A Classification of the Stable Type of Bg

We give a classification of the p-local stable homotopy type of BG , where G is a finite group, in purely algebraic terms. BG is determined by conjugacy classes of homomorphisms from p-groups into G . This classification greatly simplifies if G has a normal Sylow p-subgroup; the stable homotopy types then depends only on the Weyl group of the Sylow p-subgroup. If G is cyclic mod p then BG determines G up to isomorphism. The last class of groups is important because in an appropriate Grothendieck group BG can be written as a unique linear combination of BH 's, where H is cyclic mod p . 0. Introduction and statement of main results Let G be a finite group. In this note we give a classification of the stable homotopy type of BG in terms of G. Our analysis shows that for each prime number p, the p-local stable type of BG depends on the homomorphisms from p-groups Q into G. The suspension spectrum of BG and, in particular, its wedge summands have played an important role in homotopy theory. In a previous paper [MP], the authors have given a characterization of the indecomposable summands of BG in terms of the modular representation theory of Out(ß) modules for Q < P the Sylow subgroup of G. It is this characterization that we use to study the stable type of BG. For another such characterization see [BF]. It is known that the stable type of BG does not determine G up to isomorphism. A simple example (due to N. Minami) is given by Ô4p x Z/2 and D2p x Z/4 where p is an odd prime, Q^p is the generalized quaternion group [CE] of order 4p , and D2p is the dihederal group of order 2p . The situation is even worse for p-local classifying spaces since BG and BG/0P'(G) have isomorphic mod p homology and hence equivalent stable types. Here Op>(G) is the maximal normal subgroup of G of order prime to p . However there is a positive result in this direction, due to Nishida [N], who established the following: Suppose G\, G2 are finite groups with Sylow /^-subgroups P\, P2, then BG\ ~ BG2 stably at p implies P\ « P2. Our main result is a necessary and sufficient condition. Theorem 0.1 (Classification). For two finite groups G\, G2 the following are equivalent: (1) Localized at p, BG\ and BG2 are stably homotopy equivalent. (2) For every p-group Q, FpRep(Q, Gi) « FpRep(g, G2) Received by the editors November 20, 1991. 1991 Mathematics Subject Classification. Primary 55R35; Secondary 20J06, 55P42. The first author was partially supported by NSF Grant DMS-9007361, the second by NSF Grant DMS-880067 and the Alexander von Humboldt Foundation. 165 © 1992 American Mathematical Society 0273-0979/92 $1.00+ $.25 per page 166 JOHN MARTINO AND STEWART PRIDDY as Out(ß) modules. Rep(ß, G) = Hom(ß, G)IG with G acting by conjugation. (3) For every p-group Q, Vnj(ß,G,)«FpInj(ß,G2) as Out(ß) modules. Inj(ß, G) < Rep(ß, G) consists of conjugacy classes of injective homomorphisms. Nishida's theorem follows since the largest ß for which Inj(ß, G) is nonzero is the Sylow p-subgroup of G. We will refer to the common Sylow p-subgroup as P. It should not be concluded from (3) that the G\ conjugacy classes of a /»-subgroup correspond to the G2 conjugacy classes, although the number of classes is equal. An important application of Theorem 0.1 is to the case of a normal Sylow p-subgroup, in particular, the cyclic mod p groups. Definition 0.2. Two subgroups H, K < G are called pointwise conjugate in G if there is a bijection of sets H-^K such that a(h) = g^lhgh for gh G G depending on h G H. Alternately it is easy to see that an equivalent condition is \Hn(g)\ = \Kn(g)\ for all g G G, where (g) denotes the conjugacy class of g. Let WG(H) denote the Weylgroup of H < G, i.e., WG(H) = NG(H)/H-CG(H) where NG(H) is the normalizer and CG(H) is the centralizer of H in G. Then WG(H) < Out(H). Theorem 0.3. Suppose G\, G2 are finite groups with normal Sylow p-subgroups P\, P2. Then BG\ and BG2 have the same stable homotopy type, localized at p, if and only if P\ « P2 (« P say) and WGx(P) is pointwise conjugate to WGl(P) in Out(P). Definition 0.4. G is called cyclic mod p (or p-hypoelementary) if a Sylow psubgroup P is normal and C — G/P is a cyclic p'-group, i.e., has order prime to p. We say G is reduced if Op>(G) — 1. For a cyclic mod p group G, being reduced is equivalent to WG(P) — C. Theorem 0.5. Suppose G\, G2 are reduced cyclic mod p groups. Then BG\ ~ BG2 stably at p if and only if G\ « G2. Cyclic mod p groups are important for several reasons. Their mod p cohomology is computed as the ring of invariants H*(G) = H*(P)C . On the level of stable homotopy one also has the Minami-Webb Formula [M]: Let %(G) be the set of cyclic mod p subgroups of G. Then BG~\/f(H)/[NG(H):H] BH where H runs over the conjugacy classes of ^¡,(G) and / : WP(G) —* Z is the Möbius function given by YiJ where w means view w as an element in Out(ß). W ^ 0 mod p if and only if a G K(Q, G). For conjugacy classes ß; of /^-subgroups of G\ (or G2 ), where X is in one-to-one correspondence with a simple module of R(Qj) and the inclusion Q¡ «-> G\ (or G2 ) is in K(Qj, G¡), i I or 2, we find that the multiplicity of X in BG\ (or BG2 ) equals