ON THE COHOMOLOGY AND

HOMOTOPY OF SWAN GROUPS

John Martino and Stewart Priddy

1. Introduction.

It is classical that the mod-p cohomology ring of a finite group G maps injectively into the ring of invariants of the cohomology of a Sylow p-subgroup P, i.e.,

H^{∗}(G; F_{p}) ↣^{res} H^{∗}(P; F_{p})^{W_{G}(P)}

where W_{G}(P) = N_{G}(P)/P C_{G}(P) and cohomology is taken with simple coefficients in F_{p}. In this paper we study the class of finite p-groups P for which this map is always an isomorphism, that is, the mod-p cohomology of any finite group G with P as Sylow p-subgroup is this ring of invariants. We call such p-groups Swan groups because abelian p-groups were shown to have this property by an early result of R. Swan [Sw]. At first this may seem a highly specialized and restrictive condition, however, it is known that in an appropriate sense almost all p-groups have this property [HP].

There is a topological condition on the classifying space BP of a finite p-group P that ensures P is a Swan group. The stable homotopy type of BP decomposes, via maps which may or may not involve the transfer, into a wedge sum of indecomposable spectra. If the transfer plays no role in the stable decomposition of BP then P is a Swan group (Theorem 3.3). This condition may be viewed as a first level in studying the cohomology or homotopy of groups by considering the extent to which the transfer is involved. The question of whether the transfer is involved in the stable decomposition of BP is also essential to understanding when BP itself is indecomposable.

Besides the introduction the paper is divided into three sections. In Section 2 we use Alperin’s fusion theorem as strengthened by Goldschmidt to establish a sufficient condition (Theorem 2.3) for a p-group to be a Swan group. Swan’s original result follows directly from this result. We also show that for p odd every metacyclic p-group is a Swan group. Section 3 is more topological in nature. We establish that if the classifying space BP has no stable summands involving the transfer then P

Typeset by AMS-TEX

is a Swan group. We also give ways (Theorems 3.7, 3.8) to modify our result of the previous section to ensure BP has no such transfer summands. These results are shown to apply in various cases. In the final section we observe that almost all p-groups are Swan groups. All cohomology groups are taken with simple coefficients in F_{p}.

We are grateful to Jon Alperin, Bill Dwyer, and Richard Lyons for correspondence on this topic and we especially thank George Glauberman for supplying the proof, given in Section 2, that M_{3}(p) is a Swan group.

2. Fusion.

In this section we study the relation between fusion and Swan groups. Elements of a subgroup which are conjugate in a supergroup are said to fuse in the larger group.

For a fixed prime p, we recall that a subgroup H ≤ G is said to control strong p-fusion in G if H contains a Sylow p-subgroup of G and if Q, gQg^{−1} ≤ H for Q a p-subgroup of G then g = hc where h ∈ H, c ∈ C_{G}(Q). This notion has the following well known implication in cohomology.

Proposition 2.1. If H ≤ G controls strong p-fusion in G then

res : H^{∗}(G) ⟶^{≈} H^{∗}(H)

is an isomorphism.

Proof. It is immediate from the definition that both cohomology groups have the same stable elements in the sense of Cartan-Eilenberg [CE]. □

Corollary 2.2. P is a Swan group if N_{G}(P) controls strong p-fusion in G for each group G containing P as a Sylow p-subgroup.

Let G be a finite group with P ∈ Syl_{p}(G), fixed. An important result of Alperin’s work on fusion [A] was to show that p-fusion of subsets of P is controlled locally i.e., by subgroups N_{G}(Q) for Q a p-subgroup which is a tame intersection. We recall Q is a tame intersection if Q = P ∩ R where R ∈ Syl_{p}(G) and N_{P} (Q), N_{R}(Q) ∈ Syl_{p}(N_{G}(Q)). He also established that such Q could be required to satisfy C_{P} (Q) = Z(Q). Later Goldschmidt [Gs] refined this to require that Q also be a Sylow psubgroup of O_{p′,p}(N_{G}(Q)). We recall O_{p}(G) is the largest normal p-subgroup of G, O_{p′}(G) is the largest normal p′-subgroup of G, and O_{p′,p}(G) = π^{−1}O_{p}(G/O_{p′}(G)) where π : G → G/O_{p′}(G) is the projection. This has immediate implications for Swan groups. We adopt the notation

W_{G}(Q) = N_{G}(Q)/QC_{G}(Q).

so that W_{G}(Q) ≤ Out(Q) where Out(Q) = Aut(Q)/Inn(Q). Let Z(Q) denote the center of Q.

