Reductions to Simple Fusion Systems

We prove that if E E F are saturated fusion systems over p-groups T E S, such that CS(E) ≤ T , and either AutF (T )/AutE(T ) or Out(E) is p-solvable, then F can be “reduced” to E by alternately taking normal subsystems of p-power index or of index prime to p. In particular, this is the case whenever E is simple and “tamely realized” by a known simple group K. This answers a question posed by Michael Aschbacher, and is useful when analyzing involution centralizers in simple fusion systems, in connection with his program for reproving parts of the classification of finite simple groups by classifying certain 2-fusion systems. When p is a prime and S is a finite p-group, a saturated fusion system over S is a category whose objects are the subgroups of S, whose morphisms are injective group homomorphisms between the subgroups, and which satisfies a certain list of axioms motivated by the Sylow theorems for finite groups (Definition 1.1). For example, when G is a finite group and S ∈ Sylp(G), the p-fusion system of G is the category FS(G) whose objects are the subgroups of S, and where for each P,Q ≤ S, HomFS(G)(P,Q) is the set of those homomorphisms induced by conjugation in G. Normal fusion subsystems of a saturated fusion system are defined by analogy with normal subgroups of a group (Definition 1.4). Among the normal subsystems, we look at two particular classes: those of index prime to p (defined over the same Sylow subgroup), and those of p-power index (see the discussions before and after Lemma 1.10). A natural question arises: when E E F , under what conditions can F be “reduced” to E via a sequence of steps, where one alternates taking normal subsystems of p-power index and normal subsystems of index prime to p? Our main theorem (Theorem 2.3) says that if E E F are saturated fusion systems over p-groups T E S, such that CS(E) ≤ T , and either AutF(T )/AutE(T ) or Out(E) is p-solvable, then F can be reduced to E in the above sense. In particular, if E is the fusion system of a known finite simple group K, and is “tamely realized” by K in the sense that Out(K) surjects onto Out(E) (see Section 2), then Out(E) is solvable since Out(K) is solvable by the Schreier conjecture, and hence F reduces to E . This paper was motivated by a question posed by Michael Aschbacher. The above situation arises frequently when analyzing centralizers of involutions in simple fusion systems. If F is the centralizer of an involution and E = F ∗(F) denotes the generalized Fitting subsystem of F (see [As, Chapter 9]), then the hypothesis CS(E) ≤ T always holds, and Out(E) is solvable by Schreier’s conjecture if E/Z(E) is tamely realized by a known simple group K. Hence in this situation, Theorem 2.3 together with results in [AOV] imply that F itself is realized by a certain extension of K. (See Corollary 2.5 for a slightly more general situation where this applies.) 2000 Mathematics Subject Classification. Primary 20E25. Secondary 20D20, 20D05, 20D25, 20D45.