THE NILPOTENT FILTRATION AND THE ACTION OF AUTOMORPHISMS ON THE COHOMOLOGY OF FINITE

p–GROUPS

NICHOLAS J. KUHN

Abstract. We study H^{∗}(P), the mod p cohomology of a finite p–group P, viewed as an 𝔽_{p}[Out(P)]–module. In particular, we study the conjecture, first considered by Martino and Priddy, that, if e_{S} ∈ 𝔽_{p}[Out(P)] is a primitive idempotent associated to an irreducible 𝔽_{p}[Out(P)]–module S, then the Krull dimension of e_{S}H^{∗}(P) equals the rank of P. The rank is an upper bound by Quillen’s work, and the conjecture can be viewed as the statement that every irreducible 𝔽_{p}[Out(P)]–module occurs as a composition factor in H^{∗}(P) with similar frequency.

In summary, our results are as follows. A strong form of the conjecture is true when p is odd. The situation is much more complex when p = 2, but is reduced to a question about 2–central groups (groups in which all elements of order 2 are central), making it easy to verify the conjecture for many finite 2–groups, including all groups of order 128, and all groups that can be written as the product of groups of order 64 or less.

The the odd prime theorem can be deduced using the approach to U, the category of unstable modules over the Steenrod algebra, initiated by H.-W. Henn, J. Lannes, and L. Schwartz in [HLS1]. The reductions when p = 2 make heavy use of the nilpotent filtration of U introduced in [S1], as applied to group cohomology in [HLS2]. Also featured are unstable algebras of cohomology primitives associated to central group extensions.

1. Introduction

Fix a prime p, and let H^{∗}(P) denote the mod p cohomology ring of a finite p–group P. Interpreted topologically, H^{∗}(P) = H^{∗}(BP; 𝔽_{p}), where BP is the classifying space of P, and interpreted algebraically, H^{∗}(P) = Ext_{𝔽_{p}[P]}^{∗}(𝔽_{p}, 𝔽_{p}).

The automorphism group of P, Aut(P), acts on P, and thus via ring homomorphisms on H^{∗}(P). As the inner automorphism group, Inn(P), acts trivially on cohomology, H^{∗}(P) becomes a graded 𝔽_{p}[Out(P)]–module, where Out(P) = Aut(P)/Inn(P) is the outer automorphism group.

The ring H^{∗}(P) is known to be Noetherian, and D. Quillen [Q1] computed its Krull dimension: dim H^{∗}(P) = rk(P). Here rk(P) denotes the maximal rank of an elementary abelian p–subgroup of P.

Now let S be an irreducible 𝔽_{p}[Out(P)]–module and e_{S} ∈ 𝔽_{p}[Out(P)] an associated primitive idempotent. Then e_{S}H^{∗}(P) is a finitely generated module over the Noetherian ring H^{∗}(P)^{Out(P)} (see §4.3), and thus has a Krull dimension with evident upper bound rk(P). In the early 1990’s, J. Martino and S. Priddy [MP]

Date: May 3, 2006. 2000 Mathematics Subject Classification. Primary 20J06; Secondary 55R40, 20D15. This research was partially supported by a grant from the National Science Foundation.

asked whether the following conjecture might be true.

Conjecture A dim e_{S}H^{∗}(P) = rk(P) for all pairs (P, S).

As dim e_{S}H^{∗}(P) determines the growth of the Poincaré series of e_{S}H^{∗}(P), and this Poincaré series gives a count of the occurrences of S as a composition factor in H^{∗}(P), the conjecture is roughly the statement that every irreducible 𝔽_{p}[Out(P)]– module occurs in H^{∗}(P) with similar frequency.

The topological interpretation of H^{∗}(P) makes it evident that H^{∗}(P) is an object in K and U, the categories of unstable algebras and modules over the mod p Steenrod algebra A, and in this paper, we study Conjecture A using modern ‘U–technology’.

In summary, our results are as follows. A strong form of Conjecture A is true when p is odd. The situation is more complex when p = 2. We reduce the conjecture to a different conjecture about 2–central groups, making it easy to verify Conjecture A for many finite 2–groups, including all with order dividing 128, and all groups that can be written as the product of groups of order 64 or less.

The the odd prime theorem can be deduced using the functor category approach to U initiated by H.-W. Henn, J. Lannes, and L. Schwartz in [HLS1]. The reductions when p = 2 make heavy use of the nilpotent filtration of U introduced in [S1], as applied to group cohomology in [HLS2].

Featured here, and in a companion paper [K5], are algebras of cohomology primitives associated to central group extensions, and p–central groups: groups in which every element of order p is central.

2. Main results

2.1. Notation. Before describing our results in more detail, we introduce some notation. Throughout, V < P will denote an elementary abelian p–subgroup of P and C = C(P) < P will be the maximal central elementary abelian p–subgroup. We let c(P) be the rank of C, depth_{H^{∗}(P)} H^{∗}(P) be the depth of H^{∗}(P) with respect to the ideal of positive degree elements, and mrk(P) be the minimal rank of a maximal V < P. These are related by

c(P) ≤ depth_{H^{∗}(P)} H^{∗}(P) ≤ mrk(P) ≤ rk(P),

where the first inequality is due to J.Duflot [D], and the second follows from Quillen’s work and standard commutative algebra (see §4.6).

Given V < P, we let Aut_{0}(P, V) ◁ Aut(P, V) < Aut(P) be defined by

Aut(P, V) = {α ∈ Aut(P) | α(V) = V } and
Aut_{0}(P, V) = {α ∈ Aut(P) | α(v) = v for all v ∈ V },

Then let Out_{0}(P, V) and Out(P, V) be the projection of these groups in Out(P). Note that Out(P, V) = Aut(P, V)/N_{P} (V) and Out_{0}(P, V) = Aut_{0}(P, V)/C_{P} (V), so that there is an extension of groups

W_{P} (V) → Aut(P, V)/Aut_{0}(P, V) → Out(P, V)/Out_{0}(P, V),

where W_{P} (V) = N_{P} (V)/C_{P} (V).

Finally, if S is an irreducible 𝔽_{p}[G]–module, and M is an arbritrary 𝔽_{p}[G]– module, we say that S occurs in M if it is a composition factor.

2.2. Quillen’s approximation. Let A(P) denote Quillen’s category, having as objects the elementary abelian p–subgroups V < P, and as morphisms the homomorphisms generated by inclusions and conjugation by elements in P. Let H^{∗}(P) denote H^{∗}(P) = lim_{A(P)} H^{∗}(V). It is not hard to see the same inverse limit is attained

by using the smaller category A^{C}(P), where A^{C}(P) ⊂ A(P) is the full subcategory consisting of elementary abelian V < P containing the maximal central elementary abelian C.

The restriction maps associated to the inclusions V < P assemble to define a natural map of unstable A–algebras

H^{∗}(P) → H^{∗}(P),

which Quillen shows is an F –isomorphism. As a special case of Proposition 2.6 below, we have

dim e_{S}H^{∗}(P) ≥ dim e_{S}H^{∗}(P),

for all irreducible 𝔽_{p}[Out(P)]–modules S.

Theorem 2.1. Let S be an irreducible 𝔽_{p}[Out(P)]–module. Then e_{S}H^{∗}(P) ≠ 0 if and only if S occurs in 𝔽_{p}[Out(P)/Out_{0}(P, V)] for some maximal V < P. In that case, dim e_{S}H^{∗}(P) = max{rk(V) | S occurs in 𝔽_{p}[Out(P)/Out_{0}(P, V)]}.

This implies an observation noted earlier by Martino and Priddy [MP, Prop.4.2].

Corollary 2.2. If e_{S}H^{∗}(P) ≠ 0, then dim e_{S}H^{∗}(P) ≥ mrk(P).

Given H < G, a standard argument shows that if H is a p–group, then every irreducible 𝔽_{p}[G]–module occurs as a submodule of 𝔽_{p}[G/H]. Thus the theorem implies that if Out_{0}(P, V) is a p–group, then dim e_{S}H^{∗}(P) ≥ rk(V) for all irreducibles S. The converse will be true if P has a unique maximal elementary abelian subgroup, for example, if P is p–central: see Corollary 6.3.

One can now make effective use of group theory results about p′–automorphisms of p–groups, particularly the Thompson A × B lemma [Gor, Thm.5.3.4].

Theorem 2.3. Let P a finite p–group. If V < P is an elementary abelian subgroup, then Out_{0}(P, V) will be a p–group if Out_{0}(C_{P} (V), V) is a p–group. If V is also maximal, and p is odd, this will always be the case. Thus if p is odd, dim e_{S}H^{∗}(P) = rk(P) for all irreducible 𝔽_{p}[Out(P)]–modules S.

For 2–groups, one does not need to look far to see that story is quite different. The quaternionic group of order 8, Q_{8}, is 2–central with center C of rank 1. As Aut(Q_{8}) clearly must act trivially on the center, Out_{0}(Q_{8}, C) = Out(Q_{8}) ≃ Σ_{3}, which is not a 2–group.

This example generalizes: it is the case t = 1 in the following.

Example 2.4. Let G_{t} be the 2–Sylow subgroup of SU_{3}(𝔽_{2^{t}}). This group is 2– central with center C_{t} ≃ 𝔽_{2^{t}}, and G_{t}/C_{t} ≃ 𝔽_{2^{2t}}. The group Out(G_{t}, C_{t}) contains a cyclic subgroup of order 2^{t} + 1. Thus there is at least one irreducible 𝔽_{2}[Out(G_{t})]– module S with e_{S}H^{∗}(G_{t}) = 0.

