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Automorphisms of Fusion Systems of Sporadic Simple Groups

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Automorphisms of Fusion Systems of Sporadic Simple Groups AUTOMORPHISMS OF FUSION SYSTEMS OF SPORADIC SIMPLE GROUPS BOB OLIVER Abstract. We prove here that with a very small number of exceptions, when G is a sporadic simple group and p is a prime such that the Sylow p-subgroups of G are nonabelian, then Out(G) is isomorphic to the outer automorphism groups of the fusion and linking systems of G. In particular, the p-fusion system of G is tame in the sense of [AOV1], and ∼ is tamely realized by G itself except when G = M11 and p = 2. From the point of view of homotopy theory, these results also imply that Out ∼ Out ^ in many (but not all) (G) = (BGp ) cases. This paper is centered around the comparison of certain outer automorphism groups as- sociated to a sporadic simple group: outer automorphisms of the group itself, those of its fusion at dierent primes, and those of its classifying space completed at dierent primes. In most, but not all cases (under conditions made precise in Theorem A), these automorphism groups are all isomorphic. This comparison is important when studying extensions of fusion systems, and through that plays a role in Aschbacher's program (see, e.g., [A5]) for reproving certain parts of the classication theorem from the point of view of fusion systems. When is a nite group, is a prime, and Syl , the -fusion system of is the G p S 2 p(G) p G category FS(G) whose objects are the subgroups of G, and which has morphism sets Mor Hom some with −1 FS (G)(P; Q) = ' 2 (P; Q) ' = cx; x 2 G xP x ≤ Q : A -subgroup is called -centric in if Syl ; equivalently, if p P ≤ G p G Z(P ) 2 p(CG(P )) CG(P ) = 0 for some (unique) subgroup 0 of order prime to . The centric linking Z(P ) × CG(P ) CG(P ) p system of at is the category c whose objects are the subgroups of which are G p LS(G) S p-centric in G, and where 0 −1 Mor c (P; Q) = T (P; Q)=C (P ) where T (P; Q) = x 2 G xP x ≤ Q : LS (G) G G G Note that there is a natural functor c which is the inclusion on objects, π : LS(G) −! FS(G) and which sends the class of x 2 TG(P; Q) to cx 2 Hom(P; Q). Outer automorphism groups of these systems were dened in [BLO] and later papers (see below). We say that F = FS(G) is tamely realized by if the natural homomorphism Out Out c is surjective G κG : (G) −! (LS(G)) and splits. The fusion system F is tame if it is tamely realized by some nite group. In terms of homotopy theory, it was shown in [BLO, Theorem B] that for a nite group G and Syl , there is a natural isomorphism Out c ∼ Out ^ . Here, ^ is S 2 p(G) (LS(G)) = (BGp ) BGp the p-completion, in the sense of Bouseld-Kan, of the classifying space of G, and Out(X) means the group of homotopy classes of self equivalences of the space X. Thus FS(G) is tamely realized by if the natural map from Out to Out ^ is split surjective. G (G) (BGp ) When p = 2, our main result is easily stated: if G is a sporadic simple group, then the ∼ 2-fusion system of G is simple except when G = J1, and is tamely realized by G except when ∼ G = M11. The 2-fusion system of M11 is tamely realized by PSL3(3). 2000 Mathematics Subject Classication. Primary 20E25, 20D08. Secondary 20D20, 20D05, 20D45. Key words and phrases. Fusion systems, sporadic groups, Sylow subgroups, nite simple groups. B. Oliver is partially supported by UMR 7539 of the CNRS. 1 2 BOB OLIVER For p odd, information about fusion systems of the sporadic groups at odd primes is summarized in Table 0.1. In that table, for a given group and prime and Syl , G p S 2 p(G) • a dash means that S is abelian or trivial; • constr. means that FS(G) is constrained; and • an almost simple group L in brackets means that FS(G) is almost simple but not simple, and is shown in [A4, 16.10] to be isomorphic to the fusion system of L. For all other pairs (G; p), F is simple by [A4, 16.10], and we indicate what is known about the nature of Out Out c . In addition, κG : (G) −! (LS(G)) • a dagger (y) marks the pairs (G; p) for which S is extraspecial of order p3. G jOut(G)j p = 3 p = 5 p = 7 p ≥ 11 y M12 2 κ isom. y M24 1 [M12:2] y J2 2 constr. J3 2 constr. 