then 8.5(d) implies that G is conjugate in B to T w0 . Let T be the ƒ.x/ h i 0 abelian subgroup of index 2 in G . By 8.5(c), ŒT ;E 1, so that E T ƒ.x/ 0 0 D 0 Ä 0 and G CB .E / T A . Let R be the Sylow 2-subgroup of T , and set ƒ.x/ D 0 D 0 0 0 N NG.R/. Then 7.13 yields D

CN .E0/ .O.T 0/Cker.A /.R/ R/A0; D 0 and CN .A0/ Cker.A /.R/A0. Since x CN .A0/, 8.6(a) now yields x Cker.A /.R/. D 0 2 2 0 Thus, x centralizes the Sylow 2-subgroup RA0 of T 0A0. Since O.T 0/Cker.A /.R/ contains no nontrival 2-elements, it follows that: 0

(1) x centralizes every 2-subgroup of CN .E0/ that x normalizes.

By 5.8(a), there is a surjective homomorphism G0 Co3 whose kernel inter- ! sects B0 trivially. Then CB0 .E0/ is isomorphic to a subgroup of CCo3 .E0/. Since 7 E0 is conjugate to E in G0, and since CCo3 .E/ is of order 2 , we conclude that CB .E / is a 2-group. Since CB .E / G0 G , (1) now yields: 0 0 0 0 D \ ƒ.x/

(2) x centralizes CB0 .E0/.

If there exists F Ᏹ with F CB .E / then x ker.F / by (2) and 8.6(a). 2 4 Ä 0 0 2 Thus, we may assume:

(3) CB0 .E0/ contains no member of Ᏹ4.

Set † 0 E . Then † is a subtree of 0 containing ƒ.x/. For any d > 0 D \ 0 z denote by ƒ.d/ the subtree of † induced on the set of vertices of † at distance at z z most d from ƒ.x/, and set Z.d/ Z.ƒ.d//. Thus E Z0 G , and we D 0 D Ä ƒ.x/ claim that Z.d/ G for all d 0. Suppose false, and let d be minimal subject Ä ƒ.x/ to the condition that, for some vertex of † at distance d from ƒ.x/, we have .d 1/ z .d 1/ Z. / G . Then Z G , and thus Z G0 B B0. Now (3) — ƒ.x/ Ä ƒ.x/ Ä \ D yields Z.d 1/ E . Notice that E centralizes Z. / since E ﬁxes every edge in D 0 0 0 at every vertex of †. Thus Z. / centralizes Z.d 1/. Arguing as in the proof of 0 z 8.5(c), it follows that Z. / ﬁxes every vertex of ƒ.d 1/, and thus Z.d/ G Ä ƒ.x/ as claimed. It now follows that Z.†/ CB .E /, and then (3) yields Z.†/ E . On z Ä 0 0 z D 0 the other hand, since E is fused to E in G0 there exists F Ᏹ with E F . 0 2 4 0 Ä 944 MICHAEL ASCHBACHER and ANDREW CHERMAK

G0 Then F A . Set ‚ F 0. Since ‚ † and Z.†/ E , we conclude 2 D z \ Â z z D 0 that Z.‚/ E , and hence E E NG.F /. This is contrary to 7.10(c), and the D 0 0 proposition is thereby proved.

THEOREM 8.8. D is a signalizer functor on Ᏸ.

c Proof. Let P Ᏸ and set Y X D. We ﬁrst verify that D.P / is a 2 D \ complementary subgroup to Z.P / in CD.P /. This is the case if ZP 4, by 8.4, j j Ä so assume that ZP 8. Suppose that Ᏹ4.P / ¿ and choose F Ᏹ4.P /. Then j j ¤ 2 every subgroup of P which contains F is in Ᏸc, by 7.3(d). We have F AG by 2 7.9, so that 7.10(d) yields CG.F / F ker.F /. Since ker.F / CX .F / by D D 8.6(a), CY .F / is a subgroup of CD.F /, and we get

CD.F / F CY .F /: D In particular O.CD.F // 1, so CY .F / D.F / by deﬁnition. Thus D.F / is a D D complement to F in CD.F /. Let P1 be maximal in P subject to the conditions: F P1 and CD.P1/ Ä D Z.P1/ D.P1/. If P1 P then P1 < P2 NP .P1/ and CD.P2/ CD.P1/. ¤ WD Ä Both Z.P1/ and Y are P2-invariant, so that P2 also acts on CY .P1/ D.P1/. D Thus

CD.P2/ C .P2/ C .P2/ Z.P2/ D.P2/; D Z.P1/ D.P1/ D contrary to the maximality of P1. Thus P1 P and D.P / is a complement to D Z.P / in CD.P /. On the other hand, suppose that Ᏹ4.P / ¿. Then P D S by 8.3(c). D D \ 1 Set I C .T2/. Then CG.P / T I by 7.13(d). Since ker.A/ G0 G , D ker.A/ D Ä Ä we have CD.P / .D T/ I . If D G then D T T P O.CD.P //, D \ D \ D D while if D G0 then D T T2. Thus D.P / O.CD.P //I is a complement D \ Dc D to P in CD.P / for any P Ᏸ . g 2 g Evidently D.P / D.P / for any g ND.P; S/, so that by 2.6 it remains Dc 2 to show, for any Q Ᏸ with P Q, that .Q/ .P /. Since CY .Q/ 2 Ä Ä Â CY .P /, this amounts to showing that O.CD.Q// O.CD.P //. But O.CD.P // 2 Ä D O .CD.P //, by 7.4 if ZP 4, and by 7.6 if P D S , while in all other j j Ä2 D \ 1 cases we have just seen that O .CD.P // 1. Thus O.CD.Q// O.CD.P // as D Ä required.

9. Saturation and Theorem A

We continue the notation that was introduced at the start of Section 8.

PROPOSITION 9.1. Let D G0;G ;G . Then the fusion systems ᏰH and 2 f g ᏰK are saturated. 2-LOCALFINITEGROUPS 945

Proof. If D G0 or G then HD and KD are finite, and the result is then D immediate from 1.6(c). In the case that D G we appeal to Remark 6.4. D Until further notice, we take D G;G . 2 f g PROPOSITION 9.2. We have ᏰH ᏲS .CD.Z// and ᏰK ᏲS .NG.U //. D D D D Proof. Supposing first that ᏰH ᏲS .CD.Z//, we show that D D ᏰK ᏲS .NG.U //; D D g g as follows. Let g NG.U / and let P S with Q P S. Then .P U / g 2 Ä WD Ä D P U S, and we may therefore take U P . Since AutK .U / Aut.U /, we may Ä Ä g D write g g k with g CG.U / and k K. Set Q P 0 . Then Q CK .Z/ B, D 0 0 2 2 0 D 0 Ä D so by 8.1 we may choose g so that Q S. By assumption, we then have c c 0 0 g0 h h Ä D for some h NH .P; Q / with U U . Then h NH .U / K, and cg c c 2 0 D 2 Ä D h k D c where hk K. We are therefore reduced to proving that ᏰH ᏲS .CD.Z//. hk 2 D D Suppose that ᏰH ᏲS .CD.Z//. By 7.5 and 3.14(c) there then exist P ¤ D Ä SD, F Ᏹ3.Z.P /; U /, and g N .P; S/, such that 2 2 CD.Z/ (1) cg HomH .P; S/, … D (2) U F F g , and Ä \ g g g (3) CB .F / BD and CB .F / .BD/ . D Ä D Ä g Suppose that both F and F are conjugate to E in BD, and choose elements b gb 1 b; b0 in BD with F E F 0 . Set g0 b gb0. From the first statement g D D WD in (3), CB .E/ 0 CB .E/, and thus g normalizes SD T by 4.9(c). We may D Ä D 0 \ then adjust b in T D so that g normalizes .SD T/ w0 CS .E/. Set 0 \ 0 \ h i D D N ND.CSD .E//. It follows from 7.13(f) that N H controls strong fusion in D \ b CN .Z/, and so there exists t N H with ct cg on P . Then cg cbt.b / 1 2 \ D 0 D 0 on P , contrary to (1). g We may therefore assume that either F or F is not conjugate to E in BD. Then 7.8 yields D G , and every member of Ᏹ3.BD;U/ is fused in BD to E D or to E0, where E0 is as defined in 7.8(b). Suppose that F is fused to E and that g g F is fused to E0 in BD. A Sylow 2-subgroup of CB .F / is elementary abelian, by 7.8(d), whereas a Sylow 2-subgroup of CB .F / has exponent 4, contrary to g (3). Similarly if F is conjugate to E0 then F is not conjugate to E. Thus we are g 0 reduced to the case where both F and F are fused to E0 in B , and hence in B , by 7.8(b). 0 b gb 1 Let b; b B with F E F 0 . Then b gb ND.E /. By 7.8(e) 0 2 D 0 D 0 2 0 there exists h H such that ch cb 1gb as elements of AutD.E0/. Then cg 2 g D 0 D cbh.b / 1 in ND.F; F /, and so P F by (1). Since P CS .F / and EE0 is 0 ¤ Ä b gb0 a Sylow 2-subgroup of CH .E0/, we conclude that P , P , and A0 EE0 are a Dga Sylow in CH .E /. Thus there exist a; a H with P A P 0 . Then 0 0 2 D 0 D 946 MICHAEL ASCHBACHER and ANDREW CHERMAK

1 a ga ND.A /. Observe next that A is in the set Ᏹ deﬁned just prior to 7.11, 0 2 0 0 so that 7.11 implies there exists h H with ch ca 1ga as elements of AutD.A0/. 2 D 0 Then cg cah.a / 1 on P , contrary to (1). D 0 LEMMA 9.3. Let P SD, and assume that there exists no ND.P /-invariant Ä subgroup X of ZP with X 2 or 4. Then one of the following holds. j j D (a) P ED AD. 2 [ (b) D G and P is D-conjugate to the group E deﬁned in 7.8(b). D 0 (c) P CS .E/, and either S P or P T is homocyclic of rank 3 and Ä 1 Ä \ exponent at least 4.

G Proof. By hypothesis, ZP 8. Suppose ﬁrst that ZP > 8. Then P A D j j j j 2 by 7.9, and if P A then 7.11(c) and 7.12 show that there is an ND.P /-invariant … subgroup of P of order 2. Thus P AD, and (a) holds in this case. Also, if P 2 2 EG ED then 7.8 yields (b). We may therefore assume that P is not elementary abelian. Then ZP 8, and Z ZP . j j D h Ä By 7.7 we have .ZP / E for some h H, and by 8.1 we may choose h D 2 h so that P S. Let P0 be the group generated by the noninvolutions in P . Ä Since CS .E/ S consists entirely of involutions, P0 is a characteristic abelian 1 subgroup of P , of index at most 2, containing ZP . Since P has no characteristic subgroups of order 2 or 4, it follows that either P S or that P0 is homocyclic. D 1 Since P Ᏹ.S/, P contains a conjugate of T2, and then 4.9(c) implies that P0 Tn … D for some n 2 or P0 S . Thus, (c) holds. D 1 LEMMA 9.4. Let P SD and let X be a nonidentity ND.P /-invariant sub- Ä # group of ZP of minimal order. Then ND.P / acts transitively on X , and the following hold.

(a) There exists an element f hkh0 of D, with h; h0 HD and k KD, such f D f 2 2 that NS .P / SD and such that either X Z;U;E;A , or D G D Ä 2 f g D and X f E , where E is as deﬁned in 7.8(b). D 0 0 (b) If P is fully normalized in Ᏸ then so is P f , for any f as in (a).

# Proof. We ﬁrst show that ND.P / acts transitively on X . This is trivial if X 2, and is immediate from the minimality of X if X 4. Suppose that j j D j j D X 8. If P Ᏹ4.SD/ then 7.12 and 7.11(c) show that AutD.P / leaves no j j D 2 maximal subgroup of P invariant, while if X P Ᏹ3.D/ then 7.7 and 7.8(e) # D 2 show that ND.P / acts transitively on P . Thus, we may assume that P is not elementary abelian, and then 9.3 yields P CS .E/, and either P S or P T Ä D 1 \ is homocyclic of rank 3 and exponent at least 4. Then X E, and 7.13 implies D that ND.P / induces the full automorphism group of X. 2-LOCALFINITEGROUPS 947

We next prove (a). Let 1 x CX .NS .P // and suppose that x z. By ¤ 2 D ¤ 3.13(b) there exist elements h H and k K such that xhk z and such that hk 2 2 D NSD .P / SD. Thus, we may assume that z X, after replacing P by a suitable Ä 2 h conjugate. Then, by 7.6 through 7.9, there exists h0 HD with X 0 Z;U;E;A , h2 2 fh g or else D G and there exists h HD with X 0 E ;Az . If X 0 Az then D 0 2 2 f 0 g D 7.11(c) contradicts the minimality of X, so that this case does not arise in our context. Set Y X h0 . In order to complete the proof of (a) it sufﬁces to show that, in D each case, every 2-subgroup of NHD .Y / is fused into NSD .Y / in NHD .Y /. But in each case we have NS .Y / Syl .NH .Y /, and so the required fusion follows D 2 2 D from Sylow’s Theorem, or from 6.3 and 6.4 when D G and NS .Y / is inﬁnite. D Thus, (a) holds. Now suppose that P is fully normalized in Ᏸ, let f be given as in (a), and set Q P f . Let R Qg be a Ᏸ-conjugate of Q contained in S. Then R P fg , and D D d D d since P is fully normalized there exists d G such that R P and NS .R/ 2 D d Ä NS .P /. Also, as P is fully normalized there exists d 0 G with Q 0 P and d f 2 D d with NS .Q/ 0 NS .P /. Since NS .P / NS .Q/ we conclude that NS .Q/ 0 Ä dd 1 Ä dd 1 D NS .P /, and NS .R/ 0 NS .Q/. Since R 0 P , we have thus shown that Ä D Q is fully normalized, proving (b).