The next proposition, an easy application of properties of Lannes T_{V} –functor, shows that Theorem 2.3 can be applied to 2–groups with reduced cohomology, i.e., groups P for which H^{∗}(P) → H^{∗}(P) is monic. This includes groups built up by iterated products and wreath products of ℤ/2’s.

Proposition 2.5. If H^{∗}(P) is reduced, and V < P is a maximal elementary abelian subgroup, then C_{P} (V) = V, and thus Out_{0}(C_{P} (V), V) is the trivial group.

To say more about dim e_{S}H^{∗}(P), we now begin to take a deeper look at H^{∗}(P) using the nilpotent filtration.

2.3. A stratification of the problem. In §4, we study functors R̄_{d} : U → U. In brief, they are defined as follows. An unstable module M has a natural nilpotent filtration

⋯ ⊂ nil_{2}M ⊂ nil_{1}M ⊂ nil_{0}M = M,

and nil_{d}M/nil_{d+1}M = Σ^{d}R_{d}M, where R_{d}M is a reduced unstable module. Then R̄_{d}M is defined as the nilclosure of R_{d}M.

Henn [H] has shown that the nilpotent filtration of a Noetherian unstable algebra is finite, so only finitely many of the the modules R̄_{d}H^{∗}(P) will be nonzero. Furthemore, R̄_{0}H^{∗}(P) = H^{∗}(P), and each R̄_{d}H^{∗}(P) is a finitely generated H^{∗}(P)– module. The study of dim e_{S}H^{∗}(P) stratifies as follows.

Proposition 2.6. Let S be an irreducible 𝔽[Out(P)]–module. Then

dim e_{S}H^{∗}(P) = max_{d}{dim e_{S}R̄_{d}H^{∗}(P)}.

This will be a special case of a more general statement about K: see Proposition 4.9.

Martino and Priddy observe [MP, Prop.4.1] that the Depth Conjecture, subsequently proved by D. Bourguiba and S. Zarati [BoZ], implies that dim e_{S}H^{∗}(P) ≥ depth_{H^{∗}(P)} H^{∗}(P). We prove a stratified variant of this, and note that an analogue of Duflot’s theorem holds.

Proposition 2.7. If e_{S}R̄_{d}H^{∗}(P) ≠ 0, then

dim e_{S}R̄_{d}H^{∗}(P) ≥ depth_{H^{∗}(P)} R̄_{d}H^{∗}(P) ≥ c(P).

That dim e_{S}R̄_{d}H^{∗}(P) ≥ c(P) is also a corollary of part (c) of Theorem 2.8 below.

2.4. Computing dim e_{S}R̄_{d}H^{∗}(P). In [K5], the work of [HLS2] will allow us to write down useful formulae for R̄_{d}H^{∗}(P) analogous to

R̄_{0}H^{∗}(P) = lim_{V ∈A^{C}(P)} H^{∗}(V).

Using these, we can generalize Theorem 2.1 to a statement about dim e_{S}R̄_{d}H^{∗}(P) for all d.

To state this, we need to define the primitives associated to a central extension. The cohomology of an elementary abelian p–group is a Hopf algebra. If Q is a

finite group, and V < Q is a central elementary abelian p–subgroup, then multiplication m : V × Q → Q is a homomorphism, and the induced map in cohomology, m^{∗} : H^{∗}(Q) → H^{∗}(V × Q) = H^{∗}(V) ⊗ H^{∗}(Q) makes H^{∗}(Q) into an H^{∗}(V)– comodule. We then let P_{V} H^{∗}(Q) denote the primitives:

\begin{aligned}P_{V} H^{∗}(Q) & = \{x ∈ H^{∗}(Q) | m^{∗}(x) = 1 ⊗ x\} \\ & = Eq \{H^{∗}(Q) \begin{matrix}m^{∗} \\ ⟶ \\ ⟶ \\ π^{∗} \end{matrix} H^{∗}(V × Q)\}, \end{aligned}

where π : V × Q → Q is the projection.

As equalizers of algebra maps are algebras, P_{V} H^{∗}(Q) is again an unstable A– algebra. Note also that P_{V} H^{0}(Q) = 𝔽_{p}. It is not hard to check that P_{V} H^{1}(Q) ≃ H^{1}(Q/V) = Hom(Q/V, 𝔽_{p}), via the inflation map. The reader may find it illuminating to know that P_{V} H^{∗}(Q) is again Noetherian, and has Krull dimension equal to rk(Q) − rk(V): see [K4, K5].

Our theorem about dim e_{S}R̄_{d}H^{∗}(P) now goes as follows.

Theorem 2.8. Let S be an irreducible 𝔽_{p}[Out(P)]–module. For each d, there exists a set sup_{d}(S) ⊂ ob A^{C}(P), the d–support of S, with the following properties.

(a) sup_{d}(S) is a union of Aut(P)–orbits in ob A^{C}(P).

(b) Suppose that V_{1} < V_{2} < P, and P_{V_{1}}H^{d}(C_{P} (V_{1})) → P_{V_{1}}H^{d}(C_{P} (V_{2})) is monic. Then V_{1} ∈ sup_{d}(S) implies that V_{2} ∈ sup_{d}(S).

(c) e_{S}R̄_{d}H^{∗}(P) ≠ 0 if and only if sup_{d}(S) ≠ ∅, and, in this case,

dim e_{S}R̄_{d}H^{∗}(P) = max{rk(V) | V ∈ sup_{d}(S)}.

(d) Let V ∈ ob A^{C}(P) be maximal. Then V ∈ sup_{d}(S) if and only if S occurs in Ind_{Out(P,V)}^{Out(P)}([𝔽_{p}[Aut(P, V)/Aut_{0}(P, V)] ⊗ P_{V} H^{d}(C_{P} (V))]^{W_{P} (V)}).

When d = 0, we recover Theorem 2.1. In this case, the hypothesis in part (b) always holds. Thus maximal elements in sup_{0}(S) will always be maximal in ob A^{C}(P). Thus part (d) implies that V will be a maximal element in sup_{0}(S) if and only if S occurs in

Ind_{Out(P,V)}^{Out(P)}(𝔽_{p}[Aut(P, V)/Aut_{0}(P, V)]^{W_{P} (V)})

which rewrites as

𝔽_{p}[Out(P, V)/Out_{0}(P, V)].

Thus part (c) implies the calculation of dim e_{S}H^{∗}(P) in Theorem 2.1. Theorem 2.8 now lets us recast Conjecture A as follows.

Corollary 2.9. Conjecture A is true for a pair (P, S) if and only if there exists a V < P of maximal rank such that S occurs in

Ind_{Out(P,V)}^{Out(P)}([𝔽_{p}[Aut(P, V)/Aut_{0}(P, V)] ⊗ P_{V} H^{∗}(C_{P} (V))]^{W_{P} (V)}).

Starting from this, it is not hard to show (see Lemma 8.2) that Conjecture A for a 2–group P would be implied by

Conjecture B Let P be a finite 2–group. If V < P is a maximal elementary abelian subgroup, then every irreducible 𝔽_{2}[Out_{0}(P, V)]–module occurs in P_{V} H^{∗}(C_{P} (V))^{W_{P} (V)}.

Note that the subgroups C_{P} (V) that arise here are 2–central. Our general theory shows

Proposition 2.10. If Q is 2–central with maximal elementary abelian subgroup C, then Conjecture B is true for Q, i.e. every irreducible 𝔽_{2}[Out_{0}(Q, C)]–module occurs as a composition factor in P_{C}H^{∗}(Q).

We conjecture a subtle strengthening of this.

Conjecture C If Q is 2–central with maximal elementary abelian subgroup C, then every irreducible 𝔽_{2}[Out_{0}(Q, C)]–module occurs as a submodule of P_{C}H^{∗}(Q).

Again using the Thompson A × B lemma, we have

Theorem 2.11. Conjecture B is true for the pair (P, V) if the 2–central group C_{P} (V) satisfies Conjecture C. Thus if P is a 2–group such that C_{P} (V) satisfies Conjecture C for some V < P of maximal rank, then dim e_{S}H^{∗}(P) = rk(P) for all irreducible 𝔽_{2}[Out(P)]–modules S.

Though Out_{0}(Q, C) need not be a 2–group if Q is 2–central, its 2′ part tends to be very small, even in cases when Out(Q) is quite complicated. This makes it not hard to check the following theorem, using information about 2–groups of order dividing 64 available at the website [Ca], or, in book form, [CTVZ].

Theorem 2.12. Conjecture C is true for all 2–central groups that can be written as the product of groups of order dividing 64.

A counterexample to Conjecture A would have to be a 2–group P that is not 2–central, and having the property that the proper subgroups C_{P} (V), with V of maximal rank, are all counterexamples to Conjecture C. Thus the last theorem implies

Corollary 2.13. Conjecture A is true for all 2–groups of order dividing 128, and all 2–groups that can be written as the product of groups of order 64 or less.

We end this section with an example^{1} that illustrates the sorts of patterns that the numbers dim e_{S}R̄_{d}H^{∗}(P) can take, when P is not p–central.