Theorem 2.3. Let P be a p-group. If for every proper subgroup Q < P such that C_{P} (Q) = Z(Q),

W_{P} (Q) ∩ O_{p}(Out(Q)) ≠ 1

then P is a Swan group.

We immediately recover Swan’s result [Sw; Lemma 1] as a trivial corollary.

Corollary 2.4. An abelian p-group is a Swan group.

Remark 2.5. If Out(Q) is a p-group then the hypothesis of Theorem 2.3 is clearly satisfied since O_{p}(Out(Q)) = Out(Q). Thus the hypothesis need only be checked for proper subgroups Q < P where C_{P} (Q) = Z(Q) and Out(Q) is not a p-group. However, in the sense of U. Martin [Ma] almost all p-groups Q have Out(Q) a p-group. See Section 4 for the definition of “almost all.”

Lemma 2.6. If Q ≤ G is a p-group and C_{G}(Q)/Z(Q) is a p′-group then

C_{G}(Q) = O_{p′}(N_{G}(Q)) × Z(Q).

Proof. Since C_{G}(Q)/Z(Q) is a p′-group

1 → Z(Q) → C_{G}(Q) → C_{G}(Q)/Z(Q) → 1

splits by Zassenhaus. Thus, letting K = O_{p′}(C_{G}(Q)), we have C_{G}(Q) = K × Z(Q). It remains to show K = O_{p′}(N_{G}(Q)). From the definitions the commutator group [O_{p′}(N_{G}(Q)), Q] ≤ O_{p′}(N_{G}(Q)) ∩ Q = 1. Thus O_{p′}(N_{G}(Q)) ≤ C_{G}(Q) and O_{p′}(N_{G}(Q)) ≤ K. If n ∈ N_{G}(Q) then nKn^{−1} ≤ C_{G}(Q). Let c ∈ C_{G}(Q) and set γ = n^{−1}cn ∈ C_{G}(Q). Then cnKn^{−1}c^{−1} = nγKγ^{−1}n^{−1} = nKn^{−1}. Thus nKn^{−1} is a normal p′-subgroup of C_{G}(Q), i.e. nKn^{−1} = K. Therefore K is normal in N_{G}(Q) and K ≤ O_{p′}(N_{G}(Q)). □

Proof of Theorem 2.3. Assume G is a finite group with P ∈ Syl_{p}(G), Q < P, and C_{P} (Q) = Z(Q). Then W_{P} (Q) ≤ W_{G}(Q) ≤ Out(Q) therefore 1 ≠ W_{P} (Q) ∩ O_{p}(Out(Q)) ≤ W_{G}(Q)∩O_{p}(Out(Q)) which is a normal p-subgroup of W_{G}(Q). Thus O_{p}(W_{G}(Q)) ≠ 1 or equivalently O_{p}(N_{G}(Q)/C_{G}(Q)) > Q/Z(Q).

By Alperin, we may assume Q is a tame intersection. Therefore C_{P} (Q) ∈ Syl_{p}(C_{G}(Q)) and so C_{G}(Q)/Z(Q) is a p′-group. By Lemma 2.6

1 → Z(Q) → N_{G}(Q)/O_{p′}(N_{G}(Q)) → N_{G}(Q)/C_{G}(Q) → 1

is exact. It follows that Q is not a Sylow p-subgroup of O_{p′,p}(N_{G}(Q)). By the result of Goldschmidt only N_{G}(P) is needed to control strong p-fusion in G and so P is a Swan group. □

We now turn to showing the existence of another large class of Swan groups, namely the metacyclic p-groups for p odd, in addition to the abelian ones discovered by Swan. Except for the metacyclic groups of order p^{3} and exponent p^{2}, i.e.,

M_{3}(p) = ⟨x, y|x^{p^{2}} = y^{p} = 1, yxy^{−1} = x^{p+1}⟩

this result follows from work of J. Dietz. The result for M_{3}(p) is due to G. Glauberman who kindly supplied us with a proof.

