Proceedings of Symposia in Pure Mathematics Volume 00, 0000

Applications of stable classifying space theory to group cohomology

Stewart Priddy

0. Introduction

The 1996 AMS Summer Research Institute brought together a diverse group of mathematicians from algebra, group theory, and homotopy theory all sharing an interest in group cohomology and representation theory. Numerous interactions between these fields have occurred from the beginning and the time seemed ripe to explore some of the more recent connections. Our purpose here is to discuss some aspects of this interaction which we express diagramatically as follows

\{ \begin{matrix}Stable homotopy theory \\ of BG at the prime p \end{matrix}\} ⇄ \{ \begin{matrix}mod p cohomology and modular \\ representation theory of G \end{matrix}\}

We shall concentrate on the right arrow; thus this paper is a primer for algebraists interested in topological applications in their domain. We have tried to give enough references so that an algebraist can find the relevant background material.

The first two sections give a brief introduction to classifying spaces, their suspensions, and the topological transfer map. Analogies with the corresponding notions in group cohomology are emphasized. Section three discusses Carlsson’s solution of Segal’s Burnside ring conjecture. In sections four and five, this development leads to the use of modular representation theory to study decompositions of classifying spaces in a manner entirely analogous to decomposing modules in the essentially unique form given by the classical Krull-Schmidt theorem. It is these decompositions which give rise to new approaches in group cohomology. Sections six and seven discuss two such developments: Swan groups and the ring of universally stable elements.

All groups in this paper are finite unless otherwise stated.

\begin{matrix}c \\ ◯ \end{matrix}0000 American Mathematical Society 0082-0717/00 $1.00 + $.25 per page

1991 Mathematics Subject Classification. Primary 55R35; Secondary 20J06, 55P42.

1. Classifying spaces

From our point of view a classifying space for a group G is a topological space whose cohomology is the group cohomology of G. This can be constructed as follows: Let EG be a contractible space on which G acts freely then define BG = EG/G. Many such spaces occur naturally for example, BZ = R/Z = S^{1}, the circle and BZ/2 = S^{∞}/(Z/2), the real projective space where S^{∞} = ∪S^{n}, each sphere being embedded in the next equatorially. In general however it is useful to have a functorial model for BG so that if h : G → G′ is a homomorphism Bh : BG → BG′ is an induced map. Such a construction is classical [Mn] but algebraists may prefer a categorical treatment using simplicial sets given by Segal [S]. It is obtained by viewing G as a category with one object and morphism set G. Then BG is defined as the geometric realization of the nerve of this category.

In any event we have a functor B : {groups and homomorphisms } → {topological spaces and continuous maps }. For the algebraists the salient fact is that in singular homology (or cohomology)

H_{∗}(BG; Z) = T or_{∗}^{Z}(Z, Z) = H_{∗}(G; Z)

the group homology of G. This is immediate since the singular chain complex C_{∗}(EG) is free resolution of Z as a trivial Z(G) module, hence

T or_{∗}^{Z}(Z, Z) = H_{∗}(C_{∗}(EG) ⊗_{Z(G)} Z)) = H_{∗}(C_{∗}(BG)) = H_{∗}(BG; Z)

As expected Bh_{∗} : H_{∗}(BG : Z) → H_{∗}(BG′; Z) agrees with the usual homomorphism h_{∗} : H_{∗}(G; Z) → H_{∗}(G′; Z) in group homology.

From a homotopy theorist’s point of view BG is not very exciting since π_{1}BG = G and π_{k}BG = 0 for k > 0. Nevertheless, it is known that for any connected space there is a discrete group (usually infinite) with the same homology [KT].

Missing from this discussion however is the transfer, one of the key tools of group cohomology. In fact it is known to be impossible, in general, to construct a transfer in the category of spaces. One can remedy this by considering suspensions.

2. Suspension and transfer

In homotopy theory spaces are often equipped with a basepoint which this author views as mildly analogous to the zero of a module. For example BG comes equipped with a natural basepoint since in the simplicial construction there is a single zero simplex. Let X be a space with basepoint x_{0} and let I = [0, 1], the unit interval. We define the suspension ΣX to be

ΣX = X × I/(X × 0 ∪ X × 1 ∪ x_{0} × I)

that is X × 0 ∪ X × 1 ∪ x_{0} × I is identified to a point which becomes the basepoint of ΣX.