LEMMA 9.5. Let P SD be fully normalized in ᏰH . Assume that there exists Ä a minimal, nonidentity, ND.P /-invariant subgroup X of ZP with Z X. Then P Ä is fully normalized in Ᏸ. g g Proof. Let g ND.P; SD/ and set Q P and Y X . There then exists # 2 D D # y Y such that Œy; NSD .Q/ 1, and since ND.Q/ acts transitively on Y , by 9.4, 2 g D we may assume that y z . By 3.13(b) there exist elements h HD and k KD hk D hk 2 2d such that y z and such that NSD .P / SD. Set d hk and R Q . As D Ä D D h gd CD.Z/ it follows from 9.1 and 9.2 that there exists h0 HD with R 0 P 2 h dh 2dh D and with NS .R/ 0 NS .P /. Then Q 0 P and NS .Q/ 0 NS .P /. This Ä D Ä shows that P is fully normalized in Ᏸ.

LEMMA 9.6. Let P and Q be subgroups of S, and let x; y G such that 2 P x Q and Qy P . Then P x Q and Qy P . Ä Ä D D Proof. The map cxy P P is injective, so by 6.4 and 6.2(7), cxy is an W ! isomorphism. Thus P P xy Qy, and similarly P x Q. D D D THEOREM 9.7. Ᏸ is saturated.

Proof. Let 1 P SD and let X be a minimal, nonidentity, ND.P /-invariant ¤ Ä g subgroup of ZP . By 9.4 there is g D with NS .P / SD and with Z 2 D Ä Ä X g . Adjusting in H and appealing to 9.1, we may assume that Q P g is fully WD normalized in ᏰH . Then Q is fully normalized in Ᏸ, by 9.5, and thus Ᏸ satisfies the 948 MICHAEL ASCHBACHER and ANDREW CHERMAK saturation condition (I) in 1.5. It remains to verify the two parts of condition (II). Thus we may take P to be fully normalized in Ᏸ. y By 9.4 there exists a fully normalized conjugate P 0 P of P such that y D y X X is in Z;U;E;E ;A . Then y can be chosen so that NS .P / NS .P /, 0 WD f 0 g Ä 0 and similarly (since P is fully normalized) there exists y0 ND.P 0;P/ such that y y 2 NS .P / 0 NS .P /. By 9.6, NS .P / NS .P /, so that if P satisfies saturation 0 Ä D 0 0 condition (II) then so does P . Thus, we may assume that X Z;U;E;E ;A 2 f 0 g and, in particular, that Z X. Also, since ᏰH is saturated there exists a ᏰH - Ä conjugate P 00 of P which is fully normalized in ᏰH , and hence in Ᏸ by 9.4(b); the preceding argument then shows that NS .P / and NS .P 00/ are Ᏸ-isomorphic. We may therefore assume that P is fully normalized in ᏰH . Then P is fully normalized in ᏲSD .CD.Z//, by 9.2. If X Z;U then fusion in ND.P / is controlled by fusion in NH .P / or 2 f g D in NKD .P /, by 9.2, and since ᏰH and ᏰK are saturated, there is nothing more to prove in these cases. Thus, we may assume that X 8. By 9.3 we then have j j P X E;E0;A , or else P CSD .E/ with P T S or with P T D 2 f g Ä \ D 1 \ homocyclic and of exponent at least 4. In the first case, where P is elementary abelian, we have AutS .P / Syl .AutD.P // by 7.8 and 7.10. In the second case D 2 2 we obtain AutS .P / Syl .AutD.P // from 7.13. This establishes the saturation D 2 2 condition (IIA). g 1 Now let ˛ cg AutᏰ.P /, where g ND.P /. Set Z0 Z . By definition, g D 2 2 D we have N˛ CD.P /SD. Thus N˛ CSD .Z0/. By 3.13(b) there exists a Ä a Ä a 1 2 ND.N˛;SD/ with .Z0/ Z. Set Q P and b a g. Then b CD.Z/ and b D D D 2 Q P . As P is fully normalized in ᏲS .CD.Z//, it follows from the “standard” D D axioms for saturation in[BLO03] – equivalent to those in 1.5 – that ˇ c extends WD b to a Ᏸ-morphism c of N into SD. That is, there exists d ND.N ;SD/ such Nb ˇ 2 ˇ that d 1 centralizes Q. Set g ad. Then N D g 1g g 1ad g 1gb 1d b 1d; N D D D 1 a and so g g centralizes P . But also .N˛/ N since N Ä ˇ ab a 1ag g .N˛/ .N˛/ .N˛/ ; D D cg and AutN .P / AutS .P / by definition of N˛. Now ˛ Ä D g ad g .N˛/ .N˛/ .N / SD; N D Ä ˇ Ä and thus cg is an extension of cg from P to a Ᏸ-morphism of N˛ into SD. This N shows that Ᏸ satisfies the saturation condition (IIB), and the proof is thereby complete.

In 2.6 it was shown that if Ᏸ is saturated then a Ᏸ-signalizer functor D determines an associated centric linking system and an associated p-local group 2-LOCALFINITEGROUPS 949

Ᏻ .D/ .SD; ᏲS .D/; ᏸ /. Write Ᏻ0 .S0; Ᏺ0; ᏸ0/, Ᏻ .S ; Ᏺ ; ᏸ /, SD;D D D D D D Ᏻ .S; Ᏺ; ᏸ/, for Ᏻ .D/, when D is G0, G , G, respectively. Theorems 8.8 D SD;D and 9.7 therefore have the following immediate corollary:

COROLLARY 9.8. Ᏻ is a 2-local finite group, and Ᏻ is a 2-local group. THEOREM 9.9. For any fixed p, p 3 or 5 mod 8, and for any integer of 2n Á the form q p , there is a unique n such that the fusion system ᏲSol.q/ D D constructed in [LO02] is isomorphic to Ᏺ . Moreover the associated 2-local finite group constructed in [LO02] is isomorphic to Ᏻ .

Proof. The fusion system Ᏹ ᏲSol.q/ is constructed as D ᏲS .Hq/; ᏲS .Kq/ ; h q U;q i where Hq H is Spin .q/, Sq S is a Sylow 2-group of Hq, SU;q CS .U /, D 7 D D and Kq K , subject to a choice of embedding ˛ of Bq NH .U / in Kq such D D q that, for I NH .SU;q/; NK .SU;q/ , AutI .SU;q/ GL.3; 2/ C2. By parts D h q q i Š ˛ (a) and (d) of 9.7, the amalgam Ꮽ˛ .H B K/ is determined up to D ! isomorphism by these properties, so Ꮽ˛ Ꮽ .H > B < K /. Thus D ; D Ᏹ ᏲS .H /; Ᏺ .K / . By 1.10, ᏲS .K / ᏲS .S /; Ᏺ .K / , D h CS .U / i D h CS .U / i so Ᏹ ᏲS .H /; ᏲS .K / , and then Ᏹ ᏲS .G / Ᏺ by 3.7, completing the D h i D D proof of the first part of the theorem. The remainder of the theorem follows from the uniqueness of the 2-local finite group associated to Ᏹ, proved in[LO02]. To sum up: parts (1) and (2) of Theorem A are given by the construction of G in Section 5, while parts (3) and (4) of Theorem A are given by the preceding theorem. Part (5) is given by 7.9, 7.10, and the definition of X. Thus all parts of Theorem A have been proved.

LEMMA 9.10. Ᏺ is frc-generated. That is, Ᏺ A .P / P Ᏺfrc ; D h Ᏺ j 2 i frc where Ᏺ is the set of fully normalized radical centric subgroups in Ᏺ, and AᏲ.P / is the fusion system on P whose morphisms are the restrictions of members of AutᏲ.P / to subgroups of P . Proof. This generalization of Alperin’s Fusion Theorem is well known for saturated fusion systems on ﬁnite p-groups S; a short proof appears in Theorem A.10 of[BLO03]. A modiﬁcation of this proof when S is a discrete p-toral group appears in Theorem 3.6 in[BLO05]. We sketch that proof in our special case, where things are much easier. Pick P;P 0 S and an Ᏺ-isomorphism ' P P 0 such that ' is not in Ä frc W ! Ꮽ A .P / P Ᏺ . Then ' cg for some g G. As in the proof of D h Ᏺ j 2 i D 2 Theorem A.10 in[BLO03], using the fact that Ᏺ is saturated, we may assume 950 MICHAEL ASCHBACHER and ANDREW CHERMAK

P is fully normalized in Ᏺ. Pick P so that T2 P is maximal. If T2 P then 0 j \ j — T2 P < NT2 P .P /, and as in the proof of A.10, after we replace P by NT2 .P /P , \ \ the maximality of T2 P supplies a contradiction. j \ j As T2 P , 4.9(c) says that g NG.T2/. As S E NG.T2/, replacing P;P 0 Ä 2 1 by PS ;P 0S , we may assume S P . In particular P is centric. Finally 1 1 1 Ä choosing P S maximal, and arguing as in the proof of Theorem A.10 in j W 1j [BLO03], we ﬁrst reduce to the case where g NG.P /, and then show P is 2 radical, contradicting the assumption that cg Ꮽ. … 10. Radical centric subgroups

In this section, we determine the members of Ᏸrc, and make the necessary preparations for obtaining embeddings among the 2-local groups constructed in the preceding section.

LEMMA 10.1. Let Y H;K , and let P S with ND.P / Y . Then the 2 f g Ä Ä following are equivalent. (a) P Ᏸrc. 2 (b) P Ᏸrc . 2 Y (c) P contains every 2-element in CY .P /, and O2.OutY .P // 1. D D c Proof. As CD.P / Y , 8.2(a) says that P Ᏸ if and only if P contains Ä 2 c every 2-element in CYD .P /, and that this holds if and only if P ᏰY . Thus we c 2 may assume P Ᏸ . As ND.P / Y , we have 2 Ä O2.AutD.P // O2.AutY .P //: D D rc Thus P Ᏸ if and only if InnD.P / O2.AutD.P // if and only if InnD.P / 2 rc D D O2.AutY .P // if and only if P Ᏸ if and only if O2.OutY .P // 1, completing D 2 Y D D the proof. 0 Recall from 4.5 that B is the commuting product of “components” L1, L2, and L3, where each Li is isomorphic to SL.2; F/. rc LEMMA 10.2. Let P Ᏸ , set N NK .P /, R SD, Ji Li D, and 2 K D D D D \ J J1J2J3. For any subgroup X of KD, and any i with 1 i 3, denote by D Ä Ä Xi the projection of X J in Ji . Denote by ᏽi the set of subgroups of Ri that are \ isomorphic to the quaternion group Q8. Then the following hold:

(a) CK .P / P . Ä (b) ZP Z;U . 2 f g (c) P J P1P2P3, and for each i with 1 i 3, either Pi Ri or Pi ᏽi . \ D Ä Ä D 2 (d) Either (i) P CR.U /; R , 2 f g 2-LOCALFINITEGROUPS 951

(ii) P P J and Pi ᏽi for at least two indices i, or D \ 2 (iii) P .P J/ s for some s P CP .U /, and either P3 ᏽ3 or Pi ᏽi D \ h i 2 2 2 2 for both i 1 and 2. Moreover, if P3 R3 then O .N J3/ 1. D ¤ \ ¤ (e) P Ᏺc , and every N -invariant 2-subgroup of K is contained in S. 2 K (f) If U is a fours group in K such that ŒP; U Z, then U U . 0 0 Ä 0 D rc Conversely, any subgroup P of S which satisﬁes (c) and (d) is in ᏰK .

