ALMOST ALL EXTRASPECIAL p-GROUPS ARE SWAN GROUPS DAVID JOHN GREEN AND PHAM ANH MINH Abstract. Let P be an extraspecial p-group which is neither dihedral of * *or- der 8, nor of odd order p3 and exponent p. Let G be a finite group havin* *g P as a Sylow p-subgroup. Then the mod-p cohomology ring of G coincides with that of the normalizer NG(P). Introduction Let P be a finite p-group. Martino and Priddy call P a Swan group [?] if for every finite group G with Sylow p-subgroup P , the mod-p cohomology ring H*(G) coincides with H*(NG (P )) ~=H*(P )NG(P). In particular, if there are no so-cal* *led transfer summands in the stable decomposition of the classifying space BP , then P is a Swan group. We prove in Theorem ?? that all extraspecial p-groups are Swan groups, apart from the well-known exceptions 21+2+= D8 and (for p odd) p1+2+= E. The cases where P is the metacyclic group p1+2-= M(p3) with p odd, and where P is 21+2n-= Q8* D8* . .*.D8, were proved in [?]; the former case being due to G. Glauberman. Earlier, the p = 2 case of the theorem was published in [?], but with an incorr* *ect proof: see Remark ??. In Corollary ?? we generalize another result of Martino a* *nd Priddy, exhibiting three infinite families of Swan groups whose classifying spa* *ces do have transfer summands in their stable decompositions. Throughout this paper we denote the mod-p cohomology ring H*(G; Fp) by H*(G). A suitable reference on group cohomology is Evens' book [?]. 1. Extraspecial p-groups Recall that a p-group P is called extraspecial if its centre Z(P ), its deri* *ved subgroup P 0and its Frattini subgroup (P ) all coincide, and are cyclic of orde* *r p. So if P is extraspecial there is a central extension 1 -! Fp -! P -! V -! 1 with V an elementary abelian p-group. Hence there is a nondegenerate alternate bilinear form f on V defined by f( (g); (h)) = [g; h] for all g; h 2 P . Moreover, for p odd there is a linear form on V defined by ( g) = gp; and for p = 2 there is a quadratic form Q on V defined by Q( g) = g2 with associated bilinear form f. Conversely, such a pair (; f) determines an extraspecial p-gro* *up when p is odd, and such a Q determines an extraspecial 2-group. See [?, Ch. 6] for a reference on alternate bilinear forms, and [?, I.16] fo* *r a reference on quadratic forms in characteristic 2. Nondegeneracy means that V has even dimension, say 2n. Up to change of basis for V there are two possibilities* * for Q when p = 2, and exactly one choice of f for odd p. For p odd there are two possibilities for the pair (; f): either is identically zero, or it is not. No* *te that in ____________ Date: 27 October 1999. 1991 Mathematics Subject Classification. Primary 20J06; Secondary 20D15, 55R* *35. 1 2 D. J. GREEN AND P. A. MINH the case of nonzero , Witt's extension theorem does not hold for the pair (; f), as the restriction of f to ker() does have a kernel. So there are four types of extraspecial p-groups. In each case P is generate* *d by A1; : :;:An; B1; : :;:Bn; C, with C central of order p, [Ai; Aj] = [Bi; Bj] = 1* * and [Ai; Bj] = Cffiij. Moreover Api= Bpi= 1 for 2 i n. The four cases are: o 21+2n+= D8 * . .*.D8: here A21= B21= 1. o 21+2n-= Q8 * D8 * . .*.D8: here A21= B21= C. o p1+2n+= E * . .*.E has odd exponent p: here Ap1= Bp1= 1. o p1+2n-= M(p3) * E * . .*.E has odd exponent p2: here Ap1= C and Bp1= 1. The characteristic subgroup 1(P ) of P is the subgroup generated by all order p elements. Denote Z(1(P )) by Y . If P has odd exponent p2 then Y = is rank two elementary abelian; in all other cases, Y equals Z = Z(P ). The following result could be called Witt's theorem for extraspecial p-group* *s. Proposition 1.1. Let P be an extraspecial p-group. Suppose that H; K are sub- groups of P containing Z, and that OE: H ! K is a group isomorphism inducing the identity map on Z. If P has odd exponent p2, assume further that H \ Y = K \ Y and that OE induces the identity map on (H \ Y )=Z. Then OE extends to an auto- morphism of P . Lemma 1.2. In Proposition ??, suppose that P has odd exponent p2 and that H\Y is Z rather than Y . Then OE extends to an isomorphism from to * *which itself satisfies the conditions of Proposition ??. Proof. Since OE(C) = C it follows that h-1OE(h) lies in ker() for every h 2 H. * *Hence_ OE([h; B1]) = [OE(h); B1]. So we may set OE(B1) = B1. * *|__| Proof of Proposition ??.Denote by U; W the images in V of H; K respectively. Since OE is the identity on Z there is an Fp-vector space isomorphism