APPLICATIONS OF STABLE CLASSIFYING SPACE THEORY TO GROUP COHOMOLOGY 3

This can be visualized as follows ( where the thick line x_{0} × I is identified to a point.):

One easily checks that ΣS^{1} = S^{2} and in general ΣS^{n} = S^{n+1}. Intuitively, suspension raises the dimension of holes, more precisely, H_{k}(X) = H_{k+1}(ΣX) for k > 0. Thus suspension preserves homology and the same is true of cohomology. Additionally, suspending tends to make a space nicer or at least to push the complications into higher dimensions (as we will see below iterating the suspension does away with some difficulties altogether). However, suspension does exact a price, namely, cup products are not preserved. In fact, the cup products of positive dimensional classes in H^{∗}(ΣX) are zero. This is, admittedly, a serious loss but there are also several advantages gained by passing to suspensions. First, using the suspension parameter one can add maps and second, one can define a geometric transfer map.

In order to define addition of maps we observe that the appropriate notion of union for spaces with basepoints is wedge X ∨ Y = {(x, y) ∈ X × Y : x = x_{0} or y = y_{0}}. We can add the two inclusions of ΣX into ΣX ∨ ΣX by means of θ : ΣX → ΣX ∨ ΣX defined by

θ(x, t) = ((x, 2t), x_{0}) for t ≤ 1/2
= (x_{0}, (x, 2t − 1)) for t ≥ 1/2

Finally we define the “folding map” ∇ : Y ∨ Y → Y by ∇(x, y_{0}) = x, ∇(x_{0}, y) = y. Now combining these maps we can define a general addition operation: if f, g : ΣX → Y are two maps then f + g : ΣX → Y is the composite

ΣX →^{θ} ΣX ∨ ΣX →^{f∨g} Y ∨ Y →^{∇} Y

In homology it is not hard to see (f +g)_{∗} = f_{∗} +g_{∗}. Spanier’s book [Sp] is a general reference for homotopy theory including the suspension functor.

In order to define the transfer it is necessary to consider the collection of iterated suspensions BG, ΣBG, Σ^{2}BG, ..., Σ^{n}BG, ... from which we define the stable homotopy classes of maps

{BG, BG′} = colim_{n}[Σ^{n}BG, Σ^{n}BG′]

where [X, Y] denotes the usual homotopy classes of basepoint preserving maps. Then {BG, BG′} is an abelian group and we refer to representatives of its classes as stable maps . Now for H ≤ G there exists a stable map

tr : BG → BH

inducing the usual transfer in group homology. This is a rather mysterious map but there are two ways to view it which are intuitive. First we consider the covering projection π : EG/H → EG/G with fiber G/H. Since EG is also an H free space we may consider EG/H as a model for BH. A chain map π_{!} : C_{∗}(EG/G) → C_{∗}(EG/H) is given as follows: by subdividing EG/G sufficiently the inverse image of each simplex σ is N = |G/H| simplices {σ_{1}, ...σ_{N}} lying over it. Then π_{!}(σ) = ∑_{i=1}^{N} σ_{i}. It is well known and intuitively clear that π_{!} induces the ordinary transfer in homology. Of course we are asking for more and it is not clear in what order one should add these simplices on the level of pointset topology. Up to homotopy, addition is commutative (after two suspensions) but this is not sufficient to define a map of spaces. One way around this with considerable group theoretical appeal is to use the classical Frobenius homomorphism ϕ : G → H^{N} ⋊ Σ_{N} which in a sense keeps track of coset representatives of H in G. The induced map on the level of classifying spaces Bϕ : BG → BH^{N} ⋊ Σ_{N} can then be used to construct the desired stable map [KP],[Ad; Chpt.4].

The geometric transfer had many applications but without a precise hold on the maps {BG, BG′} progress was limited. To describe this part of the story we turn to the Burnside ring.