Example 2.14. Let P be the group of order 64 number #108 on the Carlson group cohomology website. From the information there one learns that Out(P) has order 3 · 2^{8}, c(P) = depth_{H^{∗}(P)} H^{∗}(P) = 2, while mrk(P) = rk(P) = 3. One can also deduce that 𝔽_{2}[Out(P)] has precisely two irreducibles - the one dimensional trivial module ‘1’ and a two dimensional module ‘S’. We compute the following table of nonzero dimensions, where ∅ denotes that the corresponding summand of R̄_{d}(H^{∗}(P)) is 0.

\begin{matrix}d & dim e_{1}R̄_{d}H^{∗}(P) & dim e_{S}R̄_{d}H^{∗}(P) \\ 0 & 3 & ∅ \\ 1 & 2 & 3 \\ 2 & 2 & 3 \\ 3 & 3 & 2 \\ 4 & 2 & 2 \\ 5 & 2 & 2 \\ 6 & 2 & 2 \\ 7 & 2 & ∅ \end{matrix}

1Checking Conjecture A for this group led us to the formulation and proof of Theorem 2.8.

Note that this example shows that, when d > 0, e_{S}R̄_{d}H^{∗}(P) ≠ 0 does not imply that dim s_{S}R̄_{d}H^{∗}(P) ≥ mrk(P), in constrast to Corollary 2.2. Indeed, for this group, dim R̄_{d}H^{∗}(P) = 2 < 3 = mrk(P) for d = 4, 5, 6, 7.

2.5. Organization of the paper. The rest of the paper is organized as follows. In §3, we quickly survey previous results related to Conjecture A. The nilpotent filtration is reviewed in §4, which has various relevant general results about U and K concerning both dimension and depth. In §5, we give a first proof of Theorem 2.1 using the functor category description of U/Nil introduced in [HLS1]. In §6, we then collect various results about p′–automorphisms of p–groups, and prove Theorem 2.3 and Proposition 2.5. In this section we also discuss the family of 2-central groups given in Example 2.4. In §7, we use a convenient formula for R̄_{d}H^{∗}(P) from [K5] to prove Theorem 2.8. A short proof of Proposition 2.10 also appears there. That Conjecture C implies Conjecture A is shown in the short §8, and in §9 various 2–central groups are shown to satisfy Conjecture C, including those listed in Theorem 2.12. Finally, in §10, we will discuss Example 2.14 in detail.

The author wishes to thank the Cambridge University Pure Mathematics Department for its hospitality during a visit during which a good part of this research was done. Information compiled by Ryan Higginbottom has been very useful, and was used by him in [Hi] to verify Conjecture A for all groups of order dividing 64 except for group 64#108 . Example 2.4 arose from a conversation with David Green.

3. Previous results

In the 1984 paper [DS], T. Diethelm and U. Stammbach gave a group theoretic proof that, if P is a finite p–group, every irreducible 𝔽_{p}[Out(P)]–module occurs as a composition factor in H^{∗}(P). Independently and concurrently, it was noted explicitly in [HK], and implicitly in [N], that this same result was a consequence of the proof of the Segal Conjecture in stable homotopy theory. From both [DS] and [HK], one can conclude that every irreducible occurs an infinite number of times, so that dim e_{S}H^{∗}(P) ≥ 1 for all irreducibles S.

The 1999 paper of P. Symonds [Sy] gives a second group theoretic proof that every irreducible 𝔽_{p}[Out(P)]–module occurs in H^{∗}(P), and inspection of his proof also yields Martino and Priddy’s lower bound dim e_{S}H^{∗}(P) ≥ c(P).^{2}

A proof of Conjecture A for an elementary abelian p–group V serves as a starting point for work on the general question, and goes as follows.

Let S^{∗}(V) ⊂ H^{2∗}(V) denote the symmetric algebra generated by β(H^{1}(V)) ⊂ H^{2}(V), where β is the Bockstein. Thus S^{∗}(V) is a polynomial algebra on r = rk(V) generators in degree 2. It is a classic result of Dickson [D, Cr] that the invariant ring S^{∗}(V)^{GL(V)} is again polynomial on homogeneous generators c_{V} (1), …, c_{V} (r), where c_{V} (i) has degree 2(p^{r} − p^{r−i−1}).

For any idempotent e ∈ 𝔽_{p}[GL(V)], eH^{∗}(V) is an S^{∗}(V)^{GL(V)}–module via left multiplication. Now one observes that S^{∗}(V) (and thus H^{∗}(V)) is a free S^{∗}(V)^{GL(V)}–module, and that every finite 𝔽_{p}[GL(V)]–module occurs as a submodule of S^{∗}(V) [Alp, p.45]. These facts imply that, for all irreducible 𝔽_{p}[GL(V)]– modules S, e_{S}H^{∗}(V) is a nonzero finitely generated free S^{∗}(V)^{GL(V)}–module, and thus dim e_{S}H^{∗}(V) = rk(V).

2[Sy] was written with knowledge of [HK], but apparently not of [DS].

Using [HK], this result immediately extends to all abelian p–groups.

In [MP], Martino and Priddy use the analogous action of H^{∗}(P)^{Out(P)} on eH^{∗}(P) to show that, for all nonzero idempotents e, dim eH^{∗}(P) ≥ depth_{H^{∗}(P)} H^{∗}(P). Their proof is critically dependent on an unpublished paper that was later withdrawn. However, as we note in the next section, their argument can be salvaged by using the later work of D. Bourguiba and S. Zarati [BoZ] utilizing deep properties of U and K.

Remark 3.1. Readers of [MP] will know that much of Martino and Priddy’s paper concerns a refined version of Conjecture A. The double Burnside ring A(P, P) acts on H^{∗}(P), and one can conjecture that dim eH^{∗}(P) = rk(P) for all idempotents e ∈ A(P, P) ⊗ 𝔽_{p} that project to a nonzero element under the retraction of algebras A(P, P) ⊗ 𝔽_{p} → 𝔽_{p}[Out(P)]. The paper [HK] notes that one can deduce that eH^{∗}(P) ≠ 0, and thus that dim eH^{∗}(P) ≥ 1, from the Segal conjecture; there is currently no ‘group theoretic’ proof of this fact. Partly for this reason, we have focused on the Out(P) conjecture, though some of our general theory evidently applies to the A(P, P) version.^{3}

4. Krull dimension and the nilpotent filtration of U

4.1. The nilclosure functor. Let N il_{1} ⊂ U be the localizing subcategory generated by suspensions of unstable A–modules, i.e. N il_{1} is the smallest full subcategory containing all suspensions of unstable modules that is closed under extensions and filtered colimits.

An unstable module M is called nilreduced (or just reduced) if it contains no such nilpotent submodules, or, equivalently, if Hom_{U}(N, M) = 0 for all N ∈ N il_{1}. M is called nilclosed if also Ext_{U}^{1}(N, M) = 0 for all N ∈ N il_{1}.

Let L_{0} : U → U be localization away from N il_{1}. Thus L_{0}M is nilclosed, and there is a natural transformation M → L_{0}M with both kernel and cokernel in N il_{1}.

Recall that T_{V} : U → U is defined to be left adjoint to H^{∗}(V) ⊗ . The various marvelous properties of T_{V} are reflected in similar properties of L_{0}.

Proposition 4.1. The functor L_{0} : U → U satisfies the following properties.

(a) There are natural isomorphisms L_{0}(M ⊗ N) ≃ L_{0}M ⊗ L_{0}N.

(b) There are natural isomorphisms T_{V} L_{0}M ≃ L_{0}T_{V} M.

(c) If K ∈ K, then L_{0}K ∈ K, and K → L_{0}K is a map of unstable algebras.

One approach to properties (a) and (b) is to use Proposition 5.1. See [HLS2, I.4.2] and [BrZ1] for more detail about property (c).

Given a Noetherian unstable algebra K ∈ K, we recall the category K_{f.g.} − U as studied in [HLS2, I.4]. The objects are finitely generated K–modules M whose K–module structure map K ⊗ M → M is in U, and morphisms are K–module maps in U.

3Our results do imply the refined conjecture for p–central groups at odd primes. See Remark 6.5.

Proposition 4.2. Let K ∈ K be Noetherian, and M ∈ K_{f.g.} − U. Then L_{0}K ∈ K_{f.g.} − U, and thus is Noetherian, and L_{0}M ∈ L_{0}K_{f.g.} − U.

See [HLS2, I(4.10)].

4.2. The nilpotent filtration. For d ≥ 0, let N il_{d} ⊂ U be the localizing subcategory generated by d–fold suspensions of unstable A–modules. An unstable module M admits a natural filtration

⋯ ⊆ nil_{2}M ⊆ nil_{1}M ⊆ nil_{0}M = M,

where nil_{d}M is the largest submodule in N il_{d}.

As observed in [K3, Prop.2.2], nil_{d}M/nil_{d+1}M = Σ^{d}R_{d}M, where R_{d}M is a reduced unstable module. (See also [S2, Lemma 6.1.4].)

Proposition 4.3. The functors R_{d} : U → U satisfy the following properties.

(a) There are a natural isomorphisms R_{∗}(M ⊗ N) ≃ R_{∗}M ⊗ R_{∗}N of graded objects in U.

(b) There are natural isomorphism T_{V} R_{d}M ≃ R_{d}T_{V} M.

(c) Let K ∈ K be Noetherian, and M ∈ K_{f.g.} − U. Then R_{0}K is also a Noetherian unstable algebra, and R_{d}M ∈ R_{0}K_{f.g.} − U, for all d.

For the first two properties, see [K3, §3], and the last follows easily from the first.

Now let R̄_{d}M denote the nilclosure of R_{d}(M). Thus R_{d}M ⊆ L_{0}R_{d}M = R̄_{d}M. We also note that L_{0}M = R̄_{0}M.

The previous propositions combine to prove the following.

Proposition 4.4. The functors R̄_{d} : U → U satisfy the following properties.

(a) There are natural isomorphisms R̄_{∗}(M ⊗ N) ≃ R̄_{∗}M ⊗ R̄_{∗}N of graded objects in U.