Proof. Set ᏶ Ji 1 i 3 and set PJ P1P2P3. Since ᏶ is KD- D f j Ä Ä g D invariant, PJ is N -invariant, and then PJ P by 8.2(b). Thus Pi P for all i, Ä Ä and PJ P J . D \ P P Suppose that Pi U for some i. Then P P U , and R R Ä WD h i i Ä WD h i i is a P -invariant 2-group which properly contains P . Pick r NR .P / P with 2 r2 U . Then r centralizes the chain P U 1, and since U E N we conclude 2 from 2.2 that r P . This contradiction shows that no Pi is contained in U . 2 Set N NJ .Pi /. Then N N N N is N -invariant, so by 8.2(b), i D i D 1 2 3 O2.N / Pi . Let ᏿ be the class of 2-groups each of whose ﬁnite subgroups is i D cyclic. Suppose that Pi ᏽi . There is then a unique maximal subgroup X of Pi in … ᏿ of order at least 8. Since X 8 there is also a unique maximal 2-subgroup Vi j j of Ji in ᏿ containing X. Then Vi is N -invariant, and 8.2(b) yields Vi Pi . Then i Ä also AutRi .P / O2.AutKD .P //, and Pi Ri . Thus, (c) is proved. It follows Ä D 0 from (c) that CL .Pi / Pi , so C 0 .PJ / PJ . Since any element of K B acts i Ä B Ä nontrivially on U , and since U PJ , we obtain (a) and (b). Ä Suppose next that no Pi is a quaternion group. Then for all i we have Pi D Ri ᏽi . If D G then PJ CS .U / is of index 2 in P , and (d)(i) holds. On … D D the other hand, suppose that D G. Since Ri ᏽi if D G0, we conclude that ¤ 2 D D G . By 7.8(a) there exists an element x of SD such that x induces a diagonal D outer automorphism on each Ji . Here N J PJ and PJ x E SD, so as P \ D h i is ᏰK -radical it follows that x P . Then PJ x CR.U / P , and again (d)(i) 2 h i D Ä holds. If CP .U / PJ then there exists x CP .U / J . As in the preceding para- ¤ 2 graph, D G, and for all i either Ji SL.2; F /, or D G0 and Ji ᏽi . As ¤ 2 Š D 2 ŒNi ; x P for all i, O .Ni / is not isomorphic to SL.2; 3/, and Pi Ri for all i. Ä D Thus P .R J/ x CR.U /. Once again, we obtain (d)(i). \ h i D Now assume that (d)(i) does not hold. We conclude from the discussion above that Pi ᏽi for some i and that PJ CP .U /. Suppose next that P PJ , 2 D 0 D and that neither P1 nor P2 is a quaternion group. Recall that B B s where D h i s S0, s centralizes L3, and s interchanges L1 and L2. One may now check that 2 2 N .N J/ s and that O .N J/ J3, whence P s E N . As P is ᏰK - D \ h i \ Ä h i radical, s P , contrary to the assumption that P PJ . Thus, either P1 or P2 is a 2 D quaternion group. Since K=B0 Sym.3/, a similar argument shows that for any Š 952 MICHAEL ASCHBACHER and ANDREW CHERMAK two distinct indices j and k, at least one of Pj and Pk is a quaternion group. Thus (d)(ii) holds in this case. Now suppose that P PJ . As PJ CP .U / we then have P PJ s ¤ D D h i for some s R CR.U /. Then s interchanges P1 and P2, and since some Pi 2 is a quaternion group we get either P3 ᏽ3 or Pi ᏽi for both i 1 and 2. If 2 2 D P3 R3 then (d)(iii) holds. Thus we may assume P3 R3, so that P3 ᏽ3. D 6D 2 Also R3 ᏽ3, so that in particular D G0. Let N N=PJ . If P1 R1 then 2 … 2 2 6D x D 2 D O .N / O .CN .s// O .N3/, while if P1 ᏽ1 then O .N / PJ d or D2 N Ä 2 D h i PJ d O .N3/ for a suitable element d of order 3 in N1N2. In particular if 3h i O .N / does not centralize s, then NR .P / O2.N /, again contrary to R3 P3 3 N 3 Ä 6D and P ᏷r . Thus (d)(iii) holds, completing the proof of (d). 2 D Suppose next that x is a 2-element in K U , such that ŒP; x Z. If ŒP; x Z Ä D assume that x is an involution. Since CLi .Pi / Z.Pi / U , and all involutions in 0 D Ä L3 are in Z, it follows that x B . But then x interchanges Pi and Pj for some … pair of distinct indices i and j , and so ŒP; x Z. Thus no such x exists. This — proves (f), and shows also that P Ᏺc . 2 K Let F be an N -invariant 2-subgroup of K with F S. Then FP is a 2-group. — 0 0 For any i, set Si S Li , and recall Fi , Ni are the projections of F B , N B D \ \ \ on Li . Then Fi is Ni -invariant. It is a property of the group L PSL.2; F/ that the x D intersection of any pair of distinct Sylow 2-subgroups is abelian (either cyclic or a fours group), and therefore Si is the unique Sylow 2-subgroup of Li containing 0 Pi . Thus Fi Si , and we conclude that F B S. Ä 0 \ Ä Now F .F B / t , where t acts nontrivially on ᏸ L1;L2;L3 . Thus, D \ h i WD f g there is an ordering .1 ; 2 ; 3 / of 1; 2; 3 such that t interchanges L and L , and 0 0 0 10 20 0 f g ﬁxes L3 . Suppose that P B . Then there exists s P such that s interchanges 0 — 2 L1 and L2, and since s; t is a 2-group it follows that i i for all i. Without h i D 0 loss, we may assume that P F , so that st F B0. But then t S and F S. 0 Ä 2 \ 2 Ä Thus P B , and it follows from (c) and (d) that Ni N for all i. Ä Ä 0 Suppose that P J . Then (d) implies that CR.U / P , and since P B we — Ä Ä get P CR.U /. The Frattini Argument then implies that there exists an element D x of N which permutes the components Li transitively, and then x; t is not a h i 2-group. This shows that P J , and a similar argument shows that there exists j Ä with Pj Rj . ¤ t 0 t t t Since Pi F B S, we have Pi S Si , so that Si S for all i. N Ä \ Ä Ä \ Ä Since t i is a 2-group, we obtain N1 R1 and N2 R2 . Then P3 R3 , h i 0 D 0 0 D 0 0 ¤ 0 and so N GL .3/ . Then N .N / C .O2.N //N . Since N E N we 30 2 C K 30 K 30 30 Š 2 D 2 have N CN .O .N //N3 , and since CN .O .N // is a 2-group we get N N3 P . D 0 D 0 Thus, N N1N2N3. On the other hand, there exists an element t of CK .J3 / D 0 D 0 such that t interchanges R1 and R2 . Then t P , by 7.1(b), contradicting P J , 0 0 0 0 2 Ä and completing the proof of (e). 2-LOCALFINITEGROUPS 953

It remains to establish the final statement in the lemma. Thus let P S such Ä that P satisfies (c) and (d). Then there exists Q P such that Q Qi Q8 Ä c Š for all i, and one checks that CK .Q/ U Q. Then Q Ᏸ , and hence also D Ä 2 K P Ᏸc . 2 K Set N NK .P /. As CK .Q/ U we have O.N / 1, and it remains to D D D D show that P O2.N /. If (d)(i) holds, then P R or P CR.U /, so that N R D D D D or N=R Sym.3/, and in particular P O2.N /. Suppose that (d)(ii) holds. If Š D P Q then it is easy to check that P O2.N /; so we may assume that P Q. D D ¤ Then Pj Rj ᏽj for exactly one index j , N=P Sym.3/ C2, and we are done D … Š o in this case. Finally, suppose that (d)(iii) holds, set PJ P J , and let s P J . Since D \ 2 PJ CP .U / we have N M NK .PJ /. Set M M=PJ ; it follows that N D Ä WD D x D is the preimage in M of CM .s/. Thus it suffices to show that x N . / s O2.CM .s//: hNi D x N If Ri ᏽi then D G0;G , PJ Q, R=Q is a 4-group, and every involution in 2 2 f 0 g D R=Q has nontrivial fixed points on O3.M/. Since NK .R/ R we conclude that x D D . / holds in this case. We may therefore assume that Ri ᏽi . Suppose that PJ Q. … 2 D Then M Sym.3/ Sym.3/, and P3 R3. Then (d)(iii) requires O .N3/ 1, x Š o ¤ ¤ and hence N contains an element g of J3 of order 3. Then O3.N/ 9, s is a j x j D hNi Sylow 2-subgroup of CM .O3.N //, and we have . /. x x If P3 R3 ᏽ3 then P1 and P2 are quaternion groups, M Sym.3/ C2, D … x Š o and N D12. This yields . /, so we are reduced to the case where Pi Ri ᏽi x Š 2 D … for both i 1 and 2. Then P3 R3, so O .N J3/ 1, and M N with D ¤ \ ¤ x D x s O2.M/. Again . / holds, so the proof is complete. hNi D x LEMMA 10.3. Let g G and R S such that R R1R2R3, with Ri g 2g Ä D 8 D R L Q8. Then U U Z.R/, and R is special of order 2 . \ i Š D D Proof. For each i j we have Ri Rj Ri Rj , so that R is special with 6D D center U g . Thus it remains to show U U g . Set S S=S . Then S has no i D g z D 1 z Q8-subgroups, so that R S 1, and hence U S . \ 1 ¤ Ä 1 Set Y CR.E/ and Y0 R S , and suppose first that R=Y 2. Then D D \ 1 j j Ä Y0 64, so as R has no elements of order 8 it follows that Y0 T2. Since j j D ˆ.R/ Z.R/, this is a contradiction and so we conclude that R=Y 4. Since D j j U g Y , R=Y is elementary abelian, and is then a maximal elementary abelian Ä subgroup of AutS .E/. Since the fixed point groups in E for the two maximal g elementary abelian subgroups of AutS .E/ are Z and U , we conclude U U . D rc LEMMA 10.4. Let P Ᏸ with ZP 4. Then: 2 j j Ä (a) P contains every P -invariant subgroup of D of order 4.

(b) ZP Z;U . 2 f g 954 MICHAEL ASCHBACHER and ANDREW CHERMAK

Proof. Let F be a P -invariant subgroup of D of order 4. Then ŒF; P 2, j j Ä and so 8.2(a) implies that ŒF; P ZP . If ŒZP ; F 1 then F centralizes the Ä D chain P ZP 1, whence F P by 8.2(a) and 2.2. On the other hand, suppose Ä h that ŒZP ; F 1. Then ZP 4 and ZP ˆ.P /. By 8.3(b) we have ZP U ¤ j h j D — D for some h HD. Then E CD.ZP /, and so P ZP . Thus ˆ.P / 1, so that 2 Ä ¤ ¤ ZP ˆ.P / Z, and F centralizes the chain P ZP Z 1 of characteristic \ D subgroups of P . Thus F P by 2.2, and (a) holds. Ä g g Now suppose that ZP Z. Then ZP U for some g HD, and P ¤ D 2 gk Ä CHD .U / KD by 8.3(b). By 6.4 there exists k KD with P SD, and Ä g 2 g rc Ä replacing g by gk, we may assume P SD. Then P Ᏸ by 8.3(b) and 10.1, Ä 2 K and then 10.2 implies that P contains a subgroup R satisfying the hypothesis of 1 9.3, with g 1 in the role of g. Then U U g by 10.3, proving (b). D rc D LEMMA 10.5. P Ᏸ with ZP > 4 if and only if P A or P CS .E/. 2 j j 2 D D

Proof. Let P SD. Suppose that ZP > 8. Then P CD.P / ZP Ä D j j D D 2 Ᏹ4.SD/, by 7.9. If P A then 7.10 shows that O2.AutD.P // 1, whence rc 2 D D P Ᏸ . On the other hand suppose that P A . Then D G by 5.9 and 7.9, and 2 D … D then 7.12 shows that P A for some u U Z. Now AutD.Au/ C .u/ 2 u 2 D Aut.Au/ by 7.11(c). On the other hand, the deﬁnition of Au preceding 7.11 shows that

AutT2 .Au/ is an elementary abelian subgroup of AutD.Au/ of order 8. It follows rc that O2.AutD.Au// 1. Thus P Ᏸ . Hence the lemma holds when ZP > 8, ¤ … j j so we are reduced to the case where ZP 8. rc j j D If Ᏹ4.P / ¿ then P Ᏸ by 8.3(c). Thus we may assume Ᏹ4.P / ¿. D D … ¤ Suppose ZP E . Then D G by 7.7, and D G0 by 5.9, so that D G . … 6D 6D D Then CSD .ZP / Ᏹ4.SD/ by 7.8. But in that case P ZP , and P is not centric. 2 D D Thus we may assume ZP E . Set R SD. 2 D G By 5.9, 7.7, and 7.8, HD is transitive on E HD. Since E is normal in the h\ h h Sylow 2-group R of HD, there is h HD with E ZP , P R , and ZP E R . 2 h D Ä In particular P CRh .ZP /, so that P0 P S is of index 2 in P . Ä D \ 1 Now ZP is generated by the involutions in P0, and P P0 consists entirely of involutions, and so P0 is a characteristic subgroup of P . Let R0 be the unique h conjugate of T2 in R . Then R0 centralizes the chain P R0 P 1, and so R0 P \ Ä by 2.2. Thus R0 T2 is weakly closed in P by 4.9(c), and so h; ND.P / D h h i Ä ND.T2/. Then ZP E E and AutCR.E/.P / O2.AutD.P // by 7.13. Thus rc D D Ä if P Ᏸ then P CR.E/. On the other hand if P CR.E/ then 7.13 says 2 D D P Ᏸrc. This completes the proof. 2 Recall from Section 4 that H acts on an orthogonal space V of dimension 7 over F, and that there is a distinguished basis x1; : : : ; x7 of V such that T acts on f g Fxi for each i. Deﬁne FD to be F if D G, F if D G , and F if D G0. D D 0 D 2-LOCALFINITEGROUPS 955

Let VD be the FD-span of x1; : : : ; x7 . Then the quadratic form f associated f g with V restricts to a quadratic form fD on VD, preserved by HD. Denote by ƒ.VD/ the collection of all sets ƒ of pairwise orthogonal subspaces F of VD whose sum is VD. For any subspace X of VD, denote by X the F-span of F X in V . For any ƒ ƒ.VD/ define ƒ ƒ.V / by 2 2 ƒF X F X ƒ ; D f j 2 g and define the type of ƒ to be the nondecreasing sequence .ƒ/ of integers D given by the dimensions of the members of ƒ. We will abbreviate such sequences, using exponential notation. For example, .ƒ/ 17 means that each member of D ƒ is a 1-space, while .ƒ/ 152 means that ƒ consists of five 1-spaces and one D 2-space. Write ƒ.VD; / for the set of ƒ ƒ.VD/ with .ƒ/ . For ƒ ƒ.V / 2 D 2 and X a subgroup (or subset) of H , write CX .ƒ/, NX .ƒ/ for the set of all x X h i 2 which acts on each member of ƒ, and permutes the members of ƒ, respectively.