3. The Burnside ring

In 1984, G. Carlsson published the last and principal step in G. Segal’s Burnside ring conjecture, thus giving an effective computation of {BG, BG′}. There are several versions of this conjecture; we will concentrate on the form most convenient for the applications we have in mind.

For finite groups G_{1}, G_{2} let A(G_{1}, G_{2}) be the Grothendieck group of isomorphism classes of finite G_{1} × G_{2} sets which are free as G_{2} sets. Sum is given by disjoint union of sets. It is convenient to think of these as left G_{1} sets with a free right G_{2} action which commutes with that of G_{1}.

In case G_{1} = G_{2}, this group is a ring with product X×_{G}Y where (x · g, y) is identified with (x, g · y).

Generators for A(G_{1}, G_{2}) are given by the transitive sets which may be described as

G_{1} ×_{ρ} G_{2} = (G_{1} × G_{2})/ < h, ρ(h) : h ∈ H >

where ρ : H → G_{2} is a homomorphism and H ≤ G_{1}. The usual Burnside ring A(G_{1}) is the Grothendieck group of isomorphism classes of left G_{1}-sets. There is an augmentation homomorphism ϵ : A(G_{1}, G_{2}) → A(G_{1}) given by ϵ(G_{1} ×_{ρ} G_{2}) = G_{1}/H. We define the augmentation ideal Ã(G_{1}, G_{2}) = ker(ϵ).

In terms of these generators, there is a homomorphism

Ã(G_{1}, G_{2}) → {BG, BG′}

given by

(G_{1}×_{ρ}G_{2}) − (G_{1}/H × G_{2}) ↦ (BG_{1} ⟶^{tr} BH ⟶^{Bh} BG_{2})

This map is an isomorphism after suitable completion. This fundamental result, still referred to as the Segal conjecture, is the basis or at least the motivation for the remaining sections.

Theorem 3.1. [C], [LMM] If P is a p-group

Ã(P, G) ⊗ Z_{p} ⟶^{≈} {BP, BG}

The target is a module over the p-adic integers Z_{p} so the result states an isomorphism results when the source is completed at p as well.

4. Stable splittings of BG

Suppose P is a p-group. One could ask for the possible homologies H_{∗}(BG) of groups G with P as a Sylow p-subgroup. We motivate this point of view by considering decompositions of BG into indecomposable pieces. If X is a space with basepoint it is natural (by analogy with modules) to ask if X = X_{1} ∨ X_{2} for non-trivial spaces X_{i}. If not we say X is indecomposable .

A primitive orthogonal idempotent decomposition of 1 = e_{1} + e_{2} + ... + e_{n} in {X, X} gives rise to a decomposition of X

(4.1)  X = X_{1} ∨ X_{2} ∨ ... ∨ X_{n}

into indecomposable summands. We call this a complete splitting of X . Given e_{i}, X_{i} can be recovered as X_{i} = e_{i}X, the homotopy colimit or mapping telescope

This is an iterated mapping cylinder construction [Sp] in which the cylinders {X × [2k, 2k + 1], k ≥ 0} are identified end to end by setting (x, 2k + 1) equal to (e_{i}(x), 2k + 2) for all k.

In homology H_{∗}(X_{i}) = e_{i}H_{∗}(X) so we have

H_{∗}X = H_{∗}(X_{1}) ⊕ H_{∗}(X_{2}) ⊕ ... ⊕ H_{∗}(X_{n})

Similarly e_{i} can be recovered as projection-inclusion X → X_{i} → X.

Of course (4.1) may be an infinite decomposition, however, if X = BP then the finiteness of the Burnside ring A(P, P) ⊗ Z/p implies that n is finite. Furthermore there is an analogue of the Krull-Schmidt Theorem in this context which states that this decomposition

(4.2)  BP = X_{1} ∨ X_{2} ∨ ... ∨ X_{n}

is unique up to homotopy equivalence [N].