(b) There are natural isomorphisms T_{V} R̄_{d}M ≃ R̄_{d}T_{V} M.

(c) Let K ∈ K be Noetherian, and M ∈ K_{f.g.} − U. Then R̄_{0}K is also a Noetherian unstable algebra, and R̄_{d}M ∈ R̄_{0}K_{f.g.} − U, for all d.

Henn [H] proved the following important finiteness result.

Proposition 4.5. Let K ∈ K be Noetherian, and M ∈ K_{f.g.} − U. Then the nilpotent filtration of M has finite length. Equivalently, R̄_{d}M = 0 for d >> 0.

4.3. Invariant rings as Noetherian unstable algebras.

Proposition 4.6. Let K ∈ K be Noetherian. Given any subgroup G < Aut_{K}(K), the invariant ring K^{G} is again a Noetherian unstable algebra, and K is a finitely generated K^{G}–module.

Proof. Let D(d) ∈ K denote the d^{th} Dickson algebra: in the notation of §3, D(d) is the polynomial algebra S^{∗}((𝔽_{p})^{d})^{GL_{d}(𝔽_{p})}. Then let D(d, j) ∈ K denote the subalgebra consisting of p^{j} powers of the elements of D(d). In [BoZ, Thm.A.1], Lannes shows that, if d = dim K, then there is an embedding of unstable algebras D(d, j) → K that is unique in the sense that any two such embeddings will agree after restriction to D(d, k) with k large enough. As G is necessarily finite, it follows that for large enough j, there is an embedding D(d, j) → K^{G} such that K is a finitely generated D(d, j)–module. The proposition follows. □

Our various propositions apply to the case when K = H^{∗}(P).

Corollary 4.7. Let P be a finite p–group.

(a) Both H^{∗}(P)^{Out(P)} and H^{∗}(P)^{Out(P)} are Noetherian.

(b) e_{S}H^{∗}(P) ∈ H^{∗}(P)_{f.g.}^{Out(P)} − U and e_{S}R̄_{d}H^{∗}(P) ∈ H^{∗}(P)_{f.g.}^{Out(P)} − U, for all irreducible 𝔽_{p}[Out(P)]–modules S and all d.

4.4. How to deal with odd primes. When p is odd, K ∈ K is not necessarily commutative unless K is concentrated in even degrees. To use standard definitions and results from the commutative algebra literature, and for other technical reasons, it is useful to have a systematic way of ridding ourselves of this problem.

A standard thing to do, done many times before, and going back at least to [LZ], goes as follows. Let U′ denote the full subcategory of U consisting of modules concentrated in even degrees. Given M ∈ U, we let M′ ∈ U′ denote the image of M under the right adjoint of the inclusion U′ ⊂ U: in more down-to-earth terms, M′ is the largest submodule of M contained in even degrees. It is easy to see that if K is an unstable algebra, then K′ is a subalgebra. As an example, H^{∗}(V)′ = S^{∗}(V).

Statements about K_{f.g.} − U become statements about K′_{f.g.} − U using the next result.

Proposition 4.8. If K ∈ K is Noetherian, so is K′, and K is a finitely generated K′–module.

This is [BrZ1, Lemma 5.2].

4.5. Krull dimension. We will generally work with the standard definition of Krull dimension. Given a commutative graded Noetherian 𝔽_{p}–algebra K, dim K = d if

℘_{0} ⊂ ⋯ ⊂ ℘_{d}

is a chain of prime ideals in K of maximal length. If M is a finitely generated K–module, dim M is then defined to be dim(K/Ann(M)), where Ann(M) is the annilator ideal of M.

Given a Noetherian K ∈ K and M ∈ K_{f.g.} − U, the proposition of the last subsection shows that M ∈ K′_{f.g.} − U, and thus dim M will be a well defined finite natural number.

In this situation, the Poincaré series of M, ∑_{i=0}^{∞}(dim_{𝔽_{p}} M^{i})t^{i}, will be a rational function, and dim M equals the order of the pole at t = 1. There is a third way of calculating dim K: it is the number d such that there exist algebraically independent elements k_{1}, …, k_{d} ∈ K with K finitely generated over 𝔽_{p}[k_{1}, …, k_{d}]. See [CTVZ, §10.2] for a nice discussion of these and related facts.

Proposition 2.6 of the introduction is a special case of the following.

Proposition 4.9. Let K ∈ K be Noetherian. Given M ∈ K_{f.g.} − U,

dim M = max_{d}{dim R̄_{d}M}.

Proof. As the modules Σ^{d}R_{d}M are the composition factors associated to a finite filtration of M, standard properties of Krull dimension [Mat, (12.D)] imply that dim M = max_{d}{dim R_{d}M}. Recalling that R̄_{d} = L_{0}R_{d}, the next proposition finishes the proof. □

Proposition 4.10. Given K and M as above, if M is reduced then

dim M = dim L_{0}M.

This follows from the next two lemmas.

Lemma 4.11. Ann(M) is a sub A–module of K.

Proof. We are claiming that, if kx = 0 for all x ∈ M, then, for all a ∈ A and x ∈ M, we have (ak)x = 0. Fixing a ∈ A, assume by induction that (a′k)y = 0 for all y ∈ M and a′ ∈ A with |a′| < |a|. With Δa = ∑ a′ ⊗ a′′, we then have

0 = a(kx)

= (ak)x + ∑_{|a′|<|a|}(a′k)(a′′x)

= (ak)x,

where the last equality uses the inductive hypothesis. □

Lemma 4.12. With M reduced, Ann(M) = Ann(L_{0}M).

Proof.

Ann(M) = {k ∈ K | Ak ⊗ M → M is 0}
= {k ∈ K | Ak ⊗ M → L_{0}M is 0}
= {k ∈ K | Ak ⊗ L_{0}M → L_{0}M is 0}
= Ann(L_{0}M).

Here the first and last equalities are consequences of the last lemma. The second is immediate since M is reduced, i.e. M → L_{0}M is monic. Finally, the third equality follows from the universal property of nilclosed modules, as Ak ⊗ M → Ak ⊗ L_{0}M is a N il_{1}–isomorphism. □

4.6. Depth. If M is a K–module, and I ⊂ K is an ideal, the depth of M with respect to I is defined to be the maximal length l of an M–regular sequence in I: r_{1}, …, r_{l} ∈ I such that for each i between 1 and l, r_{i} is not a zero divisor on M/(r_{1}, …, r_{i−1})M.

If K is a Noetherian unstable algebra, and M ∈ K_{f.g.} − U, we let depth_{K} M denote the depth of M with respect to the ideal of positive degree elements in K.

It is standard that depth_{K} M ≤ dim N, where N ⊂ M is any nonzero submodule. This is usually stated in the following equivalent formulation [Mat, Thm.29]: depth_{K} M ≤ dim K/℘ where ℘ is any associated prime ideal, i.e. a prime ideal arising as the annihilator of an element of M. The associated primes include all the minimal primes in the support of M [Mat, Thm.9].

Quillen [Q2, Prop.11.2] shows that the minimal primes correspond to the maximal elementary abelian subgroups, and so depth_{H^{∗}(P)} H^{∗}(P) ≤ mrk(P), as asserted in the introduction.

The work of Bourguiba, Lannes, and Zarati [BoZ] shows

Proposition 4.13. Let K ∈ K be Noetherian, G < Aut_{K}(K), and M ∈ K_{f.g.} − U. Then depth_{K^{G}} M = depth_{K} M.

The point here is that the main theorem of [BoZ] says that depth_{K} M can be calculated by using a very specific regular sequence of ‘generalized Dickson invariants’, and, as discussed in the proof of Proposition 4.6, Lannes’ theorem in the appendix of [BoZ] says that these elements will be in K^{G}.

Corollary 4.14. Let K ∈ K be Noetherian, G < Aut_{K}(K), M ∈ K_{f.g.} − U, and N be a nonzero direct summand of M, viewed as a K^{G}–module. Then dim N ≥ depth_{K} M.

Proof. dim N ≥ depth_{K^{G}} N ≥ depth_{K^{G}} M = depth_{K} M. □

Specializing further, we have

Corollary 4.15. Let P be a finite p–group, and S an irreducible 𝔽_{p}[Out(P)]– module. If e_{S}H^{∗}(P) ≠ 0, then dim e_{S}H^{∗}(P) ≥ depth_{H^{∗}(P)} H^{∗}(P). If e_{S}R̄_{d}H^{∗}(P) ≠ 0, then dim e_{S}R̄_{d}H^{∗}(P) ≥ depth_{H^{∗}(P)} R̄_{d}(P).

Duflot showed [D] that depth_{H^{∗}(P)} H^{∗}(P) ≥ rk(C) = c(P). Subsequent proofs, beginning with [BrH], have emphasized that this theorem is a reflection of the H^{∗}(C)–comodule structure on H^{∗}(P) induced by the multiplication homomorphism C × P → P. This same homomorphism also induces an H^{∗}(C)–comodule structure on R̄_{∗}H^{∗}(P), and the proof of Duflot’s theorem given in [CTVZ, Thm.12.3.3] goes through without change to prove

Proposition 4.16. depth_{H^{∗}(P)} R̄_{∗}(P) ≥ c(P).

This proposition and the preceeding corollary imply Proposition 2.7.

5. Proof of Theorem 2.1

In this section, we give a proof of Theorem 2.1 based on the identification of U/N il as a certain functor category.

Following [HLS1], let F be the category of functors from finite dimensional 𝔽_{p}– vector spaces to 𝔽_{p}–vector spaces. This is an abelian category in the obvious way: F → G → H is exact if F (W) → G(W) → H(W) is exact for all W.