Theorem 2.7 [Dietz [D], Glauberman [G]]. If P is a metacyclic p-group of odd order then P is a Swan group.

Proof. If P is a metacyclic p-group, P ≉ M_{3}(p), then P is a Swan group by [D]. This can be seen as follows: let Q ≤ P and K = Ker ϕ,

ϕ : Out(Q) ⟶ Aut(Q/Φ(Q)).

where ϕ(P) is the subgroup of P generated by commutators and p-th powers. If Q < P is a proper subgroup then W_{P} (Q) ∩ K ≠ 1 [D] so in particular, W_{P} (Q) ∩ O_{p}(Out(Q)) ≠ 1. Thus by Theorem 2.3 P is a Swan group.

The case of P ≈ M_{3}(p) is more complicated. In the following proof we will use ( )′ to denote the commutator subgroup.

Lemma 2.8. If P ≈ M_{3}(p) is a Sylow p-subgroup of G and O_{p′}(G) = 1 then there exists K ◁ G such that K ∩ P is a cyclic subgroup of P, i.e., |K| = p^{2}.

Proof. Since p > 2, P is regular, hence by Wielandt’s theorem [H, Thm. 14.4.2; Hu, Thm. IV.8.1] G′ ∩ P = N′ ∩ P where N = N_{G}(P). Let C = C_{G}(P) then N/CP ≤ Out(P) is a p′-group and hence is of order k | p − 1 since |Out(P)| = p^{r}(p − 1) for some r ≥ 0.

If k = 1, then N′ = P′ hence G′ ∩ P = N′ ∩ P = P′. Thus P/P′ ≈ G/G′ and so H_{1}(P; F_{p}) ≈ H_{1}(G; F_{p}). Tate’s theorem [T] implies P has a normal p-complement. By hypothesis, O_{p′}(G) = 1 thus P = G and the result is clear.

Now assume k > 1. Since P C centralizes P/P′, N/P C induces a p′-group of automorphisms on P/P′ ≈ (Z/p)^{2}. Let Q = Ω_{1}(P), the subgroup generated by elements of order p. Then Q char P, hence Q ◁ N. Thus N/P C stabilizes Q/P′ and hence by Maschke’s theorem it also stabilizes a complement R/P′ of Q/P′ in P/P′. Then R ≤ P is cyclic of order p^{2}. Since Aut(R) is abelian, N′ centralizes R. Thus N′ ∩ P ≤ R since C_{P} (R) = R. If N′ ∩ P = R then G′ ∩ P = R and we are done. Otherwise, if N′ ∩ P ≠ R then N′ ∩ P = P′. Thus H_{1}(P; F_{p}) ≈ H_{1}(G; F_{p}) and P = G by Tate. □

Returning to the proof of the theorem, by Corollary 2.2, it suffices to show N = N_{G}(P) controls strong p-fusion of P in G. It is sufficient to show N controls strong p-fusion of Q in N_{G}(Q) where Q ≤ P satisfies C_{P} (Q) ≤ Q and N_{P} (Q) ∈ Syl_{p}(N). The only possibilities for Q are (Z/p^{2}) and (Z/p)^{2} both of which are normal in P. Thus P ≤ N_{G}(Q) and so we may reduce to the case G = N_{G}(Q).