Over a period of years, stable splittings have been accomplished for many families of p-groups. Two early workers included Holzager [H] for Z/p and especially C. Witten [W] who considered Z/2 × Z/2. There followed a series of results based on modular representation theory and/or the Segal conjecture:

· dihedral and quaternion groups, Mitchell-Priddy [MiP1] · elementary abelian p-groups, Mitchell [Mt], [MiP2] · abelian p-groups, Harris-Kuhn [HK] · extra-special p-groups, Feshbach-Priddy [FsP], Benson-Feshbach [BF] · semi-dihedral and quasi-dihedral groups, Martino [M4] · metacyclic p-groups, Dietz [D1], [D2] · 2-groups of order ≤ 16, Martino-Priddy [MP2] · 2-groups of order ≤ 32, Catalano [Ct] · 4 × 4 upper triangular matrices over F_{2}, Martino [M3] · Sylow 2-subgroup of M_{12}, Milgram-Overton [MO]

If G is any group with P as Sylow p-subgroup then the composite

(4.3)  BG ⟶^{tr} BP ⟶^{Bπ} BG

is multiplication by [G : P] in homology. Since [G : P] is prime to p the composite induces an isomorphism in mod-p homology. In analogy with localization of modules there is a functor in homotopy theory called localization at p [Sl, BK] which retains only the properties of the space relating to p. Requiring a map between suspensions to induce an isomorphism in mod-p homology is equivalent to requiring the map induces a homotopy equivalence localized at p. Hence the composite (4.3) is a homotopy equivalence at p and we have

BP = BG ∨ Y

for some Y. Uniqueness of the decomposition (4.1) implies

BG = X_{i_{1}} ∨ X_{i_{2}} ∨ ... ∨ X_{i_{k}}

for some subcollection of the summands of (4.1).

At this stage we can rephrase our question by asking what are the possibilities for such subcollections.

Example 4.4. Let p = 2 and consider the quaternion group P = Q_{2^{n}}, n ≥ 3 of order 2^{n}. One of the early splittings [MiP1] gave

(4.5)  BP = BP SL_{2}(F_{p}) ∨ X ∨ Y

where X = Σ^{−1}BS^{3}/BN(S^{1}) is a stable space which suspends to BS^{3}/BN(S^{1}). Here N(S^{1}) is the normalizer of S^{1} in S^{3}. Y is homotopy equivalent to X. The cohomologies of these summands is easy to deduce from that of P which is classical [CE]

H^{∗}(BP) = F_{2}[x_{1}, y_{1}, w_{4}]/(x^{2} + xy + y^{2}, xy^{2} + x^{2}y)

where the subscripts indicate the cohomological degrees of elements.

H^{∗}(BP SL_{2}(F_{p})) = Λ[x^{2}y] ⊗ F_{2}[w]
H^{∗}(X) = F_{2} < x, x^{2} > ⊗F_{2}[w]
H^{∗}(Y) = F_{2} < y, y^{2} > ⊗F_{2}[w]

Using this one can show [MP1] that if n > 3 there are exactly three possible homotopy types of BG

1. BSL_{2}(F_{p})

2. BSL_{2}(F_{p}) ∨ X

3. BSL_{2}(F_{p}) ∨ X ∨ Y

Each of these occurs, e.g. G = SL_{2}(F_{p}), P GL_{2}(F_{p}), and P respectively. The three possibilities for the cohomology of G are (respectively)

1. Λ[e_{3}] ⊗ F_{2}[w_{4}]

2. F_{2}[a_{1}, w_{4}]/(a^{4})

3. F_{2}[x_{1}, y_{1}, w_{4}]/(x^{2} + xy, y^{3})

The case n = 3 is exceptional, see [MP1]. It is interesting to note that the decompositions of this example are equivalent to the Gorenstein-Walter classification of groups with quaternionic Sylow 2-subgroup [M1].

Other examples for which such a classification has been accomplished are the dihedral, semi-dihedral, quasi-dihedral [MP1] groups and the metacyclic groups for odd primes [DMP]. See also [M4].