Let l : U → F be defined by l(M)(W) = (T_{W} M)^{0} = Hom_{U}(M, H^{∗}(W))′. (Here M′ denotes the continuous dual of a profinite vector space M.) This has right adjoint r : F → U given by r(F)^{d} = Hom_{F}(H_{d}, F), where H_{∗}(W) is the mod p homology of the group W.

Proposition 5.1. The functors l and r satisfy the following properties.

(a) l is exact.

(b) The natural transformation M → r(l(M)) identifies with M → L_{0}M. In particular l(M) = 0 if and only if M ∈ N il_{1}.

(c) Both l and r commute with tensor products.

For proofs of these properties see [HLS1, K2, S2].

The proposition implies that a nilclosed module, like H^{∗}(P), is completely determined as an object in U by its associated functor. It is well known that Quillen’s work allows for the identification of l(H^{∗}(P)) = l(H^{∗}(P)). Let Rep(W, P) = Hom(W, P)/Inn(P).

Proposition 5.2. l(H^{∗}(P))(W) = 𝔽_{p}^{Rep(W,P)}.

For a simple proof, see [K1].

Proof of the first statement of Theorem 2.1. By the discussion above, an irreducible 𝔽_{p}[Out(P)]–module S occurs in H^{∗}(P) if and only if it appears in the permutation module 𝔽_{p}^{Rep(W,P)} for some W.

Let α : W → P represent an element in Rep(W, P). The stablizer of the Out(P)– orbit of this element will be Out_{0}(P, V) where V = α(W). Thus, as an Out(P)– module, 𝔽_{p}^{Rep(W,P)} is a direct sum of modules of the form 𝔽_{p}^{Out(P)/Out_{0}(P,V)}, and every such module appears as a summand of 𝔽_{p}^{Rep(W,P)} if rk W ≥ rk V.

Finally, we note that if V_{1} < V_{2} < P, then Out_{0}(P, V_{2}) ≤ Out_{0}(P, V_{1}), and thus 𝔽_{p}^{Out(P)/Out_{0}(P,V_{1})} is a quotient of 𝔽_{p}^{Out(P)/Out_{0}(P,V_{2})}. It follows that if S occurs in 𝔽_{p}^{Out(P)/Out_{0}(P,V)} for some V < P, then S occurs in 𝔽_{p}^{Out(P)/Out_{0}(P,V)} for some maximal V < P. □

We now finish the proof of Theorem 2.1 by showing that if S occurs in the permutation module 𝔽_{p}^{Out(P)/Out_{0}(P,V)} then dim e_{S}H^{∗}(P) ≥ rk V. We do this by looking more carefully at 𝔽_{p}^{Rep(W,P)} as an object in F with Out(P) action.

We begin by giving an increasing filtration of the set valued bifunctor Rep(W, P). Let Rep_{k}(W, P) ⊆ Rep(W, P) be the set of elements represented by homomorphisms with image of rank at most k. The inclusions

{0} = Rep_{0}(W, P) ⊆ Rep_{1}(W, P) ⊆ ⋯ ⊆ Rep_{rk(P)}(W, P) = Rep(W, P)

induces epimorphisms of objects in F with Out(P) action

𝔽_{p} = 𝔽_{p}^{Rep_{0}(W,P)} ← 𝔽_{p}^{Rep_{1}(W,P)} ← ⋯ ← 𝔽_{p}^{Rep_{rk(P)}(W,P)} = 𝔽_{p}^{Rep(W,P)}.

An appeal to Proposition 4.10 thus shows

Lemma 5.3. dim e_{S}H^{∗}(P) = max_{k}{dim e_{S}r(ker{𝔽_{p}^{Rep_{k}(W,P)} → 𝔽_{p}^{Rep_{k−1}(W,P)}})}.

To more usefully describe ker{𝔽_{p}^{Rep_{k}(W,P)} → 𝔽_{p}^{Rep_{k−1}(W,P)}}, we need to introduce some notation. Given V < P, let GL_{P} (V) = Out(P, V)/Out_{0}(P, V). Note that GL_{P} (V) is naturally a subgroup of GL(V) and also acts on the right of the Out(P)– set Out(P)/Out_{0}(P, V).

Lemma 5.4. There is an isomorphism of objects in F with Out(P) action,

ker{𝔽_{p}^{Rep_{k}(W,P)} → 𝔽_{p}^{Rep_{k−1}(W,P)}} ≃ ∏_{[V]} 𝔽_{p}^{Epi(W,V)×_{GL_{P} (V )}Out(P)/Out_{0}(P,V)},

where the product is over Aut(P)–orbits of elementary abelian subgroups of rank k.

Now we need to identify the nilclosed unstable module with Out(P) action associated to 𝔽_{p}^{Epi(W,V)×_{GL_{P} (V )}Out(P)/Out_{0}(P,V)}.

Let c_{V} ∈ S^{∗}(V)^{GL(V)} be the ‘top’ Dickson invariant: c_{V} is the product of all the elements β(x) ∈ H^{2}(V), with 0 ≠ x ∈ H^{1}(V). The key property we need is that η^{∗}(c_{V}) = 0 for all proper inclusions η : U < V.

Lemma 5.5 (Compare with [HLS1, proof of Thm.II.6.4]). l(c_{V} S^{∗}(V))(W) = 𝔽_{p}^{Epi(W,V)}.

Corollary 5.6. The functor l assigns to the reduced module

[c_{V} S^{∗}(V) ⊗ 𝔽_{p}^{Out(P)/Out_{0}(P,V)}]^{GL_{P} (V)},

the object in F which sends W to 𝔽_{p}^{Epi(W,V)×_{GL_{P} (V )}Out(P)/Out_{0}(P,V)}.

To finish the proof of Theorem 2.1, we are left needing to prove

Proposition 5.7. If an irreducible 𝔽_{p}[Out(P)]–module S occurs in the permutation module 𝔽_{p}[Out(P)/Out_{0}(P, V)], then

dim e_{S}[c_{V} S^{∗}(V) ⊗ 𝔽_{p}^{Out(P)/Out_{0}(P,V)}]^{GL_{P} (V)} = rk V.

Proof. The proof is similar to the proof of Conjecture A when P = V given in §3. Our first observation is that e_{S}[c_{V} S^{∗}(V) ⊗ 𝔽_{p}^{Out(P)/Out_{0}(P,V)}]^{GL_{P} (V)} is a S^{∗}(V)^{GL(V)}–submodule of the free S^{∗}(V)^{GL(V)}–module S^{∗}(V)⊗𝔽_{p}^{Out(P)/Out_{0}(P,V)}. As dim S^{∗}(V)^{GL(V)} = rk V, it suffices to show that if S occurs in the permutation module 𝔽_{p}[Out(P)/Out_{0}(P, V)], then S occurs in

[c_{V} S^{∗}(V) ⊗ 𝔽_{p}^{Out(P)/Out_{0}(P,V)}]^{GL_{P} (V)}.

The group ring 𝔽_{p}[GL_{P} (V)] occurs as a submodule of S^{∗}(V) [Alp, p.45], and thus as a submodule of c_{V} S^{∗}(V). Thus S will occur in [c_{V} S^{∗}(V)⊗𝔽_{p}^{Out(P)/Out_{0}(P,V)}]^{GL_{P} (V)} if it occurs in [𝔽_{p}[GL_{P} (V)] ⊗ 𝔽_{p}^{Out(P)/Out_{0}(P,V)}]^{GL_{P} (V)}. But this last 𝔽_{p}[Out(P)]– module rewrites as 𝔽_{p}^{Out(P)/Out_{0}(P,V)}. □

6. Proof of Theorem 2.3 and related results

Theorem 2.1 tells us that calculating dim e_{S}H^{∗}(P) amounts to understanding the 𝔽_{p}[Out(P)]–module composition factors of the permutation modules

𝔽_{p}[Out(P)/Out_{0}(P, V)],

when V < P is a maximal elementary abelian subgroup.

In this section, we use group theory to find various conditions ensuring that all irreducibles occur in this way.

6.1. Composition factors of permutation modules. We begin with an elementary, but useful, lemma about permutation modules.

Lemma 6.1. Let H be a subgroup of a finite group G. Then (a) ⇒ (b) ⇒ (c), and, if H is also normal, (c) ⇒ (a).

(a) H is a p–group.

(b) Every irreducible 𝔽_{p}[G]–module occurs as a submodule of 𝔽_{p}[G/H].

(c) Every irreducible 𝔽_{p}[G]–module occurs as a composition factor of 𝔽_{p}[G/H].

Proof. We begin by reminding the reader that permutation modules are self dual, so an irreducible S occurs as a submodule of 𝔽_{p}[G/H] if and only if S occurs as a quotient module. This happens if and only if there exists a nonzero homomorphism 𝔽_{p}[G/H] → S, which is equivalent to S^{H} ≠ 0.

If H is a p–group, then M^{H} ≠ 0 for all nonzero 𝔽_{p}[H]–modules M. Thus (a) implies (b), and (b) implies (c) is obvious.

Now suppose that H is normal. Then the composition factors of 𝔽_{p}[G/H] are just the irreducible 𝔽_{p}[G/H]–modules pulled back to G. But if H is not a p–group, then G/H has fewer p′ conjugacy classes than G, and thus fewer irreducible modules. Thus (c) implies (a) under the normality assumption. □

Remark 6.2. It is not hard to see that, if H is not normal, then neither implication (b) ⇒ (a) nor (c) ⇒ (b) need hold. For the former, let p = 3, G = SL_{2}(𝔽_{3}), and H = ℤ/2 permuting a basis for (𝔽_{3})^{2}. For the latter, let p = 3, G = Σ_{3}, and H = ℤ/2.