2 y y J4 1 [ F4(2)] 11: constr. y Co3 1 κ isom. constr. y Co2 1 κ isom. constr. Co1 1 κ isom. [SO5(5)] HS 2 constr.y McL 2 κ isom. constr.y Suz 2 κ isom. He 2 Out(L) = 1y κ isom.y Ly 1 κ isom. κ isom. 2 y y Ru 1 [ F4(2)] [L3(5):2] O'N 2 κ isom.y Fi22 2 κ isom. Fi23 1 κ isom. Fi0 2 isom. isom.y 24 κ κ F5 2 κ isom. κ isom. y F3 1 κ isom. κ isom. F2 1 κ isom. κ isom. y F1 1 κ isom. κ isom. κ isom. 13: κ isom. Table 0.1. Summary of results for odd p Here, a fusion system is constrained if it contains a normal -subgroup F = FS(G) p Q E F such that CS(Q) ≤ Q. The fusion system F is simple if it has no proper nontrivial normal fusion subsystems. It is almost simple if it contains a proper normal subsystem F0 E F which is simple, and such that CF (F0) = 1. We refer to [AKO, Denitions I.4.1 & I.6.1] for the denitions of normal p-subgroups and normal fusion subsystems, and to [A4, 6] for the denition of the centralizer of a normal subsystem. AUTOMORPHISMS OF FUSION SYSTEMS OF SPORADIC SIMPLE GROUPS 3 Thus when G is a sporadic simple group and p is an odd prime such that the p-fusion system F of G is simple, we show in all cases that F is tamely realized by G, and in fact that Out ∼ Out c except when ∼ He and (Theorem A). (G) = (LS(G)) G = p = 3 Before going further, we need to dene more precisely the automorphism groups which we are working with. All of the denitions given here apply to abstract fusion and linking systems (see, e.g., [AKO, III.4.3]), but for simplicity, we always assume that F = FS(G) and c for some nite group with Syl . L = LS(G) G S 2 p(G) Automorphisms of F = FS(G) are straightforward. An automorphism α 2 Aut(S) is fusion preserving if it induces an automorphism of the category F (i.e., a functor from F to itself which is bijective on objects and on morphisms). Set Aut(F) = Aut(S; F) = α 2 Aut(S) α is fusion preserving Out(F) = Out(S; F) = Aut(S; F)=AutF (S): Here, by denition, AutF (S) = AutG(S): the automorphisms induced by conjugation in NG(S). These groups have been denoted Aut(S; F) and Out(S; F) in earlier papers, to emphasize that they are groups of automorphisms of S, but it seems more appropriate here to regard them as automorphisms of the fusion system F (as opposed to the category F). c Now assume L = L (G). For each P 2 Ob(L), set ιP = [1] 2 MorL(P; S) (the inclusion S of P in S in the category L), and set [[P ]] = [g] g 2 P ≤ AutL(P ). Dene I c Aut(L) = Auttyp(L) = β 2 Autcat(L) β(ιP ) = ιβ(P ); β([[P ]]) = [[β(P )]]; 8 P 2 F Out(L) = Outtyp(L) = Aut(L) hcx j x 2 NG(S)i: Here, Autcat(L) is the group of automorphisms of L as a category, and cx 2 Aut(L) for x x x 2 NG(S) sends P to P and [g] to [ g]. There are natural homomorphisms κG µG Out(G) −−−−−! Out(L) −−−−−! Out(F) and κG = µG ◦ κG: ∼Out ^ = (BGp ) Here, κG is dened by sending the class of α 2 Aut(G), chosen so that α(S) = S, to the class of Aut , where and . For Aut , sends the αb 2 (L) αb(P ) = α(P ) αb([g]) = [α(g)] β 2 (L) µG class of β 2 Aut(L) to the class of g7![g] βj[[S]] [g]7!g βb = S −−−−−−! [[S]] −−−−−−! [[S]] −−−−−−! S 2 Aut(F) ≤ Aut(S): =∼ =∼ =∼ Then κG : Out(G) −! Out(F) is induced by restriction to S. See [AKO, III.4.3] or [AOV1, 1.3] for more details on these denitions. By recent work of Chermak, Oliver, and Glauberman and Lynd, the nature of µG is now fairly well known in all cases. Proposition 0.1 ([O2, Theorem C], [GlL, Theorem 1.1]). For each prime p, and each nite group with Syl , Out c Out is surjective, and is an G S 2 p(G) µG : (LS(G)) −−−! (FS(G)) isomorphism if p is odd. In fact, [O2] and [GlL] show that the conclusion of Proposition 0.1 holds for all (abstract) fusion systems and associated linking systems. When G is a sporadic simple group and p is odd, a more direct proof that µG is an isomorphism is given in [O1, Propositions 4.1 & 4.4].
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