LEMMA 10.6. Let ƒ ƒ.V / with CH .ƒ/, and set .ƒ/. Then H 2 2 D acts transitively on ƒ.V; /. Proof. Since every member of F is a square, all nondegenerate subspaces of V of a given dimension are isometric, and so the result follows from Witt’s Lemma.

rc LEMMA 10.7. Let D G ;G , let P Ᏸ with NH .P / K, and set 2 f g 2 H D — N NH .P /. Denote by Ꮾ.P / the set of N -invariant 2-subgroups of H , and set D D ˇ.P / Ꮾ.P / . Then: D h i (a) One of the following holds. 7 (1) There exists ƒ ƒ.VD; 1 / such that N NH .ƒ/ and P CH .ƒ/. 2 D D D D Moreover either 3 (i) P D , and N=P Alt.7/ if D G , while N=P Sym.7/ if Š 8 Š D 0 Š D G 0 , or ¤ 2 (ii) D G , P Z4 Q8, and N=P Sym.6/. D Š Š 5 (2) D 0, and there exists ƒ ƒ.VD; 1 2/ with N NH .ƒ/. More ¤ 2 Ä D precisely P O2.NHD .ƒ// t , where t acts as 1 on every point in ƒ D h i F and as a reﬂection on the line ` in ƒ. Further, ` is one of the lines li , 1 i 3, from Section 4, and N=P Sym.5/. Ä Ä Š (3) P O2.NH .E//. D D (b) CG.P / P . Ä F (c) ˇ.P / S. More precisely: ˇ.P / O2.NH .ƒ // in case (a)(1), Ä F D O2.NH .ƒ // t in case (a)(2), and CS .E/ in case (a)(3). h i Proof. Observe that U P by 10.4(a). Set H H=Z. We ﬁrst prove (a). Ä D 956 MICHAEL ASCHBACHER and ANDREW CHERMAK

Let Q be an NH .P /-invariant elementary abelian 2-subgroup of Z.P / containing U ; for example as U Z.P /, Q U NH .P / is such a sub- Ä U D h i group. Let ƒ ƒ.Q/ be the set of weight spaces of Q on VD; then ƒ .VD/ D F F 2 and NHD .P / NHD .ƒ/ NHD .ƒ /. Let R O2.NH .ƒ //. As NHD .P / F Ä D D Ä NH .ƒ / NH .R/, Ä F R D O2.NH .ƒ // O2.NH .P // P \ Ä D Ä D Ä by 8.2(b). On the other hand P centralizes Q and hence stabilizes each member F of ƒ . Further if AutP .Y / 1Y for each Y ƒ, then P R D, so that FÄ h i 2 Ä \ P R D O2.NH .ƒ // O2.NH .P //. D \ D D D D Suppose ƒ is of type 17. Then each Y ƒ is a 1-dimensional P -invariant or- 2 thogonal space, so that AutP .Y / 1Y , and hence P R D O2.NH .P // Ä h i D \ D D by the previous paragraph. Further, ƒ ƒ.P /, so that NHD .P / NHD .ƒ/, and F D D as P O2.NHD .ƒ //, P CHD .ƒ/. If D G it follows that (a)(1.i) holds, D D D F so we may take D G . As Œ; P 1, CH .ƒ /, so by 10.6 there exists D H D 2 h i h H , ‚ ƒ.H /, and 0 such that centralizes NH .‚/ N modulo Z, 0 2 2 F h2 D h centralizes ‚, and .; ƒ / . 0; ‚/. In particular NHD .P / NH .P /. As 0 D Š 0 centralizes ‚, r for some r R0 O2.NH .‚//. We may regard R as the 0 D 2 D 0 core of the permutation module for S7 NH .‚/=R0. Thus r is of weight 0, 2, 4, Š 2 2 or 6 in R0. If 0 then Œ; g z for g N O .N /, so that O .N / CN ./, D D 2 2 D and hence if in addition r is of weight 0 then NH .P / O .N /. In this case (a) D Š holds, so we may assume one of the remaining cases holds. Then N centralizes R0 if r Z and rz r if r Z, so that CN .r / R CN .r/ by a Frattini 0 2 2 … D 0 argument. Further, centralizes R0, so that 3 2 2 2 P CR .r/ CR .r/ D ; Z4 Q ; Z2 Q ; or Z4 Q ; Š 0 Š 0 Š 8 8 8 8 for the respective choices of r, and NHD .P /=P is isomorphic to the stabilizer in S7 of r , and hence is S7, S5 Z2, S4 S3, S6, respectively. So as R D \ D O2.NHD .P //, r is of weight 0 or 6, and hence (1) holds. Assume next that ƒ is of type 152, and let l be the line in ƒ. Then

O2.NH .P // P D Ä 5 and P acts on each member of ƒ. Let ‚ ƒ.V / be of type 1 2, lN the line in ‚, 2 and NH .‚/ N . Then N N1N2 tN , where N1 CG.lN /, N2 CN .l /, tN D D h i D D N? inverts lN? and induces a reﬂection on lN , O2.N1/ Q8D8, and N1=O2.N1/ S5. Š F Š If D G , then as above we can pick ‚ so that .; ƒ / is conjugate in H to .r; ‚/ D 2 such that centralizes O .N1/, and r ﬁxes each member of ‚. Thus r r1r2 with D r1 O2.N1/, r2 N2, and r is of weight 0, 2, or 4 in the permutation module 2 2 1 O2.N1/ for N1=O2.N1/ S5. But again if r 1 then O2.CN .r// does not Š 1 6D act on each point in ‚, contrary to an earlier remark. 2-LOCALFINITEGROUPS 957

F 2 Thus we may take ‚ ƒ and H1 O .CH .l// O2.N1/ A5=Q8D8 D D D Š Š with P1 O2.H1/ P by 8.2(b). Let H2 CH .l /. Then H2 is cyclic and D Ä D D ? P2 O2.H2/ P by 8.2(b). Also there is t NH .ƒ/ inverting l and inducing D Ä 2 D ? a reﬂection on l. Then t induces an automorphism in O2.AutHD .P // on P , so rc that t P as P Ᏸ . Indeed P P1P2 t . 2 2 H D h i If P 2 then P is elementary abelian with weight spaces of dimension 1, j 2j Ä and we obtain a contradiction from our treatment of the case ƒ of type 17. Thus P > 2, and if D G then P .q "/2=2 where q FD " 1 mod 4. j 2j D j 2j D D j j Á D ˙ It follows that 0. Next U Q P1P3 where P3 is the subgroup of P2 of 6D Ä Ä order 4, so P2CP1 .U / CS .U /. As AutCS .U /.S / E8, P4 ˆ.P2/ S . Ä 1 Š F D Ä 1 As P4 > 2, l? CVD .P4/, and then as P4 S D, l is one of the three lines j j D Ä 1 \ li from Section 4. Thus (2) holds in this case. h Suppose NH .P / B for some h HD. Arguing as in the ﬁrst few para- D Ä D 2 graphs of the proof of 9.2, there is R P such that R R1 R2 R3 with i h C Ä h C D C C C R R Li Q8. Then by 10.3, U U , contrary to our hypothesis that C D C \ Š D NH .P / K. Thus we may assume NH .P / is contained in no HD-conjugate D — D of BD. Suppose Z P0 P is normal in NH .P / with ˆ.P0/ 1. By the previous 6D Ä D D paragraph, m2.P0/ > 2, and so m2.P0/ 3 or 4. If m2.P0/ 4 then by 7.11, 7.12, D D and as NHD .P0/ acts on no 4-subgroup of P0, AutHD .P0/ CGL.P0/.z/. Thus D 7 P O2.NH .P0// by 8.2(b), and P E64, so that ƒ.P / .D/ is of type 1 , and D D Š 2 we obtain a contradiction from our treatment of this case. Therefore m2.P0/ 3. D Hence by 7.8, P0 is BD-conjugate to E or E , and in the latter case EE E16 0 0 Š is Sylow in CHD .E0/. In the latter case a P -invariant Sylow 2-subgroup P1 of

CHD .P0/ satisﬁes ŒP; P1 P0 CP .P1/; so P1 P by 2.2, and we obtain a Ä Ä Ä g contradiction from our treatment of the case m2.P0/ 4. Thus P0 E for some g g DE D g g BD. Let S1 S D. Then ŒS1 ; P S1 P NHD .P /; so S1 P by 2 D g 1 \ Ä \ Ä 2.2. Hence S1 S by 4.9(c), and so P0 E. Similarly CS .E/ P , and then D 1 D D Ä (3) holds by 8.2(b). We have reduced to the case where Z is the largest elementary abelian 2-subgroup of P normal in NHD .P /. It follows that Q is of symplectic type and hence (cf.[Asc86, 23.9]) Q Q0 Z.Q/ with Q0 extraspecial and Z.Q/ cyclic of order D 2 or 4. Further we may choose Q0 with U Q0. As Inn.Q0/ C .Q /, Ä D Aut.Q0/ 0 P Q0CP .Q0/. D As ŒE; P ŒE; S U P , E acts on P , and similarly E acts on Q0. Thus Ä Ä Ä ŒE; CP .Q0/ CU .Q0/ Z and if E does not centralize P then E does not Ä D centralize Q. Then as E acts on Q0, E induces a transvection on the orthogonal space Q0 with center U . This is impossible as U is a singular point in Q0. Thus E centralizes P , so that E P by 2.2, and hence E Z.P /; so replacing Q N .P / Ä Ä by Q E HD , we may assume E Q. h i Ä 958 MICHAEL ASCHBACHER and ANDREW CHERMAK

Finally let ‚ ƒ.V / be of type 17 such that each member of ƒF is a sum 2 of points in ‚, and let R0 O2.NH .‚//. Thus Q centralizes ‚, Q R0, and D Ä we view R as the core of the permutation module for NH .‚/=R0 S7 on ‚. 0 Š Thus we identify f Q with the points of ‚ inverted by f . Observe ƒF.E/ 2 D l0; l1; l2; l3 , where l0 CV .E/ is a point. As Z.Q/ is cyclic, for each e E Z f g D 2 there is f Q with Œe; f z, and hence f e is odd. It follows that for at 2 D j \ j least two i 1; 2; 3 , the eigenspaces of f on li are 1-dimensional. Hence ƒ is 2 f g of type 152 or 17, cases we have already handled. This completes the proof of (a). Set X CH .P /. Since Z ZP by (a), we conclude from 8.3(a) that if D D X P then X CH .P /, so that (b) holds. Thus to prove (b), it suffices to show Ä D X P . If P CSD .E/ then A P , and then X P since CH .A/ A. On the Ä D Ä F Ä 7D other hand, if P satisfies (a)(1) then X CH .ƒ /. If ƒ is of type 1 then P has F Ä 5 index at most 2 in O2.NH .ƒ // and X Z. If ƒ is of type 1 2 then P P0 t F D D h i where P0 O2.NH .ƒ / and where t inverts every element of CH .P0/. Thus D D X P in all cases, establishing (b). Ä Let Q Ꮾ.P /, R1 our candidate for ˇ.P / in (c), R2 NR1 .Q/, and Q2 2 c D D NQ.R1/. Then Q2 acts on R2. Observe that P ᏲH and ZP Z by (a) and c 2 D (b), so P Ᏺ by 8.3(a). Thus the same holds for any overgroup Q0 of P in S; so c 2 Q Ᏺ and NG.Q / H by 8.3(a). In particular Q2 and NQ.R2/ are contained 0 2 0 Ä in H . We claim that NQ.R2/ acts on R1, so that Q2 NQ.R2/. In case (a)(1), from D the treatment of that case above, each of R1 and P , and hence also R2 has ƒ as its set of weight spaces, so Q2 acts on O2.NH .ƒ// R1. Similarly in case (a)(2), D each of R1 and P , and hence also R2 has the same set of 1-dimensional weight spaces, so again Q2 acts on R3 O2.NH .ƒ//, and hence also on R1 R3 t as D D h i t P acts on Q2. Finally in case (3), Q2 acts on T2 by 4.9(c), so that Q2 acts on 2 S R2 R1. This completes the proof of the claim. 1 D Next from the structure of NH .R1/ and NHD .P /, NHD .P / acts on no nontrivial 2-subgroup of NH .R1/=R1; so Q2 R1, and hence also Q2 R2. Therefore Ä Ä as Q is finite, Q Q2 R1, completing the proof of the lemma. D Ä LEMMA 10.8. Let D G0 and H H0=Z. Write ᏾.H0/ for the set of D 0 D P0 S such that Z P and P is the radical of some proper parabolic of Ä Ä H Sp .2/ containing S . 0 Š 6 0 rc (1) If P Ᏸ and ND.P / H then P ᏾.H0/. 2 Ä 2 rc (2) ᏾.H0/ Ᏸ . Â (3) B CH .a /, where a is the involution of type a2 in Z.S /. For each 0 D 0 0 P ᏾.H0/ with O2.B0/ P , ND.P / K0. 2 Ä Ä rc (4) ND.P / H0 and P ᏲS .H 0 / for each P ᏾.H0/ with O2.B0/ P . Ä 2 0 2 — 2-LOCALFINITEGROUPS 959

(5) NH0 .A/ is the maximal parabolic isomorphic to L3.2/=E64.

(6) NH0 .E/ is the minimal parabolic not contained in B0. Proof. Assume the hypothesis of (1). Then by 10.1, P contains each 2-element in CH .P /, and O2.NH .P /=P CH .P // 1. Hence as H Sp .2/, P ᏾.H0/ 0 0 0 D 0 Š 6 2 by the Borel-Tits Theorem. Thus (1) holds. Next U is the unique normal 4-subgroup of S0, so that U is generated by the unique involution in Z.S0/ lifting to an involution of H0. As H0 is the covering group of Sp .2/ and B0 NH .U /, it follows that the ﬁrst statement in (3) holds. 6 D 0 By 10.3, O2.B0/ is weakly closed in S0 with respect to D and so the remaining statement in (3) follows. Next, by 7.10, AutD.A/ L4.2/ and so NH .A/=A L3.2/. This implies (5). Š 0 Š Also NH .A/ contains two minimal parabolics: NB .A/ and Y , where Y0 0 0 0 D NH .A/ ND.E/ NH .E/. Thus (6) holds. 0 \ D 0 Let P ᏾.H0/ with O2.B0/ P . Then NH .P / contains the minimal 2 — 0 parabolic Y0; so NH0 .P / is Y0, NH0 .A/, or the preimage of the third maximal parabolic of H0, isomorphic to S6=E32. In the ﬁrst two cases Z ZP from the 2 D action of ND.A/ on A, and in the third case P Z4 Q and again Z ZP . Thus Š 8 D ND.P / ND.Z/. As S0 ND.P / and S0 Syl2.D/, P contains each element Ä Ä c 2 in CD.P / by 5.8 and so P Ᏸ by 8.2(a). Then ND.P / H0 by 8.3(a). Also 2 c Ä S0 S 0 acts on P and as CS0 .P / P , P Ᏺ .H 0 /. Similarly ND.P / S 0 D Ä r 2 Ä H 0 and P O2.ND.P //, so that P ᏲS .H 0 / which completes the proof D 2 0 of (4). Then (2) follows from (3) and (4).