Corollary 6.3. If a p–group P has a unique maximal elementary abelian subgroup V, then every irreducible occurs in 𝔽_{p}[Out(P)/Out_{0}(P, V)] if and only if Out_{0}(V) is a p–group.

Proof. The hypothesis implies that Out_{0}(P, V) arises as the kernel of an evident homomorphism Out(P) → Aut(V), and thus is normal in Out(P). Now the lemma applies. □

6.2. p′–automorphisms of p–groups. We will use various well known results about detecting automorphisms of a p–group P of order prime to p.

The most classic is due to Burnside and Hall, and goes as follows. Let Φ(P) be the Frattini subgroup of P, so that P/Φ(P) = H_{1}(P; 𝔽_{p}).

Proposition 6.4 ([Gor, Thm.5.1.4]). The kernel of the homomorphism Aut(P) → Aut(P/Φ(P)) is a p–group.

With groups A and B as indicated, the Thompson A × B lemma ([Asch, (24.2)], [Gor, Thm.5.3.4]) immediately applies to prove the next proposition.

Proposition 6.5. Let B be an abelian subgroup of a finite p–group P, and let A = {α ∈ Aut(P) | α(g) = g for all g ∈ C_{P} (B)}. Then A is a p–group.

Corollary 6.6. For all elementary abelian V < P, the kernel of the homomorphism Aut(P, V) → Aut(C_{P} (V)) is a p–group. Thus if Aut(C_{P} (V)) is a p–group so is Aut(P, V), and if Aut_{0}(C_{P} (V), V) is a p–group so is Aut_{0}(P, V).

The next result requires that p be odd. Let Ω_{1}(P) be the subgroup of P generated by the elements of order p.

Proposition 6.7 ([Gor, Thm.5.3.10]). Let P be a p–group, with p odd. Then the kernel of the homomorphism Aut(P) → Aut(Ω_{1}(P)) is a p–group.

Corollary 6.8. Let Q is p–central, with p odd, and with maximal central elementary abelian subgroup C. Then Aut_{0}(Q, C) is a p–group.

This follows from the proposition, noting that C = Ω_{1}(Q), since Q is p–central.

Proof of Theorem 2.3. The first statement follows immediately from Corollary 6.6. Now we note that, if V < P is a maximal elementary abelian subgroup, then C_{P} (V) is p–central with V as its maximal central elementary abelian subgroup. Thus Corollary 6.8 applies to prove the second statement. □

6.3. 2–central group examples. We elaborate on Example 2.4.

Example 6.9. Let G_{t} be the 2–Sylow subgroup of SU_{3}(𝔽_{2^{t}}). This is a 2–central group of rank t, and if C_{t} is its maximal elementary abelian subgroup, Out_{0}(G_{t}, C_{t}) contains a cyclic subgroup of order 2^{t} + 1, and thus is not a 2–group.

This will be a consequence of a few facts^{4} about G_{t}.

Let q = 2^{t}. Given a ∈ 𝔽_{q^{2}}, let ā = a^{q}, so that 𝔽_{q} identifies with the set of b ∈ 𝔽_{q^{2}} such that b + b̄ = 0. Then G_{t} ⊂ SU_{3}(𝔽_{q}) ⊂ GL_{3}(𝔽_{q^{2}}) can be described as

G_{t} = \{A(a, b) = ( \begin{matrix}1 & a & b \\ 0 & 1 & ā \\ 0 & 0 & 1 \end{matrix} ) : a, b ∈ 𝔽_{q^{2}} with b + b̄ = aā\} .

The center C_{t} of G_{t} is the set of matrices of the form A(0, b): note that then b ∈ 𝔽_{q}. All other elements have order 4, and thus G_{t} is 2–central sitting in the central extension

C_{t} → G_{t} → G_{t}/C_{t},

with C_{t} ≃ 𝔽_{q} and G_{t}/C_{t} ≃ 𝔽_{q^{2}}.

The normalizer of G_{t} in SU_{3}(𝔽_{2^{t}}) is the semidirect product G_{t} >◁T_{t}, where T_{t} is the set of matrices

\{D(c) = ( \begin{matrix}c & 0 & 0 \\ 0 & c̄c^{−1} & 0 \\ 0 & 0 & c̄^{−1} \end{matrix} ) : c ∈ 𝔽_{q^{2}}^{×} \} .

Direct computation shows that

D(c)A(a, b)D(c^{−1}) = A(c̄^{−1}c^{2}a, c̄cb) = A(c^{2−q}a, c^{q+1}b).

From this, one deduces that T_{t} → Aut(G_{t}/C_{t}), and thus T_{t} → Aut(G_{t}), is monic if gcd(q − 2, q^{2} − 1) = 1. This is the case unless q = 2, i.e., t = 1. Meanwhile, the kernel of the homomorphism T_{t} → Aut(C_{t}) identifies with the kernel of the multiplicative norm 𝔽_{q^{2}}^{×} → 𝔽_{q}^{×}, and thus is cyclic of order q + 1. Thus, if t ≥ 2, Aut_{0}(G_{t}, C_{t}) = ker{Aut(G_{t}) → Aut(C_{t})} contains a cyclic group of order (q + 1).

(When t = 1, G_{t} = Q_{8}, and so Aut_{0}(G_{t}, C_{t}) also contains a group of order (q + 1) = 3.)

Remark 6.10. The group G_{2} has been of interest to those studying group cohomology. Even though it is 2–central, a presentation of its cohomology ring is remarkably nasty to write down: see the calculations for group number # 187 of order 64 in [Ca, CTVZ]. It is the smallest group with nontrivial products in its essential cohomology [Gr]. The calculation of the nilpotent length of p–central groups in [K4, K5] shows that its nilpotent length is 14, which seems likely to be maximal among all groups of order 64. Presumably, the groups G_{t} for larger t are similarly interesting.

4The authors are very grateful to David Green for showing us his unpublished ‘Appendix B’ to [Gr], which is our source of information about these groups.

6.4. p–groups with reduced cohomology. Recall that P has reduced cohomology if and only if H^{∗}(P) is detected by restriction to the various V < P, i.e., H^{∗}(P) → H^{∗}(P) is monic.

We restate Proposition 2.5.

Proposition 6.11. If H^{∗}(P) is reduced, and V < P is a maximal elementary abelian subgroup, then C_{P} (V) = V.

Recalling that C_{P} (V) is p–central if V < P is maximal, the proposition is a consequence of the next two lemmas.

Lemma 6.12. If H^{∗}(P) is reduced, so is H^{∗}(C_{P} (V)) for any elementary abelian V < P.

Lemma 6.13. If Q is a p–central p–group with reduced cohomology, then Q is elementary abelian.

Proof of Lemma 6.12. Properties of the functor T_{V} imply that if M ∈ U is reduced so is T_{V} M: recalling that T_{V} preserves monomorphisms, property (b) of Proposition 4.1 impies this. Now one uses that, given V < P, H^{∗}(C_{P} (V)) is a (canonical) direct summand in T_{V} H^{∗}(P). (See [HLS2, I(5.2)] for a calculation of T_{V} H^{∗}(P).) □

Proof of Lemma 6.13. Let C be the center of Q. The hypothesis is that H^{∗}(Q) → H^{∗}(C) is monic, which implies that the inflation map H^{∗}(Q/C) → H^{∗}(Q) is zero. But H^{1}(Q/C) → H^{1}(Q) is always monic, so we conclude that H^{1}(Q/C) = 0. But, since Q/C is a p–group, this means that Q/C is trivial, i.e., C = Q. □

6.5. A remark about odd prime p–central groups. Using a result of Henn and Priddy, Theorem 2.3 has an addendum. For odd prime p–central groups P, the refinement of Conjecture A described in Remark 3.1 holds: indeed, dim eH^{∗}(P) = rk(P) for all idempotents e ∈ A(P, P)⊗𝔽_{p} that project to a nonzero element under the retraction of algebras A(P, P)⊗𝔽_{p} → 𝔽_{p}[Out(P)]. Here dimension is defined via the Poincaré series, and we remind the reader that A(P, P) is the double Burnside ring.

The argument goes as follows. Given an irreducible 𝔽_{p}[Out]–module S, let e_{S} ∈ 𝔽_{p}[Out(P)] and ẽ_{S} ∈ A(P, P) ⊗ 𝔽_{p} be the associated primitive idempotents^{5}. Then e_{S}H^{∗}(P) = ẽ_{S}H^{∗}(P) ⊕ M^{∗} where M^{∗} is a finite direct sum of modules of the form ẽ_{T} H ∗ (Q) for appropriate pairs (Q, T). By [HP, Prop.1.6.1 and Lem.2.1], if P is an odd prime p–central group, the only Q’s that can occur here will be proper retracts of P. But such groups will necessarily have strictly smaller ranks. Thus dim M^{∗} < rk(P), and so dim ẽ_{S}H^{∗}(P) = dim e_{S}H^{∗}(P) = rk(P).