Let C = C_{G}(Q) ◁ N then Q = C_{P} (Q) = C ∩ N_{P} (Q) ∈ Syl_{p}(C). Now Q is in the center of C hence Q ≤ ZN_{C}(Q). By Burnside’s theorem [Gr, Thm. 7.4.3], C has a normal p-complement, O_{p′}(C), which is characteristic in C. Since C ◁ G, O_{p′}(C) ≤ O_{p′}(G) = 1, therefore C = Q. Hence

P/Q ≤ G/Q = N/C ≤ Aut(Q).

If Q ≈ Z/p^{2}, Aut(Q) is abelian so P/Q◁G/Q. Thus P ◁G and the result follows. If Q ≈ (Z/p)^{2}, Aut(Q) ≈ GL_{2}(F_{p}). By Lemma 2.8 there exits K ◁ G with

K ∩P ≈ Z/p^{2}. Hence G/Q stabilizes K ∩P ≈ Z/p^{2}. In terms of invertible matrices G/Q is upper-triangular. Since such matrices have a normal Sylow p-subgroup of order p and since P/Q ≈ Z/p we have P/Q ◁ G/Q and P ◁ G. □

Remark. For p = 2 this result fails by considering P ≈ D_{8}, the dihedral group of order 8, a Sylow 2-subgroup of Σ_{4}. Then, W_{Σ_{4}}(P) = 1, while H^{1}(Σ_{4}) = Z/2 ≠ H^{1}(P) and so P is not a Swan group.

Topologically there is a close relationship between fusion and homotopy equivalence. Let BG_{p}^{∧} denote the p-completion of Bousfield-Kan.

Theorem 2.9 [MP4, Cor. 1.2]. Let G, G′ be two finite groups. Then BG_{p}^{∧} is homotopy equivalent to BG′_{p}^{∧} if and only if there is an isomorphism f : P ⟶^{≈} P′, P ∈ Syl_{p}(G), P′ ∈ Syl_{p}(G′) satisfying the following conditions:

(1) if Q_{1} and Q_{2} are subgroups of P then Q_{1} and Q_{2} are G-conjugate if and only if f(Q_{1}) and f(Q_{2}) are G′-conjugate

(2) W_{G}(Q) and f^{−1}W_{G′}(f(Q))f are conjugate in Out(Q) for every subgroup Q of P.

In the next section we study Swan groups from the point of view of their classifying spaces.

3. Transfer summands.

In this section we study when the transfer is involved in the stable decomposition of the classifying space.

If P is a Sylow p-subgroup of G then N_{G}(P)/P has order prime to p. Thus to say P is a Swan group is to require an isomorphism

res : H^{∗}(G) ⟶^{≈} H^{∗}(N_{G}(P))

for every G with a Sylow p-subgroup isomorphic to P. Equivalently on the level of classifying spaces, the inclusion is required to induce a homotopy equivalence

(∗)  Bi : BN_{G}(P)_{p}^{∧} ⟶^{≃} BG_{p}^{∧}

completed at p. For the classifying spaces of finite groups localization and completion at p agree. For our purposes it is more convenient to work in the stable category. Equation (∗) holds unstably if and only if it holds stably. For the remainder of this section we shall work stably and assume all spectra have been localized at p.

BN_{G}(P) ⟶^{Bincl} BG ⟶^{tr} BN_{G}(P)

is, in cohomology, multiplication by the index [G : N_{G}(P)]. So BG is a summand of BN_{G}(P). Similarly BN_{G}(P) is a summand of BP. It is known that BP decomposes into indecomposable summands

BP ≃ X_{1} ∨ X_{2} ∨ ⋯ ∨ X_{n}

unique up to equivalence and order of summands [N, MP2]. Thus we are reduced to studying when BN_{G}(P) and BG have the same summands.

A summand X of BP originates in BQ, for Q ≤ P if X is a summand of BQ but not a summand in BR for R < Q [MP2]. Every summand of BP originates in BQ for some subgroup Q unique up to isomorphism. We recall from [MP3] that furthermore BP has three mutually exclusive types of indecomposable summands X; namely those which

(1) originate in BP,

(2) involve transfer, i.e., there is an equivalence of the form

X → BP ⟶^{tr} BQ ⟶^{Br} BR → X

where Q < P and r is a retraction. Moreover there is no such equivalence involving transfer to a subgroup of Q,

(3) do not involve transfer, i.e., there is no equivalence of the form given in (2) but there is one of the form

X → BP ⟶^{Br} BR → X

where r is a retraction.