PROPOSITION 10.9. Let P Ᏸrc. Then 2 (a) There is no nontrivial ND.P /-invariant subgroup of CG.P / of odd order. (b) P Ᏺc. 2 (c) One of the following holds. (1) P AD, 2 (2) P CSD .E/, or D rc (3) P Ᏸ for some Y H;K , and NG.P / Y . 2 Y 2 f g Ä Conversely, every subgroup P of SD which satisﬁes one of the conditions in (c) is in Ᏸrc.

Proof. Suppose ﬁrst that ZP 2. Then ND.P / H by 8.3(a), so that rc j j D Ä P ᏰH by 10.1. Assume that ND.P / B. Then 10.7(b) and 10.8(4) say that 2 — c CG.P / P , so that (a) and (b) hold and P Ᏺ . Then (3)(c) follows from 8.3(a). Ä 2 H Suppose next that either ZP 2 and ND.P / B, or ZP 4. In the j j D Ä j j D latter case, ZP U and ND.P / K by 10.4 and 8.3(b); certainly ND.P / K D Ä rc Ä in the former case. Thus in any event, ND.P / K, so that P Ᏸ by 10.1. Then Ä 2 K 960 MICHAEL ASCHBACHER and ANDREW CHERMAK

c CK .P / P by 10.2(a), and P ᏲK by 10.2(e). Then as U P , CH .P / Ä 2 c Ä Ä CH .U / K; so CH .P / CK .P / P , and hence P Ᏺ . Now it follows from Ä Ä Ä 2 H parts (a) and (b) of 8.3 that CG.P / P and (a)-(c) hold. Ä c If ZP 8 then P CS .E/ by 9.5. Then A P , and so P Ᏺ by 8.3(d). j j D D D Ä 2 Also, it follows from 7.13 that (a) holds, and we have (a) through (c). D If ZP > 8 then P A by 9.5. Then 8.3(d) yields (b), and 7.10 yields (a). j j 2 Thus, we are reduced to establishing the final statement in the theorem. D rc If P A or P CSD .E/ then P Ᏸ , and ND.P / H K, by 7.10 2 D rc 2 — [ and 7.13. Finally assume P ᏰY for some Y H;K with NG.P / Y . Then 2 2 f c g cÄ 10.2, 10.7, and 10.8 yield ZP Z;U and P Ᏺ , so that P Ᏺ by 8.2(a), 2 f g 2 Y 2 and P Ᏸrc by 10.1. 2 rc PROPOSITION 10.10. Let P Ᏸ , let Ꮾ.P / be the set of finite ND.P /-in- 2 variant 2-subgroups of G, and set ˇ.P / Ꮾ.P / . Then ˇ.P / S, and one of D h i Ä the following holds. (1) P AD and ˇ.P / P . 2 D (2) P CS .E/ and ˇ.P / CS .E/. D D D rc rc (3) There exists Y H;K such that P Ᏸ , ND.P / Y , ˇ.P / ᏲS .Y / , 2 f g 2 Y Ä 2 and NG.ˇ.P // Y . Ä Proof. Set N ND.P /, let R be a finite N -invariant 2-subgroup of G con- D taining P , and set R0 NR.P /. D D If P A then N NG.P / by 7.10; so R0 O2.N / P , and (1) holds. 2 D Ä D Suppose that P CSD .E/. Define Ri by Ri NR.Ri 1/ for i 1, and set D D Mi NG.Ri /. Set M NG.CS .E//. By 7.13, CM .E/ X O2.M /, where X D D D is a free normal subgroup of M , M=CM .E/ GL.3; 2/, and M .X CS .E//N . Š D As R0 is N -invariant, we get R0 CS .E/, and R0 T w0 for some k 2. By Ä D kh i 4.9(a) and 7.13, NG.Ri / M . Then a straightforward induction argument yields Ä Ri CS .E/ for all i, and thus ˇ.P / CS .E/. Since CS .E/ is the union of finite Ä Ä N -invariant subgroups, we conclude that ˇ.P / CS .E/, and (2) holds. D By 10.9(c) we are reduced to the case where there exists Y H;K with rc 2 f g P Ᏸ and NG.P / Y . Suppose further that ˇ.P / S, and set M NG.ˇ.P //. 2 Y Ä Ä D As P ˇ.P /, and since P Ᏺc by 10.9(b), we conclude ˇ.P / Ᏺc and Z Ä 2 2 ˇ.P / Ä ZP . If N H and N K then 10.7 yields ZP Z, and so Z Z, and then Ä — D ˇ.P / D M H by 8.3(a). On the other hand, suppose that N K. Then U is the unique Ä Ä 4-group in K which centralizes P=Z, by 10.2(f). Since we are assuming that ˇ.P / S, it follows that also U is the unique 4-group in ˇ.P / which centralizes Ä ˇ.P /=Z, and hence U is the unique 4-group in ˇ.P / which is normal in M . Then 8.3 says M is contained in H or K, and since NH .U / K we get M K. Thus, Ä Ä M Y . Ä 2-LOCALFINITEGROUPS 961

We now show that ˇ.P / S. If N H and N K, this follows from Ä Ä — 10.7(c) and 10.8(4). Suppose N K, and set R1 R K. Then R0 R1 S by Ä c D \ c Ä Ä 10.2(e). Also, 10.2(e) says that P Ᏺ , and hence R1 Ᏺ . As U is the unique 2 K 2 K normal 4-subgroup of R1, we have NR.R1/ NR.U /. Since ZR ZP U , Ä 1 Ä Ä it follows from 8.3 that NR.R1/ is contained in H or K, and since NH .U / K Ä we get NR.R1/ K. Then R R1, and so R S for each R Ꮾ.P /. That is, Ä D Ä 2 ˇ.P / S. Ä In order to complete the proof of (3), it remains to show that ˇ.P / Ᏺr . Let 2 Y Q be the preimage in M of O2.AutM .ˇ.P ///. As M Y we have .ˇ.P // Ä D O.CG.ˇ.P /// 1, by 8.4 and 10.9(a). Thus Q is a 2-group, Q Ꮾ.P /, and D 2 Q ˇ.P / Ᏺr as required. D 2 11. Theorem B and embeddings

We begin the section with a reﬁnement of Theorem 5.8.

THEOREM 11.1. Let G be the group Co , identify S with a Sylow 2-sub- x0 3 0 group of G0 as in Theorem 5.8, and set Ᏺ0 ᏲS .G0/. Let G0 G0 be the x x D 0 x W ! x canonical homomorphism g g induced by the inclusion maps of H0 and K0 W 7! N into G , and let be the 2-local ﬁnite group .S ; ; c / associated with G as x0 ᏳN 0 0 Ᏺx 0 ᏸN 0 x0 in Proposition 2.7. Then there is an isomorphism (in the sense of 2.10)

.˛; ˇ/ Ᏻ0 Ᏻ0; W ! N in which ˛ Ᏺ0 Ᏺ0 is the identity map on objects and, on morphisms, ˛ cg cg ; W ! N W 7! N and where ˇ ᏸrc ᏸrc is the identity map on objects, and W 0 ! N 0 ˇP;Q Morᏸ0 .P; Q/ Morᏸ .P; Q/ W ! N0 is given by 0.P /g g for P and Q in ᏸ0 and g NG .P; Q/. 7! N 2 0 Proof. Recall from the discussion following 8.4 that there is a surjection G0 G0, where may be regarded as the “identity map” on H0 K0, and W ! x [ ker.A/ ker./ M . As ker.A/ ker./ M ker./, 8.7 yields X0 ker./. D j D c j Ä Â Thus 0.P / ker./ for any P Ᏺ0. Then since O.CG .P // 1, the lemma Ä 2 x0 D follows from 5.8 and the last paragraph of 2.13. We may now establish Theorem B. Part (1) of Theorem B follows from the construction of G0 in 5.8, and part (3) follows from 10.1. Part (2a) holds since H0 and K0 are ﬁnite while the nontrivial elements of X are torsion-free. Thus it only remains to verify part (2b) of Theorem B. As we just saw during the proof of 11.1, there is a surjective homomorphism G0 G0 Co3 induced by the inclusion maps of H0 and K0 into G0. Set W ! x D x M0 NG .A/, and denote by C the colimit of the subgroup amalgam deﬁned by WD x0 z the inclusion maps among the intersections of the members of ᏹ H0;K0;M0 . WD f g 962 MICHAEL ASCHBACHER and ANDREW CHERMAK

Then there is a surjective homomorphism ˇ C G0 and a surjection ı G0 C W z ! x W ! z with ıˇ . Set M NG.A/. As seen in the proof of 7.5, M G0 and A M D D G0 Ä W ! M0 is the restriction of to M . Thus ker.ı/ ker.A/ , and then part (2b) of D h i Theorem B follows from 8.7.

i 1 mi For any i > 0, set mi 2 , i , Gi G , Si S Gi , and D D 0 D i D \ Ᏺi ᏲS .Gi /. Write ƒ for the poset ގ, under the usual total ordering. There is D i then a directed system of embeddings of fusion systems

F .i;j Ᏺi Ᏺj /i j ƒ; D W ! Ä 2 where i;j is an inclusion map, and Ᏺ0 ᏲS .G0/ is the fusion system of Co3. D 0 Let i Ᏺi Ᏺ be the inclusion, and observe that j i;j i for i j . For W ! rc ı D Ä i; j ƒ with i j and P Ᏺ , write Ꮾi .P / for the set of finite NG .P /-invariant 2 Ä 2 i i 2-subgroups of G, and set ˇi .P / Ꮾi .P / and ˇi;j .P / ˇi .P / Gj . D h i D \ LEMMA 11.2. Let i; j ƒ with i j , and let P Ᏺrc. Then 2 Ä 2 i rc (a) ˇi .P / Ᏺ , 2 rc (b) ˇi;j .P / Ᏺ and 2 j (c) ˇi;i .P / P . D Proof. Set D Gi , Ᏸ Ᏺi , N ND.P /, and ˇ ˇi . Also set Q ˇ.P /, D D D D D c D Gj , Ᏸ Ᏺj , P ˇi;j .P / Q D, and N ND.P/. Since P Ᏺ z D z D z D D \ z z D z z 2 by 10.9(b), it follows from 8.2(a) that P Ᏸc and that Q Ᏺc. If P AD or z 2 z 2 2 P C .P / then Q P or C .E/ by 10.10. Then 10.5 says Q Ᏺrc, P P SD S z D Drc 2 D or CS .E/, and P Ᏸ . That is, the lemma holds in these two cases. j z 2 z By 10.9 we may assume that ND.P / Y for some Y H;K , and that rc Ä 2 f g P ᏰY . Then (a) is given by 10.9 and 10.10. In order to complete the proof of 2 r (b), it remains to show that P Ᏺj . Let R be the pre-image in Y of O2.AutN .P //. z 2 z z As in the final lines of the proof of 10.10, we find that j .P/ 1; hence R is a z D 2-group, and since R is N -invariant we get R Q. Then R P and P Ᏺr as Ä D z z 2 j required. Finally suppose i j . Here P P and N .P / N .P/, and since P Ᏺr z Gi Gi z i D Ä Ä 2 c we get AutP .P / Op.AutGi .P // Inn.P /. Then P PCP .P / P as P Ᏺi , z Ä D z D z D 2 and (c) holds.