7. Proof of Theorem 2.8

7.1. A formula for R̄_{d}H^{∗}(P). Recall that R̄_{0}H^{∗}(P) = lim_{V ∈A^{C}(P)} H^{∗}(V). Using the fact that morphisms in A(V) factor as inner automorphisms composed with inclusions, this rewrites as follows: there is a natural isomophism

R̄_{0}H^{∗}(P) = Eq \{[∏_{V} H^{∗}(V)]^{Inn(P)} \begin{matrix}μ \\ ⟶ \\ ⟶ \\ ν \end{matrix} ∏_{V_{1}<V_{2}} H^{∗}(V_{1})\} ,

5Thus ẽ_{S} projects to e_{S}, and, in the terminology of [MP], e_{S}BP is the super dominant summand associated to the dominant summand ẽ_{S}BP.

where the products are over objects and inclusions in A^{C}(P), and μ and ν are induced by 1 : H^{∗}(V_{1}) → H^{∗}(V_{1}) and η^{∗} : H^{∗}(V_{2}) → H^{∗}(V_{1}) for each inclusion η : V_{1} < V_{2} in A^{C}(P).

We begin this section with a formula for R̄_{d}H^{∗}(P) that generalizes this.

The ingredients of our formula are the following.

First, recall our notation from the introduction: if W is a central elementary abelian p–subgroup of Q, then P_{W} H^{∗}(Q) denotes the primitives in the H^{∗}(W)– comodule H^{∗}(Q).

An automorphism α : P → P induces an isomorphism of unstable algebras P_{α(V)}H^{∗}(C_{P} (α(V))) → P_{V} H^{∗}(C_{P} (V)). Thus P_{V} H^{∗}(C_{P} (V)) is an Aut(P, V)– module, and the full automorphism group Aut(P) acts on the product

∏_{V ∈A^{C}(P)} H^{∗}(V) ⊗ P_{V} H^{d}(C_{P} (V)).

Similarly, an inclusion η : V_{1} < V_{2} induces an inclusion C_{P} (V_{2}) < C_{P} (V_{1}), and thus a map of unstable algebras η_{∗} : P_{V_{1}}H^{∗}(C_{P} (V_{1})) → P_{V_{1}}H^{∗}(C_{P} (V_{2})). We will also denote by i : P_{V_{2}}H^{∗}(C_{P} (V_{2})) → P_{V_{1}}H^{∗}(C_{P} (V_{2})) the evident inclusion of primitives.

Starting from formulae in [HLS2], in [K5], the author will deduce the following formula for R̄_{d}H^{∗}(P).

Proposition 7.1. R̄_{d}H^{∗}(P) is naturally isomorphic to the equalizer

Eq \{[∏_{V} H^{∗}(V) ⊗ P_{V} H^{d}(C_{P} (V))]^{Inn(P)} \begin{matrix}μ \\ ⟶ \\ ⟶ \\ ν \end{matrix} ∏_{V_{1}<V_{2}} H^{∗}(V_{1}) ⊗ P_{V_{1}}H^{d}(C_{P} (V_{2}))\} ,

where the products are over objects and proper inclusions in A^{C}(P), and μ and ν are induced by

1 ⊗ η_{∗} : H^{∗}(V_{1}) ⊗ P_{V_{1}}H^{d}(C_{P} (V_{1})) → H^{∗}(V_{1}) ⊗ P_{V_{1}}H^{d}(C_{P} (V_{2}))

and

η^{∗} ⊗ i : H^{∗}(V_{2}) ⊗ P_{V_{2}}H^{d}(C_{P} (V_{2})) → H^{∗}(V_{1}) ⊗ P_{V_{1}}H^{d}(C_{P} (V_{2}))

for each proper inclusion η : V_{1} < V_{2} in A^{C}(P).

Otherwise said, x = (x_{V}) ∈ [∏_{V} H^{∗}(V) ⊗ P_{V} H^{d}(C_{P} (V))]^{Inn(P)} is in R̄_{d}H^{∗}(P) exactly when the components are related by (1 ⊗ η_{∗})(x_{V_{1}}) = (η^{∗} ⊗ i)(x_{V_{2}}), for each proper inclusion η : V_{1} < V_{2} in A^{C}(P).

7.2. R̄_{d}H^{∗}(Q) when Q is p–central. The formula for R̄_{d}H^{∗}(Q) simplies dramatically if Q is p–central, as the first product in the equalizer formula is indexed by one element, and the second is empty.

Proposition 7.2. Suppose Q is p–central with maximal central elementary abelian subgroup C. Then

R̄_{d}H^{∗}(Q) ≃ H^{∗}(C) ⊗ P_{C}H^{d}(Q).

As a consequence we can prove Proposition 2.10, which said that if P is 2–central, then every irreducible 𝔽_{2}[Out_{0}(Q, C)]–module occurs as a composition factor in P_{C}H^{∗}(Q).

Proof of Proposition 2.10. Every irreducible 𝔽_{2}[Out(Q)]–module occurs in H^{∗}(Q), so the same is true for irreducible 𝔽_{2}[Out_{0}(Q, C)]–modules. Thus every irreducible 𝔽_{2}[Out_{0}(Q, C)]–module occurs in R̄_{∗}H^{∗}(Q) ≃ H^{∗}(C) ⊗ P_{C}H^{∗}(Q). But, by definition, H^{∗}(C) is a trivial 𝔽_{2}[Out_{0}(Q, C)]–module, so the irreducibles must occur in the other factor, P_{C}H^{∗}(Q). □

Remark 7.3. In [K5], we will prove that if Q is p–central, the finite unstable algebra P_{C}H^{∗}(Q) ⊂ H^{∗}(Q) identifies with the subalgebra of locally finite elements: x ∈ H^{∗}(Q) such that Ax ⊂ H^{∗}(Q) is finite.

7.3. Proof of Theorem 2.8. It will be useful to let M^{∗}(V) denote P_{V} H^{∗}(C_{P} (V)). Similarly, given V_{1} < V_{2}, let M^{∗}(V_{2}, V_{1}) = P_{V_{1}}H^{∗}(C_{P} (V_{2})).

Given V < P, define ρ_{V} to be the composite

R̄_{d}H^{∗}(P) ⊆ [∏_{U} H^{∗}(U) ⊗ M^{d}(U)]^{Inn(P)} ⟶ [H^{∗}(V) ⊗ M^{d}(V)]^{W_{P} (V)},

where the first map is the inclusion of 𝔽_{p}[Out(P)]–modules given in Proposition 7.1, and the second is the canonical projection of 𝔽_{p}[Out(P, V)]–modules. Then let

ρ̃_{V} : R̄_{d}H^{∗}(P) → Ind_{Out(P,V)}^{Out(P)}([H^{∗}(V) ⊗ M^{d}(V)]^{W_{P} (V)})

to be the 𝔽_{p}[Out(P)]–module map adjoint to the 𝔽_{p}[Out(P, V)]–module map ρ_{V}.^{6}

Definition 7.4. Given an irreducible 𝔽_{p}[Out(P)]–module S, and d ≥ 0, define

sup_{d}(S) = {V ∈ ob A^{C}(P) | S occurs in ρ̃_{V} (R̄_{d}H^{∗}(P))}.

It is easy to check that V ∈ sup_{d}(S) if and only if ρ_{α(V)}(e_{S}R̄_{d}H^{∗}(P)) ≠ 0 for some α ∈ Aut(P). Equivalently, V ∈ sup_{d}(S) if and only if there exists x = (x_{U}) ∈ e_{S}R̄_{d}H^{∗}(P) with x_{α(V)} ≠ 0 for some α ∈ Aut(P).

We will show that sup_{d}(S) has the properties described in Theorem 2.8.

Clearly, given α ∈ Aut(P), V ∈ sup_{d}(S) implies that α(V) ∈ sup_{d}(S). Thus sup_{d}(S) is a union of Aut(P) orbits in ob A^{C}(P), i.e., property (a) of Theorem 2.8 holds.

Also evident is the first statement of property (c): sup_{d}(S) ≠ ∅ if and only if e_{S}R̄_{d}H^{∗}(P) ≠ 0.

Not much harder is the proof of property (b), which we restate here as a lemma.

Lemma 7.5. Given η : V_{1} < V_{2}, suppose that η_{∗} : M^{d}(V_{1}) → M^{d}(V_{2}, V_{1}) is monic. Then V_{1} ∈ sup_{d}(S) implies that V_{2} ∈ sup_{d}(S).

Proof. Suppose x = (x_{U}) ∈ e_{S}R̄_{d}H^{∗}(P) has x_{α(V_{1})} ≠ 0, with α ∈ Aut(P). Let α(η) : α(V_{1}) < α(V_{2}) be the inclusion. Then η_{∗} monic implies that α(η)_{∗} is also monic. Since (α(η)^{∗} ⊗ i)(x_{α(V_{2})}) = (1 ⊗ α(η)_{∗})(x_{α(V_{1})})), it follows that x_{α(V_{2})} ≠ 0. □

Now suppose e_{S}R̄_{d}H^{∗}(P) ≠ 0. As e_{S}R̄_{d}H^{∗}(P) embeds in ∏_{V} ρ_{V} (e_{S}R̄_{d}H^{∗}(P)),

dim e_{S}R̄_{d}H^{∗}(P) = max{dim ρ_{V} (e_{S}R̄_{d}H^{∗}(P)) | V ∈ sup_{d}(S)}.

Furthermore, dim ρ̃_{V} (e_{S}R̄_{d}H^{∗}(P)) = max{dim ρ_{α(V)}(e_{S}R̄_{d}H^{∗}(P)) | α ∈ Aut(P)}, and there is an obvious bound: dim ρ_{V} (e_{S}R̄_{d}H^{∗}(P)) ≤ dim H^{∗}(V) = rk(V). Thus the next two lemmas prove properties (c) and (d) of Theorem 2.8.

6Recall that if H is a subgroup of a finite group G, induction Ind_{H}^{G} is both left and right adjoint to restriction Res_{H}^{G}.

Lemma 7.6. If V is maximal in sup_{d}(S), then dim ρ̃_{V} (e_{S}R̄_{d}H^{∗}(P)) ≥ rk(V).