Without loss of generality we may assume that in (2) and (3) X originates in BR. In the terminology of [MP3] summands of types (1) and (3) are linked in BP while those of type (2) are linked in BQ. We shall call a summand of type (2) a transfer summand .

In general an indecomposable summand X which is linked in BQ, Q ≤ P, corresponds to a simple R(Q)-module M where R(Q) is a certain finite ring over F_{p} associated with the endomorphisms of Q (see [MP3; Section 2]). In fact R(Q) is a quotient ring of F_{p}ÕEnd(Q), where OEnd(Q) = End(Q)/Inn(Q) and F_{p}ÕEnd(Q) is the contracted monoid (i.e. no zero homomorphism) ring. By an abuse of the notation of [MP3] we denote by \overline{J}_{τ} (Q) the kernel of F_{p}ÕEnd(Q) ⟶ R(Q).

The group ring F_{p}Out(Q) ⊆ R(Q) is a retract. Again by an abuse of the notation of [MP3] we denote the kernel of R(Q) ⟶ F_{p}Out(Q) by J(Q). Thus there is a split short exact sequence of the form

1 → J(Q) → R(Q) →^{↶} F_{p}Out(Q) → 1.

A primitive idempotent e ∈ R(Q) restricts to either a primitive idempotent or zero in F_{p}Out(Q). Furthermore, if e restricts to a primitive idempotent the corresponding simple module is merely the F_{p}Out(Q)-module with J(Q) acting as zero. A simple module where the corresponding conjugacy class of primitive idempotents restricts non-zero in F_{p}Out(Q) is in correspondence with a summand of type (1), while if the idempotent restricts to zero the simple module is in correspondence with a summand of type (3). Hence the summands of type (1) are in one-to-one correspondence with simple F_{p}Out(Q)-modules.

The following result is useful in determining summands in BG.

Theorem 3.1 [MP3; Thm. 3.4]. Let X be an indecomposable summand of BP which is linked in BQ, Q ≤ P with corresponding simple R(Q)-module M and set

\overline{W} = ∑ c_{x}

where x ∈ N_{G}(Q)/Q. Then X is a summand of BG if and only if \overline{W} M ≠ 0. Furthermore, M may be taken to be a simple F_{p}Out(Q)-module if X originates in BQ.

Remark 3.2. We note that \overline{W} M = 0 unless C_{P} (Q) = Z(Q). Therefore we may restrict our attention to summands linked in BQ for such self-centralizing subgroups Q (i.e. C_{P} (Q) = Z(Q)).

Corollary 3.3. If P is an abelian p-group then BP has no transfer summands.

Proof. Let Q ≤ P. Since P is abelian, c_{x}(q) = q for x ∈ P. Thus if Q is proper \overline{W} = 0 and the result follows by Theorem 3.1. □

Remark 3.4. John Harris and Nick Kuhn in their paper [HK] effectively show that if P is abelian then BP has no transfer summands.

We can now establish the connection between summands and the notion of a Swan group.

Theorem 3.5. If BP has no transfer summands then P is a Swan group.

Proof. By the hypothesis all the summands of BP are linked in BP. Therefore we can apply Theorem 3.1 with Q = P. However,

N_{N_{G}(P)}(P) = N_{G}(P)

therefore BN_{G}(P) and BG have the same summands. Hence res : H^{∗}(G) ⟶^{≈} H^{∗}(N_{G}(P)) is an isomorphism. □

Remark 3.6. The converse of Theorem 3.5 is not true. Let P be a metacyclic pgroup of odd order. Then P is a Swan group by Theorem 2.7. If P ≉ M_{3}(p) then a careful perusal of [D] shows BP has no transfer summands. However, if P ≈ M_{3}(p) then BP does have transfer summands [D], but P is still a Swan group. In fact the “Steinberg” summand L(2) of B(Z/p)^{2} is such a summand. To see this let Q = ⟨x^{p}, y⟩ ∈ P then Q ≈ (Z/p)^{2} is self-centralizing in P of maximal p-rank. By [P, Thm. 0.1] this is a necessary and sufficient condition for L(2) to be a summand of BP coming by transfer.