LEMMA 11.3. Let , 0, and i i be the signalizer functors on Ᏺ, Ᏺ0, D rc and Ᏺi , respectively, given by 8.8. Let i; j ƒ with i j , and let P Ᏺ . Then 2 Ä 2 i i .P / .ˇi .P // Gi j .ˇi;j .P // Gi : D \ D \ Proof. Recall that by deﬁnition,

.P / O.CG.P //CX.P / and i .P / O.CGi .P //CX Gi .P /: D D \ 2-LOCALFINITEGROUPS 963

But O.CG.P // O.CGi .P // 1 by 10.9(a), so that i .P / .P / Gi . If G D D D \ P A i then ˇi .P / ˇi;j .P / P , and the lemma follows from the preceding 2 D D observation. If P CSi .E/ then ˇ.P / CS .E/ and ˇi;j .P / CSj .P / by D D D rc 10.10, and the lemma then follows from 7.13(d). If P ᏲSi .Y Gi / for some rc 2 \ Y H;K then also ˇ.P / ᏲS .Y / by 10.10, and i .P / .ˇ.P // 1. The 2 f g 2 D D lemma holds trivially in this case, and there are no more cases to consider, by 10.9.

rc rc LEMMA 11.4. Ᏺ ˇi .P / P Ᏺ ; i 0 . D f j 2 i g rc rc Proof. Let Ꮾ ˇi .P / P Ᏺi ; i 0 . Then Ꮾ Ᏺ by 11.2(a). Let rc GD f W 2 g Â P Ᏺ . If P A then P Gi for some i, and P ˇi .P/. If P CS .E/ then z 2 z 2 z Ä z D rc z z D P ˇ0.CS .E// by 10.10. Suppose that P ᏲS .H / , such that NG.P/ H z D 0 z 2 z Ä and NG.P/ K. The possibilities for NG.P/ are listed in 10.7(a), and we shall z — z deal with them case by case. In case (1) and (2) of 10.7(a), NG.P/ NH .ƒ/ for some ƒ ƒ.V; /, with 7 5 z D 2 1 or 1 ; 2. Let Q 2.O2.NH .ƒ///. Then Q is finite; so Q Gi for some D z D z z Ä i > 0. Further ƒ ƒ.Q/ is the set of weight spaces of Q on V , and as Q Gi , F D z z z Ä7 ƒ ‚ , where ‚ is the set of weight spaces for Q on Vi VF . If 1 , let D z D D P Q P , while if 15; 2, let P O .N .‚// t , where t is as in 10.7(a). z z 2 Hi D D rc D D h i Then P Ᏺ and P ˇi .P/ by 10.7(c). 2 i z D z It remains to consider case (3), where P O2.NH .E//. We take P z D D O .N .E//, obtaining ˇ .P / P ; again from 10.7(c). 2 H G1 1 z \ D rc By 10.9(c) we may now assume that NG.P / K, so that P ᏲS .K/ . By 0 Ä z 2 10.2 we have P B P1P2P3, where P P L is either a quaternion group z \ D z z z zk D z \ k or a Sylow 2-subgroup of Lk. Since K is locally finite, and since 10.2 shows that NG.P /=P is finite, we may choose i sufficiently large so that P P Gi has z z WD z \ the following properties:

(1) For all k for which P Syl .L / we have P L 16, and for all other zk 2 2 k j \ kj k we have P L P . \ k D zk (2) NG.P/ NK .P/ NG .P /P . z D z D i z Set N N .P / and N N .P/. It follows from (1) and (2), and from the K Gi z K z D \ D rc ﬁnal statement in 10.2, that P ᏲS .K Gi / and that N N . As N N , 2 i \ Ä z Ä z P ˇi .P /, so that it remains to show ˇi .P / P . z Ä Ä z Let P R Ꮾ.P / and set R0 R K. Then R S by 10.2(e). As Ä 2 D \ Ä N N .P /P , R ; P is an N -invariant 2-group, so since P Ᏺr we conclude z Gi z 0 z z z D h i c c 2 that R0 P . By 10.9(b), P Ᏺ , so R0 Ᏺ . By 10.2(f), U is the unique normal Ä z 2 2 fours group in R0, and since ZR ZP it follows from 8.3 that NG.R0/ K. 0 Ä Ä Then R0 R, and the proof is complete. D 964 MICHAEL ASCHBACHER and ANDREW CHERMAK

12. Limits, and Theorem C

Our aim in this section is to introduce limits of directed systems of p-local groups, and to obtain Theorem C as an application. See for example[Jac80, 2.5] for a discussion of directed systems and their limits. Theorem D will then be obtained as a corollary to[LO02, Th. 4.5]. Let .ƒ; / be a directed set. For ƒ, write ƒ./ for ƒ .A Ä 2 f 2 j Ä g subset of ƒ is closed if ƒ./ for all . In particular, each of the sets Â 2 ƒ./ is closed. Recall the notion of “pre-local group” from 2.4. Fix a prime p, and assume that for each ƒ we are given a pre-local group Ᏻ .S ; Ᏺ ; ᏸ /, where each 2 D S is a p-group. We write Ᏹ for Obj.ᏸ/, and given subgroups P and Q of S we write Hom.P; Q/ for HomᏲ .P; Q/, and Mor.P; Q/ for Morᏸ .P; Q/ if P and Q are in Ᏹ . Assume that for all pairs .; / with in ƒ, we are given Ä an embedding . ; ˇ / Ᏻ Ᏻ; ; ; W ! of pre-local groups (cf. 2.10). We may write simply ˇ; for the pair .;; ˇ;/. We assume further that

G .ˇ; Ᏻ Ᏻ/ ƒ D W ! Ä 2 is a directed system of pre-local groups. That is, we have ˇ; ˇ ˇ when- ı ; D ; ever in ƒ, and each ˇ is the “identity morphism” on Ᏻ , consisting Ä Ä ; of a pair of identity functors. Let S S be the limit of the ƒ-directed system of p-groups WD 1 .; S S/ : W ! Ä By[Jac80, 2.8], the limit exists and is a group, and there are monomorphisms S S, compatible with the monomorphisms . We may then view all of W ! ; these monomorphisms as ordinary inclusion maps, in order to obtain the following result: S LEMMA 12.1. S ƒ S, and S is a p-group. D 2 Let P and Q be subgroups of S. For any Inj.P; Q/ and any ƒ, deﬁne 2 2 to be the restriction of to P S , and set \ ƒ ƒ Hom .P S ;Q S / : D f 2 j 2 \ \ g Deﬁne Hom .P; Q/ to be the set of all Inj.P; Q/ such that ƒ contains a 1 2 nonempty closed subset of ƒ. As ƒ is a directed set, each pair of elements of ƒ has an upper bound, and it follows that the composition of Hom .P; Q/ 2 1 with Hom .Q; R/ is in Hom .P; R/. Thus we may form the category Ᏺ , 2 1 1 1 whose objects are the subgroups of S, and whose morphisms are given by the sets 2-LOCALFINITEGROUPS 965

Hom .P; Q/. Moreover, if P and Q are subgroups of S and Hom.P; Q/, 1 2 then Hom .P; Q/. This natural inclusion of sets may be denoted 2 1 Hom.P; Q/ Hom .P; Q/: W ! 1 Allowing P and Q to vary over the set of all subgroups of S, is then an embedding of fusion systems (cf. 2.9). We record this in the following lemma, whose proof is straightforward and left to the reader.

LEMMA 12.2. Ᏺ is a fusion system on S and Ᏺ Ᏺ is an embedding 1 W ! 1 of fusion systems.

Recall that Ᏹ is the set of objects of the linking system ᏸ . For P Ᏹ there 2 is then a subgroup ˇ.P / of S defined by [ ˇ .P / ˇ .P /: D ; ƒ./ 2 Note that for any ƒ./, we have ˇ .P / ˇ.ˇ .P //. Set 2 D ; Ᏹ ˇ.P / ƒ; P Ᏹ : 1 D f j 2 2 g Definition 12.3. Let Ꮿ be a category and .C ; c / a directed D ;W Ä system in Ꮿ. A family † . C C ƒ/ of morphisms in Ꮿ is said to be D W ! W 2 compatible with if for all in ƒ, c . Ä D ı ; Now specialize to the case where Ꮿ is the category of sets. A compatible family † is said to be nearly injective on if [ C .C /; D ƒ 2 and for each ƒ, whenever a; b C with .a/ .b/, then there exists 2 2 D ƒ./ with ˇ .a/ ˇ .b/. For example if C C is injective for 2 ; D ; W ! each ƒ, then † is nearly injective. 2 Now take .G/ to be .Ᏹ ; ˇ /, and observe is a directed D ;W Ä system in the category of sets, if we regard ˇ; as the function from Ᏹ to Ᏹ defined by ˇ P ˇ .P /. We say that is nearly injective if †./ is nearly ;W ! ; injective on .G/, where †./ .ˇ Ᏹ Ob.ᏸ / ƒ/. Similarly define D W ! 1 W 2 G to be nearly injective if .G/ is nearly injective.

LEMMA 12.4. Assume † . Ᏹ C ƒ/ is nearly injective on .G/ D W ! W 2 and .ı Ᏹ D ƒ/ is a family of functions compatible with .G/. D W ! W 2 (1) Suppose ; ƒ, P Ᏹ , and Q Ᏹ such that ˇ .P / c ˇ.Q/. Then 2 2 2 D D there exists ƒ./ ƒ./ such that R ˇ .P / ˇ .Q/. Moreover 2 \ D ; D ; ˇ.R/ c. D 966 MICHAEL ASCHBACHER and ANDREW CHERMAK

(2) ı C D is a well-defined function, where ı.c/ ı .P / for ƒ and W ! D 2 P Ᏹ such that .P / c. 2 D (3) If is also nearly injective on .G/ then ı is a bijection. Proof. As ƒ is directed there is ƒ./ ƒ./. Set P ˇ .P / and 2 \ 0 D ; Q ˇ;.Q/. Then 0 D ˇ.P / ˇ .P / P ˇ.Q/ ˇ.Q /: 0 D D O D D 0 As † is nearly injective, there exists with ˇ;.P / R ˇ;.Q /. Then 0 D D 0 ˇ .P / ˇ;.ˇ .P // R, and similarly ˇ;.Q/ R. Since ˇ.R/ ; D ; D D D ˇ.ˇ .P // ˇ .P / c, (1) holds. ; D D To see that ı is well defined, suppose that ı .P / ı.Q/ for some and some D Q Ᏹ. Choose as in (1). Then ı .P / ı.ˇ .P // ı.ˇ;.Q// ı.Q/ 2 D ; D D and so is well defined, establishing (2). Assume the hypothesis of (3). Then by (2) applied to , the map ˛ .P / W 7! ı.P / is a well-defined function from D to C , and visibly ˛ is an inverse for ı; so (3) holds. We now assume that G is nearly injective, and define a category ᏸ whose 1 set of objects is Ᏹ , and which will be shown to be the direct limit of the directed 1 system .ˇ; ᏸ ᏸ/ of categories. W ! Ä Let P; Q Ᏹ . Then there exist ƒ, and P;Q Ᏹ, such that P ˇ.P / y y 2 1 2 2 y D and Q ˇ .Q/. In the following discussion, leading up to 12.5, we take , P , y D and Q to be fixed. Define Mor .P; Q/ to be the set of equivalence classes Œf of y y mappings 1 [ f Mor.ˇ .P /; ˇ .Q// W f ! ; ; f 2 where

(i) f is a nonempty closed subset of ƒ./,

(ii) f ./ Mor.ˇ .P /; ˇ .Q// for all , 2 ; ; 2 f (iii) ˇ;.f .// f ./ whenever , D Ä and where two such mappings f and f 0 are defined to be equivalent if they agree on a nonempty closed set. There is then a well-defined composition Mor .Q; R/ Mor .P; Q/ Mor .P; R/ 1 y y 1 y y ! 1 y y for any R Ᏹ . Namely, one may assume chosen so that also R ˇ.R/ for y 2 1 y D some R Ᏹ. Then, for any Œg Mor .ˇ.Q/; ˇ.R//, define Œg Œf to be 2 2 1 the equivalence class of the mapping g f , where .g f /./ g./f ./. This D defines the category ᏸ . Notice, using 12.4 and increasing if necessary, that 1 2-LOCALFINITEGROUPS 967 these definitions are independent of the choice of P , Q, and R. Thus ᏸ is well defined. 1 For any Mor.P; Q/, there is an element Œf of Mor .P; Q/, defined 2 1 y y by f ./ ˇ;. / for any ƒ./. The map ˇ Ᏹ Ᏹ extends to a D 2 W ! 1 functor ˇ ᏸ ᏸ , where ˇ is defined on morphisms by ˇ. / Œf . W ! 1 D LEMMA 12.5. .ˇ ᏸ ᏸ / ƒ is the direct limit of the nearly injective W ! 1 2 directed system L .ˇ; ᏸ ᏸ/ of categories. D W ! Ä

Proof. Let . ᏸ Ꮿ/ ƒ be a family of functors compatible with the W ! 2 directed system L of categories. By 12.4.2, we can define a function Ᏹ W 1 ! Obj.Ꮿ/ by .P/ .P / for P Ᏹ , where P Ᏹ and P ˇ.P /. y D O 2 1 2 y D A similar argument allows us to define on morphisms: Let Œf Mor .P;Q/ 2 1 y y and pick a representative f of Œf . We may choose so that P ˇ .P / and y D Q ˇ .Q/ for some P;Q Ᏹ , and so that . Setting f ./, we have y D 2 2 f D Mor .P; Q/ and ˇ . / Œf . We now “define” .Œf / to be . /. As in 2 D the preceding paragraph, if . / ./ for some and some ᏸ-morphism , D we may replace and by their images under the maps ˇ; and ˇ;, and reduce to the case where . Then ˇ . / ˇ ./ Œf , whence f ./ , D D D D D and is well defined on morphisms. It is now straightforward to check that is a functor, and is then visibly the unique functor such that ˇ for all ƒ. D ı 2 There is a functor ᏸ Ᏺ , defined as follows. As a map from 1W 1 ! 1 Obj.ᏸ / to Obj.Ᏺ / we take to induce the identity map on Ᏹ . As a map of morphisms,1 define1 .Œf / P1 Q by 1 1 W y ! y .Œf / x .f .//.x/ 1 W 7! for any such that x ˇ .P /. The definition is independent of , and 2 f 2 ; the verification that .Œf / is in Hom .P; Q/ is straightforward, as is the veri- y y fication of functoriality.1 1 Next, define a family of monomorphisms of groups ı ı .ı P Aut .P // ; P y ᏸ y P Ᏹ D 1 D y W ! 1 y2 1 as follows. Let P Ᏹ with P ˇ .P /, and let x P . Then x ˇ .P / P 2 y D 2 y 2 ; D for some ƒ./. Define ıP .x/ to be Œgx, where gx./ ˇ;.ı.x// for 2 y D ƒ./. The verification that each ıP .x/ is in Autᏸ .P // and that ıP is a 2 y 1 y y monomorphism reduces to the corresponding facts concerning the family ı of monomorphisms associated with Ᏻ. Henceforth ᏸ will denote the triple consisting of the category ᏸ , the functor , and the collection1 ı of monomorphisms. 1 1 1 968 MICHAEL ASCHBACHER and ANDREW CHERMAK