ae: U ! W induced by OE which respects the alternate bilinear form f on V . If p is 2 then ae respects the quadratic form Q. Since Witt's extension theo* *rem holds for Q (see [?, p. 36]), we may extend ae to a Q-orthogonal transformation ae of V . Using the standard generators for P we may lift ae to an automorphism* * OE0 of P . If h 2 H then OE0(h) = OE(h)Cr for some r 2 Z=p. Since P has enough inner automorphisms, we may assume that OE0extends OE. To be more precise: pick h1; : :;:hm 2 H whose images under :P ! V constitute a basis for U. Since the alternate bilinear form f on V is nondegene* *rate we can pick g1; : :;:gm 2 P such that f( (gi); (OEhj)) = ffiij. Hence conjugat* *ion by gi fixes OE(hj) for j 6= i and sends OE(hi) to OE(hi)C. So we can correct OE* *0 by an inner automorphism of P to ensure that OE0= OE on H. Now suppose that p is odd. As Witt's extension theorem holds for f (see [?, * *6.9]), we may extend ae to a transformation of V that respects f. But then ae respects too: this is trivial in the exponent p case, as is then zero. In the exponent* * p2 case, we may assume by Lemma ?? that H; K contain Y and that ae fixes (B1). So ae respects by Lemma ?? below. As in the p = 2 case we can now lift ae_to_an automorphism of P which extends OE. |__| Lemma 1.3. Suppose P is an extraspecial p-group with odd exponent p2. Taking C as basis for Z ~=Fp, we may assume that ( A1) = 1. Then for any v 2 V we have (v) = f(v; (B1)). Proof. Each element g of P has canonical form Bs11. .B.snn. Ar11. .A.rnn._Ct._T* *hen gp = Cr1 = [g; B1]. |__| ALMOST ALL EXTRASPECIAL p-GROUPS ARE SWAN GROUPS 3 2. Local subgroup structure Throughout this section G is a finite group with extraspecial Sylow p-subgro* *up P . Lemma 2.1. Suppose that P is not one of D8, E, M(p3). Then for any order p element g of P , the centralizer of g in P has the same Frattini subgroup as P * *itself. Proof. If n 2 then CP (g) is not abelian. In Q8, all order p elements are cent* *ral._ The other three groups really are exceptions: take g = B1. |_* *_| Lemma 2.2. Suppose that the centralizer of each exponent p element of P has Z as its Frattini subgroup. Then Zg = Z for every g 2 G such that P \P gcontains * *Z. Moreover, such g factorize as g = g1g2 with g1 2 NG (P ) and g2 2 CG (Z). Proof. Observe that P is a Sylow p-subgroup of CG (Z), and so all Sylow p-subgr* *oups of CG (Z) have Frattini subgroup Z. Now set R equal to P g\ CG (Z). By assump- tion the Frattini subgroup of R is that of P g, namely Zg. But R is contained i* *n a Sylow p-subgroup of CG (Z). We conclude that Zg = Z. Therefore P gis itself a Sylow p-subgroup of CG (Z), and so P g= P hfor_some_ h 2 CG (Z). Take g1 = gh-1 and g2 = h. |__| Lemma 2.3. Suppose that P has odd exponent p2. Then Y g= Y for every g 2 CG (Z) such that Y P \ P g. Moreover, such g factorize as g1g2, where g1 lies * *in CG (Z) \ NG (P ) and g2 2 CG (Z) \ NG (Y ) acts trivially on Y=Z. Proof. Set D1 equal to CG (Y ), which contains 1(P ). Since the centre of P is * *cyclic and 1(P ) is maximal in P , it follows that 1(P ) is a Sylow p-subgroup of D1. Now let R be P g\ D1, the centralizer of Y in P g. Since 1(P ) has exponent p, * *so does R. As g centralizes Z, we deduce that R is a maximal subgroup of P g. But the only exponent p maximal subgroup of P gis 1(P g), which has centre Y g. So g normalizes Y . Now set D2 = {h 2 CG (Z) \ NG (Y ) | g acts trivially on Y=Z}. Then P is a Sylow p-subgroup of D2, and P gis too since g lies in CG (Z) \ NG (Y ). So_P_g=* * P h for some h 2 D2. Take g1 = gh-1 and g2 = h. |__| 3. Stability conditions The following elementary reformulation of the usual stability condition is n* *ot new, but does not appear to be widely known. Lemma 3.1. Let P be a Sylow p-subgroup of a finite group G. The cohomology class x 2 H*(P; Fp) lies in Im ResGPif and only if x is an NG (P )-invariant and g * CorPP\PgResPP\Pgg (x) = 0 for all g 2 G - NG (P ). Proof. If x comes from H*(G), then it certainly satisfies both conditions. Con- versely, observe that the conditions combined with the Mackey formula mean that_ ResGPCorGP(x) = |NG (P ) : P |x. |__| Let P be a finite p-group. The ring of universally stable elements I(P ) was de* *fined in [?] as the subring of H*(P; Fp) given by " I(P ) = Im ResGP; G where G ranges over all finite groups with Sylow p-subgroup P . The following observation appears in [?]. Recall that Op(G) is the subgroup generated by all p0-elements of G. Lemma 3.2. I(P ) H*(P )Op(Out(P)). 