There is a useful criterion for determining when an element (e.g. \overline{W}) is in the Jacobson radical. Let G be a finite group and K ⊴ G be a normal subgroup. We recall that the augmentation ideal I(K) is nilpotent if and only if K is a psubgroup [Pa; p. 70]. Now suppose H be a p-subgroup of G. Denote \overline{H} = ∑_{h∈H} h. If H ∩ O_{p}(G) ≠ 1 then \overline{H} ∈ Rad(F_{p}G). This follows since the augmentation ideal I(O_{p}(G)) is nilpotent. If H ∩ O_{p}(G) ≠ 1 then \overline{H} ∈ I(O_{p}(G)) ⊆ Rad(F_{p}G).

The Frattini subgroup Φ(P) of P is generated by commutators and p-th powers so that P/Φ(P) = P/[P, P] ⊗ Z/p. Since Φ(P) is characteristic we have an exact sequence

1 → K → Out(P) ⟶^{ρ} Aut(P/Φ(P))

where K = ker(ρ). By a well-known theorem of P. Hall [Hu], K is a p-group. So the augmentation ideal I(K) is nilpotent. We are now ready to establish the main results of this section: group theoretic criteria for the classifying space to have no transfer summands.

Theorem 3.7. Let P be a finite p-group. If for each proper subgroup Q such that C_{P} (Q) = Z(Q),

[P, P] ∩ Q ≤ Φ(Q)

then BP has no transfer summands.

Theorem 3.8. Let P be a finite p-group. If for each proper subgroup Q where C_{P} (Q) = Z(Q), W_{P} (Q) ∩ O_{p}(Out(Q)) ≠ 1 and either

(1) Q has no non-trivial retracts or

(2) Out(Q) → Aut(Q/[Q, Q]) is surjective, then BP has no transfer summands.

Remark 3.9. The hypotheses of both Theorems 3.7 and 3.8 imply the hypothesis of Theorem 2.3. This is clear for Theorem 3.8. If [P, P] ∩ Q ≤ Φ(Q) then as will be observed in the proof of Theorem 3.7, W_{P} (Q) ∩ K ≠ 1 and K ≤ O_{p}(Out(Q)).

Proof of Theorem 3.7. Consider the induced homomorphism of semigroup rings

ϕ : F_{p}ÕEnd(Q) → F_{p}Ẽnd(Q/Φ(Q)).

The simple modules of a semigroup ring are in one-to-one correspondence with the simple modules of the maximal subgroup rings [HK]. Further, if S ≤ Out(Q) is a maximal subgroup then there is a retract T of Q so that S = Out(T). Since T is a retract, Φ(T) = Φ(Q) ∩ T. Thus ϕ restricts to the induced homomorphism F_{p}Out(T) → F_{p}Aut(T/Φ(T)). As has already been observed the kernel of this map is a nilpotent ideal. So the simple modules of F_{p}ÕEnd(Q) are in one-to-one correspondence with the simple modules of Im ϕ. Thus the kernel of ϕ is nilpotent.

Let M be a simple R(Q)-module. By the hypothesis, c_{x}(q) = q mod Φ(Q) for x ∈ N_{P} (Q). Since W_{P} (Q) is a p-group, a simple calculation shows that \overline{W} = ∑_{x∈W_{P} (Q)} c_{x} ∈ F_{p}Out(Q) is in I(K), the augmentation ideal of K = ker(ρ : Out(Q) → Aut(Q/Φ(Q)). Here we have used that W_{P} (Q) = N_{P} (Q)/Q since by hypothesis C_{P} (Q) = Z(Q). Since I(K) ⊆ Ker ϕ is nilpotent, \overline{W} generates a nilpotent ideal in R(Q). Therefore \overline{W} M = 0 and the result follows from Theorem 3.1. □

Proof of Theorem 3.8. If Q has no non-trivial retracts then every summand linked in BQ originates in BQ, so the only simple modules of R(Q) are the simple modules of F_{p}Out(Q).