LEMMA 12.6. Let P; Q Ᏹ . y y 2 1 1 (a) P acts semiregularly on Mor .P; Q/ via x Œf ıP .x / Œf . y 1 y y W 7! y (b) The orbits of Z.P/ on Mor .P; Q/ are the fibers of . y 1 y y 1 Proof. As ıP P Autᏸ .P/ is a monomorphism, the action in (1) defines O W O ! 1 O an injective representation of P on Mor .P; Q/. Let Œf Mor .P; Q/, ƒ, O 1 O O 2 1 y y 2 and P;Q Ᏹ with P ˇ .P / and Q ˇ .Q/. Let x P , and suppose that Œf 2 y D y D 2 y is a fixed point for ı .x/. Without loss of generality, we may assume that x P P 2 and that . Sety f ./ and . /. Then ı .x/ , and hence 2 f D D ;P D by conditions (B) and (C) in 2.4, .ı .x/ / cx. Thus as is injective, D ;P D it follows that x Z.P /, and then 2.4(A) yields x 1. Thus (a) holds. 2 D We next prove (b). Let Œf ; Œh Mor .P; Q/. We may assume that ƒ./ 2 1 y y D f h. Then Œf Œh if and only if for each ƒ./, .f .// D 1 D 1 2 D .h.//. As Ᏻ is a pre-local group, this holds if and only if there exists z 2 Z.P/ with h./ ı;P .z/ f ./. (cf. 2.4(A)). But D ı;P .z/ f ./ h./ ˇ .h.// ˇ .ı .z / f .// D D ; D ; ;P ˇ .ı .z // ˇ .f .// ı;P .z / f ./: D ; ;P ; D By (1) applied in Ᏻ this holds if and only if z z . Therefore we have shown D that Œf Œh if and only if for all ƒ./, z z Z.P/ and h./ 1 D 1 2 D 2 D ı;P f ./. Since P is the union of the groups P for ƒ./, we conclude y 2 that Œf Œh if and only if z z Z.P/ and Œh ıP .z/ Œf Œf z. 1 D 1 D 2 O D D Thus (b) holds. O The fusion system Ᏺ is in general “too large”, in various ways. In particular, 1 need not map morphism sets in ᏸ onto homomorphism sets in Ᏺ , and as 1 1 1 a result these homomorphism sets are in general too large for Ᏺ to serve as the fusion system in the limit of the direct system G. 1 The smallest fusion system on S containing .Mor .P;b Q//b for all Pb 1 1 1 and Qb in Ᏹ will be denoted Im. /. If each Ᏻ is a p-local finite group then 1 1 Im. / will contain .Ᏺ/ for all , as a consequence of Alperin’s Fusion The- orem.1 More generally, set

ᏲG Im. /; .Ᏺ/ ƒ : D h 1 j 2 i Set Ᏻ .S ; Ᏺ ; ᏸ / and ᏳG .S ; ᏲG; ᏸ /: 1 D 1 1 1 D 1 1 Let Hom .P; Q/, with P and Q in Ᏹ . By 2.4(A) there is Mor .P; Q/ 2 2 such that . / . Let in ƒ. Condition (MG2) in 2.10 yields: D Ä Ä (12.7) .ˇ . // P .ˇ;.ˇ . /// P .ˇ . //: ; j D ; j D ; 2-LOCALFINITEGROUPS 969

Deﬁnition 12.8. We say that the element of ƒ./ is -good provided that, for all ƒ./, .ˇ . // is the unique member of Hom.P;Q/ which 2 ; restricts to .ˇ;. // on P. Denote by E the set of all -good elements of ƒ.

LEMMA 12.9. (a) Either of the following conditions implies that Ᏻ is a pre-local group. 1 (1) Mor .P; Q/ Hom .P; Q/ is a surjection, for all P; Q Ᏹ . 1W 1 y y ! 1 y y y y 2 1 (2) For every , and for every Ᏺ-morphism , E is nonempty.

(b) If Ᏻ is a pre-local group, then so is ᏳG. 1 Proof. In order to show that Ᏻ is a pre-local group, we must verify conditions 1 (A) through (C) in 2.4, and that Ᏹ Ᏺc . Under the hypothesis of (1), Condition 1 Â (A) is an immediate consequence of 12.6;1 we leave the remaining verifications in (1) to the reader. Assume next that Ᏻ is a pre-local group. Then 1 .Mor .P; Q// Hom .P; Q/; 1 1 O O D 1 O O and so HomᏲG .P; Q/ .Mor .P; Q//. The argument of the preceding para- y y D 1 1 y y graph then yields (b). Finally assume the hypothesis of (2). Let Hom .P; Q/. We may assume 2 1 y y ƒ, and thus the restriction of to P is in Hom.P;Q/ for all ƒ./. 2 1 2 Further, we may assume that Ᏹ . Choose . /, and consider the 2 2 map f f on ƒ./ defined before 12.5, such that Œf Hom .P; Q/ and D 2 1 O O ˇ . / Œf . Thus f ./ ˇ . /, and so .f .// .ˇ . // on P . D D ; D ; Thus as E , .f .// , so that .Œf / . Thus we have verified 2 D 1 D the hypothesis of (1). Therefore by (1), Ᏻ is a pre-local group, and the proof of (a) is complete. 1

LEMMA 12.10. Assume that G is nearly injective.

(a) If ᏳG is a pre-local group then ˇ . ; ˇ / Ᏻ ᏳG is an embedding D W ! of pre-local groups, and .ˇ Ᏻ ᏳG/ ƒ is the direct limit of G in the W ! 2 category of pre-local groups and embeddings. (b) Assume that there exists a pre-local group Ᏻ .S; Ᏺ; ᏸ/, and a family † of z D z z z embeddings .˛ ; / Ᏻ Ᏻ; ƒ; D W ! z 2 of pre-local groups compatible with G. Assume that the following conditions hold for all . S (i) S ƒ ˛.S/. z D 2 (ii) For each Ᏺ -morphism , E ¿. ¤ 970 MICHAEL ASCHBACHER and ANDREW CHERMAK

(iii) For all P Ᏹ , 2 [ .P / ˛.ˇ .P //: D ; Ä

(iv) For each P; Q Ᏹ and HomᏲ.P; Q/, there exist ƒ and P;Q Ᏹ z z 2 z z 2 z z z 2 2 such that .P / P , .Q/ Q, and such that for each ƒ./, D z D z 2

˛.ˇ;P/ ˛.Hom.ˇ;.P /; ˇ;.Q///: zj 2 (v) has the Alperin generation property with respect to (cf. 2.14). Ᏺz Ᏹz (vi) † is nearly injective on .G/.

Then ᏳG is a pre-local group, ᏳG is the limit of G, and Ᏻ ᏳG as pre-local z Š groups.

Proof. Assume that ᏳG is a pre-local group. We check that ˇ satisfies the conditions (MG1) through (MG3) in 2.10. We have (MG1) since, for any P Ᏹ , 2 [ P ˇ .P / ˇ .P / ˇ .P /: D ; Ä ; D ƒ./ 2 Let Mor.P; Q/. Then .ˇ. // P . / by definition of , and 2 1 j D 1 thus (MG2) holds. Condition (MG3) is the assertion that ˇ ı ı on ı ;P D ˇ.P / P , which holds by definition of ıP . Thus ˇ is a morphism of pre-local groups. Recall that ˇ . / Œf , where f ./ ˇ . / for all ƒ./. In particular D D ; 2 by (ii), ˇ is injective as a mapping from Mor.P; Q/ into ᏸ -morphisms, and 1 ˇ is therefore an embedding. Let satisfy the initial hypothesis in (b), but for the moment do not assume Ᏻz the conditions (i) through (vi) in (b). Define the functor ᏸ ᏸ as in the proof W 1 ! z of 12.5. That is, .P/ .P / and .Œf / .f .//, where ˇ .P / P and y D D D y ˇ .f .// Œf . Then is the unique functor satisfying ˇ for all ƒ. D D ı 2 Let ƒ. Define ˛0 S S by ˛0 S ˛. As each ˛ is injective, 2 W 1 ! z j D ˛0 is an injective homomorphism. For any pair of subgroups P;Q of S, and any Hom .P; Q/, define ˛./ to be the homomorphism 2 ᏲG 1 ˛ ˛0 P ˛0 Q˛0: 0 ı ı W ! Then ˛ is a morphism of fusion systems if and only if the following condition holds for all P;Q S : Ä 1 . / ˛./ HomᏲ.˛0.P /; ˛0.Q//. 2 z If P;Q S and Hom .P; Q/, then . / holds as ˛ is a morphism of fusion Ä 2 systems. If P;Q Ᏹ , then .ˇ.f // for some f Mor.P;Q/. Write 2 1 D 1 2 for the restriction of to P. Then on P: . .ˇ .f /// . .f // ˛ . .f // ˛ . // ˛ ˛ 1; z D z D D D ı ı 2-LOCALFINITEGROUPS 971 and as this holds for all in a closed subset of ƒ, we conclude that . .ˇ.f /// 1 z D ˛ ˛ ˛./, and we again obtain . /. Since, by definition, ᏲG is generated ı ı D by such morphisms, . / holds in general, and ˛ is a morphism of fusion systems. We leave it to the reader to check that .˛; / satisfies the axioms in Definition 2.10, and hence is a morphism of pre-local groups, yielding (a). Now assume all of the hypotheses of (b). Then by (b)(ii) and 12.9, ᏳG is pre- local group. Then (a) says that ᏳG is the limit of G, and supplies the morphism .˛; / Ᏻ Ᏻ described above. By (b)(i) and the definition of ˛, ˛0 S S D W 1 ! z W 1 ! z is an isomorphism of groups.

The key step in the proof of (b) is to show that each of the sets HomᏲ.P; Q/, z z z with P and Q in Ᏹ, lies in the image of ˛. So let P; Q E and HomᏲ.P; Q/. z z z z z 2 z z 2 z z z Let , P , and Q be as in (iv). By (iii),

[ . / ˛.ˇ .P // ˛.ˇ .P // .P / P; D ; D D z Ä so that, in particular, Ᏹ ˛.Ᏹ /. z Â 1 For ƒ./ write P for ˇ .P / and Q for ˇ .Q/, and set 2 ; ; z D ˛.P/. Then (iv) says that for all such there exists Hom.P;Q/ zj 2 with ˛./ . By (ii), we may assume is chosen so that E . Let 1 D z 2 . / and set . /, . /, and ˇ . /. Then 2 z D z D z z D ; arguing as in the proof of (a), we get ˛.. // on ˛.P/, so that, in z D particular, on ˛ .P /. But also . / . / on P . As E z D z z D D 2 we conclude that . / is the unique extension of to P. Then

˛./ ˛.. // z D D D z on ˛.P/. That is . Thus . .ˇ. /// ˛. .ˇ. // on ˛.ˇ.P/ z D z z D z D 1 P , so indeed is in ˛.HomG.P ;Q //. D z z Next ˛.HomG.P; Q// HomᏲ.˛.P /; ˛.Q// as ˛ is a morphism of fusion Â z systems; further, this map is injective by deﬁnition. In particular ˛.Im. // Ᏺ 1 Â z and of course ˛. .Ᏺ // Ᏺ, so that by deﬁnition of ᏲG, Â z

˛.ᏲG/ ˛. Im. /; .Ᏺ/ ƒ / ˛.Im. //; ˛..Ᏺ// ƒ Ᏺ: D h 1 j 2 i Dh 1 j 2 i Â z

By the preceding paragraph, HomᏲ.˛.P /; ˛.Q// ˛.HomG.P; Q//, and so z Â AᏲ.P/ ˛.ᏲG/ (cf. 2.14). Thus by (v), ˛.ᏲG/ Ᏺ. Therefore ˛ induces a bijec- z z Â D z tion HomG.P; Q/ HomᏲ.P; Q/ for all P and Q in Ᏺ, and hence ˛ ᏲG Ᏺ ! z z z z z z W ! z is an isomorphism of fusion systems. 972 MICHAEL ASCHBACHER and ANDREW CHERMAK

By (a), ᏳG Ᏻ is an embedding, and ˇ for . In W ! z D ı ; Ä order to show that is an isomorphism, it remains to show that defines a bijection Ᏹ Ᏹ, and defines bijections on morphism sets. The first condition is a 1 ! z consequence of (vi) and 12.4(c), so it remains to verify the second condition.

Let Morᏸ.P; Q/ be an ᏸ-morphism. Set . /. As ˛ is an iso- z 2 z z z z z D z z morphism of fusion systems we may choose P; Q, and HomG.P; Q/ so that y y 2 y y ˛./ . Choose 1./. Then . / lies in the -fiber over , as satisfies D z 2 z z (MG2) and ˛.P/ P . As1 Ᏻ is a pre-local group there then exists y Z.P/ such y D z z 2 z 1 that . / ıP .y/ . Let x be the element of Z.P/ which is mapped to y by D z z z y ˛0. As satisfies (MG3) we obtain .ı .x/ / , and thus is surjective on P ı D z morphism sets. y Finally, let Œf ; Œh Mor .P; Q/ with .Œf / .Œh/. As ˛ is injective 2 1 y y D on homomorphisms, it follows from (MG2) that Œf and Œh lie in the same - 1 fiber, and thus Œh ıP .z/ Œf for some z Z.P/. We may choose so that D y 2 y z P S P , and also so that for suitable representatives f and 2 y \ WD 2 f \ h h of the given morphisms. Then z Z.P / and 2 .Œf / .Œh/ .h.// .ı .z/ f .// .ı .z// .f .// D D D ;P D ;P ıP .˛0.z// .Œf /: D z z

As ıP deﬁnes a free action of Z.P/ on Morᏸ.P; Q/, we conclude that z 1, and z z z z z z D that is bijective on morphism sets. Thus is an isomorphism of categories, and (b) is proved. Theorem C is the following result:

THEOREM 12.11. Take ƒ to be the set of nonnegative integers, and for each i ƒ let Ᏻi .Si ; Ᏺi ; ᏸi / and Ᏻ .S; Ᏺ; ᏸ/ be the 2-local groups defined prior 2 D D rc rc to 11.2. Let and i be the signalizer functors in 11.3, and let ˇi;j Ᏺi Ᏺj and rc rc W ! ˇi Ᏺ Ᏺ be the mappings defined prior to 11.2. Then the following hold: W i ! (a) For each i; j ƒ, with i j , the mappings ˇi;j and ˇi extend to embeddings 2 Ä ˇi;j .i;j ; ˇi;j / Ᏻi Ᏻj and ˇi .i ; ˇi / Ᏻi Ᏻ D W ! D W ! of 2-local groups, and ˇj ˇi;j ˇi . ı D (b) G .ˇi;j Ᏻi Ᏻj /i j ƒ is a directed system of embeddings of 2-local WD W ! Ä 2 finite groups.

