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