Suppose now that Out(Q) → Aut(Q/[Q, Q]) is surjective so the Jacobson radical maps into the Jacobson radical. Consider the induced homomorphism

ϕ : F_{p}ÕEnd(Q) ⟶ F_{p}Ẽnd(Q/[Q, Q]).

The same argument as in the proof of Theorem 3.7 shows that the kernel of ϕ is nilpotent (merely replace Φ(−) with [−, −] throughout). Therefore, since R(Q) is a quotient of F_{p}ÕEnd(Q), we have that the induced homomorphism

ϕ̃: R(Q) ⟶ Im ϕ/ϕ(\overline{J}_{τ} (Q))

has a nilpotent kernel.

We have the following string of implications:

W_{P} (Q) ∩ O_{p}(Out(Q)) ≠ 1 ⟹ \overline{W} ∈ Rad(F_{p}Out(Q))

⟹ ϕ(\overline{W}) ∈ Rad(F_{p}Aut(Q/[Q, Q])) (by hypothesis)

⟹ ϕ(\overline{W}) ∈ Rad(F_{p}Ẽnd(Q/[Q, Q]))
⟹ ϕ̃(\overline{W}) ∈ Rad(Im ϕ/ϕ(\overline{J}_{τ} (Q)))
⟹ \overline{W} ∈ Rad(R(Q))

The first implication follows from the remarks before Theorem 3.7. Harris and Kuhn [HK] show that if A is abelian a simple module of F_{p}Ẽnd(A) restricts to a simple module of F_{p}Aut(A). Hence if x ∈ Rad(F_{p}Aut(A)) then x ∈ Rad(F_{p}Ẽnd(A)), the third implication. The fourth follows since Im ϕ/ϕ(\overline{J}_{τ} (Q)) is a subquotient of F_{p}Ẽnd(Q/[Q, Q]). The last implication follows since Ker ϕ̃ is nilpotent.

In either case if \overline{W} ∈ Rad(F_{p}Out(Q)) then \overline{W} M = 0 for every simple R(Q)- module M. So by Theorem 3.1 BP has no transfer summands. □

Example 3.10. As an application of Theorem 3.7 we consider the extra-special 2groups of quaternion type E(2n)^{−} [Q,FP]. These groups are characterized by a central extension

1 → Z/2 → E(2n)^{−} → (Z/2)^{2n} → 1

with extension class ∑ x_{i}^{2}+∑ x_{i}x_{j}, e.g. E(2)^{−} ≈ Q_{8}, the quaternion group of order 8. Since Z(P) = [P, P] = Z/2, it is easy to see that every proper self-centralizing subgroup Q satisfies the hypothesis of the theorem. Thus E(2n)^{−} is a Swan group.

Example 3.11. The non-abelian group P ≈ Z/4 ⋊ Z/4 (i.e. P ≈ 16Γ_{2}c_{2} in the notation of Hall-Senior [HS]) satisfies the hypothesis of Theorem 3.8 but not the hypothesis of Theorem 3.7. There is a proper subgroup Q ≈ Z/2 × Z/4 such that C_{P} (Q) = Z(Q), where [P, P] ∩ Φ(Q) = 1 but [P, P] ∩ Q ≈ Z/2. So the hypothesis of Theorem 3.7 does not hold while the hypothesis of Theorem 3.8 clearly holds since all the proper subgroups of P are abelian. The stable homotopy type of BP is analyzed in [MP2].

4. Almost all p-groups are Swan groups.

In this section we apply some previous results on the indecomposability of classifying spaces to observe that almost all p-groups are Swan groups.