(c) The direct limit ᏳG of G, in the category of pre-local groups and embeddings, admits the structure of a 2-local group isomorphic to Ᏻ. rc rc Proof. Write ᏸi and ᏸ for the restriction of the centric linking systems ᏸi rc rc and ᏸ to radical centric linking systems on Ᏺi and Ᏺ , respectively. 2-LOCALFINITEGROUPS 973

First, 11.2 and 11.3 show that the mappings ˇi and ˇi;j are well deﬁned and satisfy conditions (10) and (20) in Proposition 2.12, relative to the appropriate signalizer functors. Then 2.11 and 2.12 yield embeddings of 2-local groups, as in rc (a), in which ˇi and ˇi;j act on ᏸi -morphisms via

ˇi i .P /g .ˇ.P //g and ˇi;j i .P /g .ˇj .P //g W 7! W 7! rc for P Ᏺ and g NG .P; S/. In order to check that ˇj ˇi;j ˇi in the category of 2 i 2 i ı D 2-local groups, it suffices to check the equality on objects. This follows from 10.10 in cases (1) and (2) of 10.9(c), and from 10.7 in case (3) when Y H. Suppose D Y K. By construction, NGi .P / NGj .ˇi;j .P //, so that .ˇj ˇi;j /.P / ˇi .P /. D Ä ı Ä rc We check from 10.2 that no proper NGi .P /-invariant subgroup of ˇi .P / is in Ᏺ , completing the proof of (a). The equality ˇj ˇi;j ˇi implies that ı D .ˇ ˇi;j /.P / ˇ .ˇi;j .P // ˇj .ˇi;j .P // G ˇi .P / G ˇ .P /; j;k ı D j;k D \ k D \ k D i;k so that ˇj;k ˇi;j ˇi;k. Also ˇi;i 1 by 11.12(c). Hence (b) holds. ı D D rc We next check that the hypotheses of 12.10(b) are satisfied with Ᏻ, ᏸi , ˇi , i ƒ, in the roles of Ᏻ, Ᏹ, , ƒ, respectively. 2 z z 2 First, S is the union of the groups Si for i ƒ, so that condition (i) of 12.10(b) 2 holds. Second, for any cg Hom .P; Q/ we have g Gi for some i, so that D 2 Ᏺ 2 Sj HomᏲj .P Sj ;Q Sj / for all j i, and hence condition (iv) holds. j 2 rc \ \ Let P Ᏺ , and for j i set Pj ˇi;j .Pi /. We have Pj ˇi .Pi / Gj for 2 i D D \ such j , and S is the union of its subgroups Sj , and so condition (iii) of 12.10(b) holds. We claim that there exists j i such that CG.Pj / centralizes P for all k k j . Suppose first that ZP 4. Then from the proof of 10.9, CG.Pi / Pi . But j i j Ä Ä Z.ˇi .Pi // is finite, and P ˇi .P / G for k i and so, from (ii), we may choose k D \ k j i with Z.Pj / Z.ˇi .Pi //. Thus CG.Pj / Z.Pk/ for k j . On the other D G D hand if ZP > 4 then Pi A or Pi CS .E/ by 10.5. In the first of these two j i j 2 D i cases, Pi P for all k i, while in the second CG.Pi / EC .T2/ CG.P / D k D .A/ D k for any k i, by 7.10 and 7.13. Thus the claim is established. We now verify condition (ii) of 12.10(b). Let Hom .Pj ;Qj / and sup- 2 Ᏺj pose that for some k j , we have 1; 2 HomᏲk .Pk;Qk/ extending . Then r 2 1 is the restriction of cg to P for some gr G, r 1; 2. Then g1g CG.Pj / r k 2 D 2 2 Ä CG.P / by the claim, so 1 2. This yields (ii). k D Condition (v) of 12.10(b) follows from 9.10. Finally each ˇi is injective on objects, since ˇi .P / Si P by 11.12(c). Thus G is nearly injective, and condition \ D rc (vi) of 12.10(b) holds. Therefore we conclude from 12.10(b) that .S; Ᏺ; ᏸ / ᏳG Š as pre-local groups, via the isomorphism .˛; / constructed in the proof of rc D 12.5 and 12.10. In particular ˛.Ᏹ/ Ᏺ , so as ˛ ᏲG Ᏺ is an isomorphism of D W ! 974 MICHAEL ASCHBACHER and ANDREW CHERMAK

rc fusion systems, Ᏹ Ᏺ , and ᏲG is saturated as Ᏺ is saturated. Transferring the D G centric linking system ᏸ on Ᏺ to ᏲG via ˛, we may regard ᏳG as a 2-local group, and the isomorphism of pre-local groups is then also an isomorphism of 2-local groups. This completes the proof of (c). Theorem D is essentially the following result. We thank Ran Levi and Bob Oliver for guiding us through a proof.

THEOREM 12.12. The 2-completed nerve ᏸG ^ is homotopy equivalent to j j2 B DI.4/. cc c c Proof. Let ᏸm be the full subcategory of ᏸm whose objects are centric in Ᏺn rc cc for all n m. Then Ᏺm Obj.ᏸm / for all m, by 10.9(b). Set m ˇm;m 1, and Â D C consider the diagram of categories and functors

rc m rc ᏸm ᏸm 1 ? ! ?C . / m? ?m 1 y y C cc cc m cc ᏸm ᏸm 1; ! C where is in every instance an inclusion functor. We claim that this diagram com- cc cc mutes up to a natural homomorphism m m 1 m. Since m is W ı ! C ı ı the identity map on objects, what this means is that for all P;Q ᏸrc , there are 2 m ᏸm 1-morphisms P and Q such that, for each Morm.P; Q/, the following C 2 diagram commutes.

cc .m. // P Q ? ! ? . / P ? ?Q y y m 1. . // m.P / C m.Q/: ! rc Indeed, for any R ᏸm deﬁne R to be m 1.R/. Recall that m.P /g 2 C cc D for some g NGm .P; Q/, by the deﬁnition of ᏸm. The functor sends to 2 cc regarded as an element of Morm 1.P; Q/. That is, we have . / m 1.P /g, C D C and hence (in our mix of left- and right-hand notation, as set forth in 1) cc .m. // Q m 1.P /gm 1.Q/ m 1.P /g; D C C D C while also

P m 1. m. // m 1.P /m 1. m.P //g m 1.P /g: C D C C D C Thus is a natural transformation as desired, and the claim is proved. Recall that the nerve of a small category Ꮿ is a simplicial set (or equiva- lently, the topological realization of a simplicial set) whose k-simplices are chains 2-LOCALFINITEGROUPS 975

.˛0; : : : ; ˛ / of composable morphisms in Ꮿ. If f Ꮿ Ᏸ is a functor of small k W ! categories, then there is a continuous map f Ꮿ Ᏸ of spaces, given by j jW j j ! j j .˛0; : : : ; ˛k/ .f .˛0/; : : : ; f .˛k//. 7! rc cc Set Xm ᏸ , Ym ᏸ , and consider the following diagram of spaces D j mj D j m j and continuous maps.

1 2 3 X1 j j X2 j j X3 j j ::: ? ! ? ! ? ! . / f ? ?f ?f 1y y 2 y 3 cc cc cc f1 f2 f3 Y1 Y2 Y3 :::: ! ! ! cc cc cc Here, we are taking fm m and fm m . Each fm may be viewed as cc WD j j WD j j inclusion, since m induces an inclusion of Morm.P; Q/ into Morm 1.P; Q/ for cc C any P;Q ᏸm . Similarly, since each m is an embedding, m induces an injec- 2 rc tive mapping Morm.P; Q/ Morm 1.P; Q/ for P;Q ᏸm, and hence m is ! C 2 j j injective. There is then no harm in viewing each m as an ordinary inclusion j j of topological spaces (and in adjusting the vertical arrows by suitable homeomor- phisms, to compensate for this). The direct limit X of the top row in . / is then the union of the spaces Xn. No such adjustment is necessary for the bottom row, whose union we denote by Y . ^ It is the content of[LO02, Prop. 4.3] that the 2-completion .Y /2 of Y is B DI.4/, up to homotopy equivalence. That this is so requires some explana- tion, since the union taken in[LO02] is that of a somewhat different collection of spaces than Ym m>0. Namely, Levi and Oliver choose a sequence .ni /i>0 of f g positive integers, so that each ni divides ni 1 and so that every positive integer C divides some ni . They then show that B DI.4/ is the 2-completion of the union ni cc of the spaces ᏸSol.q / , for any odd prime power q. We may take q p, and j j i D may take the sequence .ni /i>0 so that 2 is the highest power of 2 dividing ni . ni ni Then ᏲSol.p / is a fusion system over the Sylow 2-subgroup Si of Spin7.p /. i ni 2 ni As the 2-shares of Spin7.p / and Spin7.p / are the same, Spin7.p / has the 2i same Sylow 2-subgroup as Spin7.p /. By a result in[COS06] the fusion sys- i ni 2 tems ᏲSol.p / and ᏲSol.p / are isomorphic, and their corresponding linking systems are then isomorphic[LO02, Lemma 3.2]. Thus, the union of the nerves ni cc ᏸSol.p / is homeomorphic to Y , and .Y /^ may be identified with B DI.4/. j j 2 Since it is obvious from the definitions that the nerve of an increasing union of categories is the union of the nerves, it follows from 12.5 that the space X is homeomorphic to ᏸG . Thus, it remains only to show that X is homotopy j j equivalent to Y . The existence of a natural transformation as in . / implies that each of the squares in . / commutes up to homotopy (cf.[Dwy01, Prop. 5.2]. For 976 MICHAEL ASCHBACHER and ANDREW CHERMAK any m, the inclusion of the simplicial set Xm in Xm 1 is a CW-pair, and so the C homotopy extension property for CW-pairs[Hat02, Prop. 0.16] implies that f2 may be replaced by a map f20 which is homotopic to f2 and which extends f1. We continue up the chain, replacing fm 1 by a map fm0 1 which is homotopic to C C fm 1 and extends fm0 . Now define f X Y to be the union of the maps fm0 . C W ! Observe that every finite subcomplex of the CW-complex X (or Y ) is contained in some Xi (or Yi ). Since every compact subset of a CW-complex is contained in a finite subcomplex, every compact subset of X or Y is contained in some Xi or Yi . From this, it follows directly from the definition of homotopy groups that for each n, n.X/ is the direct limit of the n.Xi /, and similarly for n.Y /. That each fm (and hence also each fm0 ) is a homotopy equivalence is given by[BCG C05, Th. B], and thus n.f / is an isomorphism for all n. Then f is a homotopy equivalence by Whitehead’s Theorem, and the proof is complete. We close with an example. Example 12.13. Let p be a prime and let G G.F / be a Chevalley group D over the algebraic closure F of Fp of Lie rank l. Let † be the set of positive integers, partially ordered by n m if n divides m. Set I 1; : : : ; l and let Ä D f g .PJ ¿ J I/ be the set of proper parabolic subgroups of G over a fixed j ¤ Â Borel subgroup B PI . For J I let SJ be the unipotent radical of PJ , and set D p Â S SI . Let 1 a a be the Frobenius map on F , and regard 1 also as a field D W 7! k automorphism of G. For k 1 set , and let G G be the group of k D 1 k D k fixed points of on G. Set S S G , S SJ G , and let k k D \ k J;k D \ k Ᏻ ᏳS .G / .S ; Ᏺ ; ᏸ / k D k k D k k k be the p-local finite group associated with Gk. rc p By Borel-Tits, Ᏺk .SJ;k J I/, and k.P / O .CGk .P // 1 for rc D W Â WD D all P Ᏺ . When k divides j , we have the inclusion map ˇ Ᏻ Ᏻj with 2 k k;j W k ! ˇ .S / SJ;j . It follows from 2.11 that G .Ᏻ ; ˇ k j / is a directed k;j J;k D WD k k;j W Ä system of p-local finite groups. Further one can check that the hypotheses of 12.10(b) are satisfied by G, so, as in the proof of Theorem 12.11, the limit ᏳG rc rc of G is isomorphic to Ᏻ.G/ .ᏲS .G/; Ᏺ .G/; ᏸ/, where Ᏺ .G/ has object set D S S .SJ J/, Mor.SJ ;SK / Hom.SJ ;SK / NG.SJ ;SK /, and and ı are identity W D D maps.

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(Received January 12, 2006) (Revised December 14, 2006)

E-mail address: [email protected] MATHEMATICS 253-37, CALIFORNIA INSTITUTEOF TECHNOLOGY,PASADENA CA 91125, UNITED STATES E-mail address: [email protected] DEPARTMENT OF MATHEMATICS,KANSAS STATE UNIVERSITY, 138 CARDWELL HALL, MANHATTAN KS 66506-2602, UNITED STATES