As with modules finding decompositions such as (4.1) may be daunting; however, the outer automorphism groups Out(Q), Q ≤ P provide an entrée. Let Inj(Q, G) be the set of conjugacy classes of injective homomorphisms Q → G with G acting by conjugation. It is a basic fact of group homology that conjugation acts as the identity in homology. The corresponding fact in topology states that applying the classifying space functor to an inner automorphism gives a map homotopic to the identity [S]. Thus there is a map of modules

(4.6)  Z_{p}Inj(Q, G) →^{B} {BQ, BG}

Define J(Q, G) ≤ {BQ, BG} to be the submodule generated by maps BQ → BG which factor

BQ → BR → BG

where R ⪇ Q. The Segal conjecture Th. 3.1 immediately implies

(4.7)  {BQ, BG}/J(Q, G) = Z_{p}Inj(Q, G)

Thus (4.6) is a split injection onto a direct summand of {BQ, BG}. Considering the special case G = Q we note that (4.6) is a map of rings

(4.8)  Z_{p}Out(Q) →^{B} {BQ, BQ}

We say that an indecomposable summand X of BQ originates in Q if it is not a summand of some BR for R ⪇ Q. This is equivalent to saying X = eBQ for a primitive idempotent which is a summand of of an idempotent in the image of (4.8). For discussions of original or dominant summands as G. Nishida called them, see [N,BF,MP3].

Since primitive idempotents of Z_{p}Out(Q) correspond bijectively with simple F_{p}Out(Q) modules, we conclude that each original summand X of BQ corresponds uniquely to a simple F_{p}Out(Q) module which we denote by M = M_{X,Q}. This gives a one-to-one correspondence {M = M_{X,Q}} ⇆ {X = X_{M,Q}} between isomorphism classes of simple F_{p}Out(Q) modules and homotopy classes of original summands of BQ.

In Example 4.4, Out(Q_{8}) = GL_{2}(F_{2}). Thus there are two simple modules: the trivial module 1 and the standard two dimensional vector space V over F_{2} which is projective. There is a primitive idempotent decomposition 1 = e_{0} + e_{1} + e_{2} where e_{0} corresponds to 1 and e_{1}, e_{2} correspond to V. The corresponding original summands of BQ_{8} are BSL_{2}(F_{q}), X, Y respectively (see [P1], [MiP]). Thus for BQ_{8} every indecomposable summand is original. For n > 3 only the appropriate BSL_{2}(F_{q}) originates in BQ_{2^{n}}, the rest originate in BQ_{8}.

For a general group the complete stable splitting can be described in terms of summands originating in various subgroups.

Theorem 4.9. [BF], [MP2] If G is a finite group then localized at p there is a complete splitting

BG = ⋁ m_{X}X m_{X} ∈ Z^{+}

together with an algorithm for determining the multiplicities m_{X} of X. Here X = X_{M,Q} is the original summand of BQ corresponding to the simple F_{p}Out(Q) module M and Q runs over the p-subgroups of P ∈ Sly_{p}(G).

Two recent survey articles on stable splittings are [M2], [B].

5. Determining the stable type of BG

It is clear from the Segal conjecture that certain group theoretic information should determine the stable homotopy type of BG at p. Before continuing with applications to cohomology we pause to make this relation explicit. It should be reassuring to algebraists to know that only p-local group theoretic information is involved.

The first result in this direction was due to Nishida.

Theorem 5.1. [N] If BG is stably equivalent to BG′ at p then P ≈ P′ where P ∈ Syl_{p}(G), P′ ∈ Syl_{p}(G′).

The converse is false, of course, since any normal p′ group H ≤ G gives rise to a map BG → BG/H which induces an isomorphism in mod-p homology and therefore a stable homotopy equivalence localized at p. The following necessary and sufficient conditions depend on conjugacy classes of homomorphisms of p-groups into G.

Theorem 5.2. [MP3] For finite groups G, G′, the following are equivalent

1. BG and BG′ are stably equivalent at p.

2. For every p-group Q

F_{p}Rep(Q, G) ≈ F_{p}Rep(Q, G′)

are isomorphic as Out(Q) modules where Rep(Q, G) = Hom(Q, G)/G.

3. For every p-group Q

F_{p}Inj(Q, G) ≈ F_{p}Inj(Q, G′)

are isomorphic as Out(Q) modules.