Let Ω_{k}(P) ≤ P be the subgroup generated by elements of order p^{k} or less.

Theorem 4.1 [HP; Thm. 1.9, Prop. 1.6.1]. Let P be a p-group. If [P, Ω_{k+1}(P)] ≤ Ω_{k}(P), k ≥ 0 then P is a Swan group. If p > 2, the conclusion holds if [P, Ω_{1}(P)] = 1, i.e., if elements of order p are central.

Remark 4.2. The extra-special 2-groups of quaternion type E(2n)^{−} do not satisfy the hypothesis of Theorem 4.1 when n ≥ 2. Theorems 3.7 and 4.1 are distinct criteria for BP to have no transfer summands.

We recall the notion of almost all for p-groups: The Frattini series of G

G_{1} ▷ G_{2} ▷ ... ▷ G_{n} ▷ G_{n+1} ▷ ...

is given by G_{1} = G, G_{n+1} = (G_{n})^{p}[G, G_{n}], n ≥ 1. A group is said to have Frattini length n if G_{n+1} = 1 ≠ G_{n}.

Then G_{2} = Φ(G) is the Frattini subgroup of G and G/G_{2} = H_{1}(G; F_{p}) is an elementary abelian p-group. If G is a p-group then d, the minimum number of generators of G, is the rank of G/G_{2}.

Following Martin [Ma] we define “almost all”.

Let A_{d,n} be the (finite) set of isomorphism classes of p-groups of Frattini length n generated by a minimum of d elements. Given a property S of groups (invariant under isomorphisms) let S_{d,n} ⊂ A_{d,n} be the subset of elements having property S. Then we say almost all p-groups of Frattini length n have property S if

lim_{d→∞} \frac{|S_{d,n}|}{|A_{d,n}|} = 1.

It is known [Ma] that |A_{d,n}| → ∞ as soon as n ≥ 2. Martin originally used this definition to show almost all p-groups have Aut(P) a p-group.

Corollary 4.3. Almost all p-groups are Swan groups.

Proof. By [HP;Thm. 1.8], almost all p-groups satisfy the hypothesis of Theorem 4.1. Thus by Theorem 3.3 almost all p-groups are Swan groups. □

Remark 4.4. In fact more is true. Namely, for almost all non-abelian p-groups P, the stable classifying space BP is indecomposable. This follows from [HP; Cor.

1.5]. We recall that a summand X is called indecomposable if X ≠ Y ∨ Z for non-trivial summands Y, Z. Thus if BP is indecomposable and G is a finite group with P ∈ Syl_{p}(G) we have BP ≃ BG_{p}^{∧} and, in particular, H^{∗}(P) ≈ H^{∗}(G). The properties of indecomposable BP can be summarized in the terms used above as follows.

Proposition 4.5. Let P be a p-group. Then BP is indecomposable if and only if (1) Out(P) is a p-group,

(2) P has no non-trivial retracts,

(3) BP has no transfer summands.

Proof. This is a restatement of Theorem 1.7 of [HP] using the notion of transfer summand. More directly, condition (1) implies there is only one summand originating in BP since the trivial module is the only simple module of F_{p}G when G is a finite p-group. Condition (2) implies there are no summands of type (3). So if there are no transfer summands BP is indecomposable. The other direction is clear. □

Closely related to the concept of a Swan group is the idea of p-nilpotence. A finite group G is p-nilpotent if it has a normal p-complement. By a theorem of Tate [T] G has normal p-complement if and only if H^{∗}(G; F_{p}) ≈ H^{∗}(P; F_{p}) where P is a Sylow p-subgroup of G. Indeed, every finite group G with a Sylow p-subgroup isomorphic to P is p-nilpotent if and only if P is a Swan group with automorphism group Aut(P) a p-group.

References

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John Martino  Stewart Priddy
Department of Mathematics and Statistics  Department of Mathematics
Western Michigan University  Northwestern University
Kalamazoo, MI 49008  Evanston, IL 60208
email: martino@wmich.edu  email: s priddy@math.nwu.edu