4 D. J. GREEN AND P. A. MINH Proof. Pick any outer automorphism of order prime to p, and lift it to an autom* *or-_ phism ff of the same order. Let G be the semidirect product P o . * *|__| It is immediate that if P is a Swan group then equality holds in Lemma ??. The main result of this paper is: Theorem 3.3. All extraspecial p-groups P apart from D8 and E are Swan groups. If P is D8por E, then the universally stable elements for P are strictly contai* *ned in H*(P )O (Out(P)). Proof. See [?] for a proof that M(p3) is a Swan group. The groups D8 and E are treated in Lemma ?? below. So we may assume that P satisfies the hypotheses of Lemma ?? and hence those of Lemma ??. Let G be a finite group with extraspecial Sylow p-subgroup P , and let x 2 H* **(P ) be an NG (P )-invariant. We show that the conditions of Lemma ?? are satisfied. Let g be an element of G-NG (P ). If P g\P is elementary abelian but not maximal in P , then corestriction from P g\ P to P is zero: for corestriction from any * *group H to H x Cp is zero. We may therefore assume that P g, P both contain Z = (P ). Write H for P \ P gand L for gP \ P . By Lemma ?? we deduce that g normalizes Z. Moreover, since x is invariant under NG (P ), we may in fact assume that g stabilizes Z. * * If P has odd exponent p2 we deduce further by Lemma ?? either that g normalizes Y and can be taken to act trivially on Y=Z, or that H \ Y = L \ Y = Z. So we can now apply Proposition ?? and deducegthat conjugation cg: H ! L extends togan automorphism OE of P . Then ResPHg* = ResPHOE*, which means that Cor PHResPHg* = 0 as H is a proper subgroup of P . |___| Lemma 3.4. If P is D8 or E, then the inclusion in Lemma ?? is strict. Proof. Let G be GL 3(Fp). Then the upper triangular matrices with onesionjthe diagonal form a Sylow p-subgroup isomorphic to P . We may take B1 = 100011001 i1 01j i0 10j and C = 001001. Pick g = 100001. Then F = P \ P gequals , a maximal elementary abelian subgroup. Let fi; fl be the dual basis for F *. Taking the first Chern class is an Fp-linear monomorphism from F *to H2(F ). Let ae be an ordinary representation of P with character O, the sum of all p2 l* *inear characters. These linear characters restrict to F as scalar multiples of fi, ea* *ch scalar multiple being the image of p characters. So by the Whitney sum formula, the to* *tal Chern class c(ae) restricts to F as follows: Y ResPFc(ae) = (1 + fi)p: 2Fp This equals 1 - fip(p-1). Set j equal to cp(p-1)(ae) in H2p(p-1)(P ). Then j li* *es in H*(P )Op(Out(P)), since O is an invariant of Aut(P ). But ResPF(j) = -fip(p-1)* *is__ distinct from g* ResPF(j) = -flp(p-1), so j is not stable. * * |__| Suppose that P is a p-group whose classifying space does not have any transfer summands in its stable splitting. Then P is a Swan group by Theorem 3.5 of [?]. Martino and Priddy give one counterexample to the converse: M(p3). Corollary 3.5. Let P be 21+2n+with n 2; or p1+2n+with p odd and n 2; or p1+2n-with p odd and n 1. Then although P is a Swan group, the stable splitting of BP involves a transfer summand. Proof. F = is a self-centralising, maximal elementary abelian * *sub- group. By Theorem 0.1 of [?], the Steinberg summand L(n + 1) of BF is a_transfe* *r_ summand of BP . |__| ALMOST ALL EXTRASPECIAL p-GROUPS ARE SWAN GROUPS 5 Remark 3.6. Ogawa's proof of Theorem ??hinges around the following claim: if M is a maximal subgroup of an extraspecial 2-group P 6= D8, then P acts trivially* * on H*(M). As observed in [?], the proof of this claim in [?] uses inflation incorr* *ectly. We shall now see that the claim is false for 21+2n+. Let P be D8*. .*.D8, or p1+2n+if p is odd. Let M be CP (B1), an index p subg* *roup of P . Then F = is maximal elementary abelian in M and in P . Let fi1; : :;:fin; fl be the dual basis of F *, which we embed in H2(F ) by tak* *ing the first Chern class. Then S(F *) H*(F ). Define i 2 H2pn-1(M) to be the Evens norm NMF(fl), and set g = A1. Let U be the subspace of F *spanned by fi2; : :;:* *fin. By standard properties of the Evens norm map (see [?, Ch. 6]), we have: Y Y ResMF(j) = (fl + u) and so g* ResMF(j) = (fl + fi1 + u) : u2U u2U These restrict to as flpn-1 and (fl + fi1)pn-1 respectively. So ResMF(* *g*j) differs from ResMF(j), which means that j is not invariant under A1. Dept of Mathematics, Univ. of Wuppertal, D-42097 Wuppertal, Germany E-mail address: green@math.uni-wuppertal.de Dept of Mathematics, University of Hue, 27 Nguyen Hue, Hue, Vietnam