As an Out(Q) module F_{p}Inj(Q, G) is a sum of permutation modules; this is clear from the decomposition

Inj(Q, G) = ⨆ Out(Q)/W_{G}(Q′)

where Q′ runs over the conjugacy classes of subgroups isomorphic to Q and W_{G}(Q′) = N_{G}(Q′)/Q′ · C_{G}(Q′) ≤ Out(Q′) is the Weyl group of Q′ in G. Thus the stable type of BG ultimately depends on p-local information coming from Weyl groups. As a corollary we have the following strengthen of Nishida’s result

Corollary 5.3. If BG ≅ BG′ then G and G′ have the same number of conjugacy classes of subgroups isomorphic to Q for every finite p-group Q.

In case P is a normal Sylow p-subgroup of G then the single Weyl group W_{G}(P) determines the stable type of BG. This can be made precise with the following definition.

Definition 5.4. Subgroups H, K ≤ G are said to be pointwise conjugate in G if there exists a bijection of sets α : H → K such that α(h) = g_{h}^{−1}hg_{h} for g_{h} ∈ G depending on h.

Theorem 5.5. [MP3] Suppose P is a normal Sylow p-subgroup of both G and G′. Then BG is stably equivalent to BG′ iff W_{G}(P) is pointwise conjugate to W_{G′}(P) in Out(P).

A moment’s reflection shows that H is pointwise conjugate to K in G iff

|H ∩ (g)| = |K ∩ (g)|

for every g ∈ G. Thus by the Mackey formula there is an equality 1_{H}^{G} = 1_{K}^{G} of modular characters. This is the key ingredient in the proof of Theorem 5.5.

Example 5.6. Let p, l be distinct primes with V be a vector space of dimension N = l^{n} over F_{p}. Any two l-groups W_{1}, W_{2} of order N can be embedded in GL(V) via the regular representation hence we have two groups G_{i} = V ⋊ W_{i}, i = 1, 2. If W_{1}, W_{2} have exponent l then they are pointwise conjugate by the uniqueness of cycle decomposition in Σ_{N} ≤ GL(V). Thus BG_{1} is stably equivalent to BG_{2} at p. In particular

H^{∗}(V)^{W_{1}} ≈ H^{∗}(V)^{W_{2}}

as modules over the Steenrod algebra A [SE]. This means, for example, f(x^{p}) = f(x)^{p} if f is the indicated isomorphism. However, it is known that they are not isomorphic as algebras over A [MP3]. See Lee [L] for other examples of this phenomenon.

We recall that a group G is called cyclic mod p if a Sylow p-subgroup P is normal and G/P is cyclic. Let C_{p}(G) denote the collection of cyclic mod p subgroups of G. Then one has the following Minami-Webb formula [Mi]

BG = ⋁_{(H)} \frac{f(H)}{[N_{G}(H) : H]} BH

stably localized at p, where H runs over the conjugacy classes of C_{p}(G) and f : C_{p}(G) ⟶ Z is the Möbius function given by

∑_{J≤K} f(K) = 1

for J, K ∈ C_{p}(G). This expression is taken in a Grothendieck group, that is, multiplying through by a common denominator and moving the negative terms to the left side results in a valid formula

Example 5.7. We consider the Minami-Webb decomposition of BΣ_{6} at p = 2 [Wb].

BΣ_{6} = B(D_{8} × Z/2) + B(A_{4} × Z/2) − B(Z/2)^{3}

It follows from this decomposition and the complete splittings of BD_{8} and B(Z/2)^{2} [MiP] that

BΣ_{6} = B(P SL_{2}(F_{2}) × Z/2)

stably at 2. Thus the cohomologies of these groups are isomorphic as modules over A. Moreover, the cohomology algebras of these groups is known [AMM],[MiP]; inspection shows they are isomorphic as algebras over F_{2}

H^{∗}(BΣ_{6}) ≈ H^{∗}(B(P SL_{2}(F_{p}) × Z/2)) ≈ F_{2}[w_{1}, w_{2}, w_{3}, u_{3}]/(u_{3}^{2} + u_{3}(w_{3} + w_{2}w_{1}))

However it is easily seen that they are not isomorphic as algebras over A (any isomorphism of algebras fails to commute with the Steenrod operations [MP5]).

6. Swan groups

Suppose P is a Sylow p-subgroup of G. We recall that restriction defines an injection

(6.1)  H^{∗}(BG) →^{i^{∗}} H^{∗}(BP)^{W_{G}(P)}

Definition 6.2. P is called a Swan group if i^{∗} is an isomorphism for all groups G with P ∈ Syl_{p}(G).

The motivation for this definition is the following classical result of Swan [S].

Theorem 6.3. If P is an abelian p-group then P is a Swan group.

It is well known in group theory that (6.1) is an isomorphism iff P controls strong p-fusion in the sense of Alperin. Using Goldschmidt’s refinement of Alperin’s Fusion Theorem [A,Gs] we have the following generalization of Swan’s Theorem.

Theorem 6.4. [MP4] P is a Swan group if for every self-centralizing Q ⪇ P

W_{P} (Q) ∩ O_{p}(Out(Q) ≠ 1

Example 6.5. It follows that Q_{8} is a Swan group (the only proper selfcentralizing subgroups are Z/4’s for which the desired condition is easily seen to hold). On the other hand P = Q_{2^{n}}, n > 3, is not a Swan group since Out(P) is a 2-group which implies W_{G}(P) = 1 whereas (6.1) fails to be an isomorphism for G = SL_{2}(F_{p}), p an odd prime such that P ∈ Syl_{2}(G). Similarly one can see D_{8} is not a Swan group by considering Σ_{4}.

In addition to abelian p-groups there are other large classes of Swan groups

Theorem 6.6. [DMP],[G] If p is an odd prime then every metacyclic p-group is a Swan group.

Theorem 6.7. [HP] A p-group P with a central Ω series is a Swan group.

.

We recall Ω_{k}(P) =< x ∈ P : x^{p^{k}} = 1 > and require [P, Ω_{k+1}(P)] ≤ Ω_{k}(P) for all k ≥ 0. In fact initial conditions suffice to insure centrality.

Corollary 6.8. [HP] For p odd, if every element of order p is central, then the Ω series for P is central. For p = 2, we require Ω_{2}(P) to be in the center [R].

As previously observed [HP], in the probabilistic sense of Martin [Ma] almost all p-groups satisfy the conditions of Theorem 6.7, thus almost all p-groups are Swan groups.

Some summands of X of BP involve the transfer, i.e., there exists an equivalence

X → BP ⟶^{tr} BQ ⟶^{Br} BR → X

where Q ⪇ P and r : Q → R is a retraction. We call such X transfer summands of BP.

Theorem 6.9. [MP4] If BP has no transfer summands then BP is a Swan group.

We conclude by discussing two notions related to Swan groups.

7. Universally stable elements and principal original summands

For a fixed p-group P we consider those elements in H^{∗}(BP) which are common to all H^{∗}(BG) for P ∈ Syl_{p}(G). To make this precise let

I(P) = ⋂_{G}Im(i^{∗} : H^{∗}(BG) → H^{∗}(BP))

where G runs over all groups with P ∈ Syl_{p}(G). Then I(P) is called the ring of universally stable elements by analogy with the stable elements of Cartan-Eilenberg [CE].

Theorem 7.1. [EP] I(P) is a finitely generated F_{p} algebra with Krull dimension equal to that of H^{∗}(BP).

Theorem 7.2. [EP] If V is an elementary abelian p-group then

I(P) = H^{∗}(V)^{GL(V)}

unless p = 2 and rank(V) = 2.

Now let X_{0} = X_{1,P} denote the summand of BP which originates in BP and corresponds to the trivial F_{p}Out(P) module. X_{0} is called the principal original summand of BP. On the level of spaces it is related to I(P) in the following sense.

Theorem 7.3. [N],[M5] If P ∈ Syl_{p}(G) then the principal original summand X_{0} is a summand of BG.

There is a strong connection between these two notions. For most groups (in the probabilistic sense) I(P) = H^{∗}(X_{0}). This is also known to be true for certain groups but not others [EP].

Other information on I(P) can be found in [LSY],[TY].

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Department of Mathematics, Northwestern University, Evanston, IL 60208

    E-mail address: s priddy@math.nwu.edu