Lemma 7.7. If V is maximal in ob A^{C}(P), then S occurs in ρ̃_{V} (R̄_{d}H^{∗}(P)) if and only if S occurs in Ind_{Out(P,V)}^{Out(P)}([𝔽_{p}[Aut(P, V)/Aut_{0}(P, V)] ⊗ M^{d}(V)]^{W_{P} (V)}).

To prove these, we need to introduce yet more notation.

Given V < P, let Ess^{∗}(V) = ker{M^{∗}(V) → ∏_{V <U} M^{∗}(U)}, where the product is over all proper inclusions. Then α_{∗}(Ess^{∗}(V)) = Ess^{∗}(α(V)) for all α ∈ Aut(P). In particular, Ess^{∗}(V) is an 𝔽_{p}[Aut(P, V)]–submodule of M^{∗}(V).

Lemma 7.8. There is a map of Out(P)–modules

f̃_{V} : Ind_{Out(P,V)}^{Out(P)}([c_{V} H^{∗}(V) ⊗ Ess^{d}(V)]^{W_{P} (V)}) → R̄_{d}H^{∗}(P)

such that ρ̃_{V} ∘ f̃_{V} is the canonical inclusion induced by

c_{V} H^{∗}(V) ⊗ Ess^{d}(V) ⊆ H^{∗}(V) ⊗ M^{∗}(V).

Proof. It is equivalent to define an Out(P, V)–module map

f_{V} : [c_{V} H^{∗}(V) ⊗ Ess^{d}(V)]^{W_{P} (V)} → R̄_{d}H^{∗}(P)

such that ρ̃_{V} ∘f_{V} is an appropriate inclusion. Given x ∈ [c_{V} H^{∗}(V)⊗Ess^{d}(V)]^{W_{P} (V)} define f_{V} (x) ∈ [∏_{U} H^{∗}(U) ⊗ M^{d}(U)]^{Inn(P)} by letting its U^{th} component, f_{V} (x)_{U}, be (α^{−1∗} ⊗ α_{∗})(x) if U = α(V) with α ∈ Inn(P), and 0 otherwise.

We claim that f_{V} (x) is, in fact, in R̄_{d}H^{∗}(P). For given η : V_{1} < V_{2} with V_{1} conjugate to V, (1 ⊗ η_{∗})(f_{V} (x)_{V_{1}}) = 0 because η_{∗}(Ess^{d}(V)) = 0. Similarly, given η : V_{1} < V_{2} with V_{2} conjugate to V, (η^{∗} ⊗ i)(f_{V} (x)_{V_{2}}) = 0 because η^{∗}(c_{V}) = 0.

Finally, checking that ρ̃_{V} ∘ f_{V} is what it should be is straightforward. □

Next we observe that Ind_{Out(P,V)}^{Out(P)}([H^{∗}(V) ⊗ M^{d}(V)]^{W_{P} (V)}) is a H^{∗}(V)^{GL(V)}– module with the property that each nonzero cyclic S^{∗}(V)^{GL(V)}–submodule is free.

Proof of Lemma 7.6. If V is maximal in sup_{d}(S), then, after possibly replacing V by α(V) for some α ∈ Aut(P), there exists x ∈ e_{S}R̄_{s}H^{∗}(V) with x_{V} ≠ 0 in [H^{∗}(V) ⊗ M^{d}(V)]^{W_{P} (V)}, and V maximal with this property. Reasoning as in the proof of Lemma 7.5, it follows that x_{V} ∈ [H^{∗}(V) ⊗ Ess^{d}(V)]^{W_{P} (V)}. By the proof of Lemma 7.8,

c_{V} S^{∗}(V)^{GL(V)}ρ̃_{V} (x) ⊂ ρ̃_{V} (R̄_{d}H^{∗}(P)).

By the observation above, we also have

c_{V} S^{∗}(V)^{GL(V)}ρ̃_{V} (x) ⊂ e_{S} Ind_{Out(P,V)}^{Out(P)}([H^{∗}(V) ⊗ M^{d}(V)]^{W_{P} (V)}).

Thus c_{V} S^{∗}(V)^{GL(V)}ρ̃_{V} (x) is contained in the intersection of these, i.e.

c_{V} S^{∗}(V)^{GL(V)}ρ̃_{V} (x) ⊂ ρ̃_{V} (e_{S}R̄_{d}H^{∗}(P)).

But, as a graded 𝔽_{p}–vector space, c_{V} S^{∗}(V)^{GL(V)}ρ̃_{V} (x) is just the (|c_{V} | + |x|)^{th} suspension of S^{∗}(V)^{GL(V)}, and so

rk(V) = dim c_{V} S^{∗}(V)^{GL(V)}ρ̃_{V} (x) ≤ dim ρ̃_{V} (e_{S}R̄_{d}H^{∗}(P)).
□

Proof of Lemma 7.7. If V is maximal in A^{C}(P), then the product defining Ess^{∗}(V) is empty, so Ess^{∗}(V) = M^{∗}(V). From Lemma 7.8, we thus deduce that we have inclusions of 𝔽_{p}[Out(P)]–modules

Ind_{Out(P,V)}^{Out(P)}([c_{V} H^{∗}(V) ⊗ M^{d}(V)]^{W_{P} (V)}) ⊂ ρ̃_{V} (R̄_{d}H^{∗}(P))

and

ρ̃_{V} (R̄_{d}H^{∗}(P)) ⊂ Ind_{Out(P,V)}^{Out(P)}([H^{∗}(V) ⊗ M^{d}(V)]^{W_{P} (V)}).

Thus the next lemma will finish the proof. □

Lemma 7.9. Let S be an irreducible 𝔽_{p}[Out(P)]–module, and M be an 𝔽_{p}[Aut(P, V)]– module. Then the following are equivalent.

(a) S occurs in Ind_{Out(P,V)}^{Out(P)}([c_{V} H^{∗}(V) ⊗ M]^{W_{P} (V)}).

(b) S occurs in Ind_{Out(P,V)}^{Out(P)}([H^{∗}(V) ⊗ M]^{W_{P} (V)}).

(c) S occurs in Ind_{Out(P,V)}^{Out(P)}([𝔽_{p}[Aut(P, V)/Aut_{0}(P, V)] ⊗ M]^{W_{P} (V)}).

Proof. Noting once again that multiplication by c_{V} is a monomorphism on the module in (b), we see that (a) and (b) are equivalent.

To simplify notation, let G = Aut(P, V)/Aut_{0}(P, V) and W = W_{P} (V). Thus W < G < GL(V). The equivalence of (b) and (c) follows immediately from the following claim: if T is an irreducible 𝔽_{p}[G/W]–module, then T occurs in (H^{∗}(V)⊗ M)^{W} if and only if T occurs in (𝔽_{p}[G] ⊗ M)^{W}. The claim is a consequence of the well known fact that, for any G < GL(V), every finite 𝔽_{p}[G]–module embeds in S^{∗}(V) [Alp], and thus also in H^{∗}(V). Thus T occurs in (H^{∗}(V) ⊗ M)^{W} if and only if T occurs in (N ⊗ M)^{W} for some finite 𝔽_{p}[G]–module N, and this happens if and only if T occurs in (a direct sum of copies of) the module (𝔽_{p}[G] ⊗ M)^{W}. □

8. Conjecture C implies Conjecture A

If H is a subgroup of G, and M is an 𝔽_{p}[H]–module, it is easy to verify the following: if every irreducible 𝔽_{p}[H]–module occurs in M then every irreducible 𝔽_{p}[G]–module occurs in Ind_{H}^{G} M. In light of Corollary 2.9, Conjecture A would thus be implied by

8.1. If V < P is maximal, then every irreducible 𝔽_{p}[Out(P, V)]–module occurs in (𝔽_{p}[Aut(P, V)/Aut_{0}(P, V)] ⊗ P_{V} H^{∗}(C_{P} (V)))^{W_{P} (V)}.

The 𝔽_{p}[Out(P, V)]–module (𝔽_{p}[Aut(P, V)/Aut_{0}(P, V)] ⊗ P_{V} H^{∗}(C_{P} (V)))^{W_{P} (V)} contains 𝔽_{p}[Aut(P, V)/Aut_{0}(P, V)]^{W_{P} (V)} ⊗ P_{V} H^{∗}(C_{P} (V))^{W_{P} (V)} as a submodule, and this submodule rewrites as

Ind_{Out_{0}(P,V)}^{Out(P,V)}(P_{V} H^{∗}(C_{P} (V))^{W_{P} (V)}).

Conjecture B is the statement that every irreducible 𝔽_{p}[Out_{0}(P, V)]–module occurs in P_{V} H^{∗}(C_{P} (V))^{W_{P} (V)}. Thus Conjecture B would imply 8.1, and we have proved

Lemma 8.2. If a pair (P, V) satisfies Conjecture B, then P satisfies Conjecture A.

Now we prove that Conjecture C implies Conjecture B, establishing Theorem 2.11. More precisely, let V < P be maximal, and assume that every irreducible 𝔽_{2}[Out_{0}(C_{P} (V), V)]–module occurs as a submodule in P_{V} H^{∗}(C_{P} (V)). We will show that then every irreducible 𝔽_{2}[Out_{0}(P, V)]–module occurs as a composition factor in P_{V} H^{∗}(C_{P} (V))^{W_{P} (V)}.

The key here is again to use Corollary 6.6, our application of the Thompson A×B lemma, which tells us that the kernel of Aut_{0}(P, V) → Aut_{0}(C_{P} (V), V) is a 2–group. Thus our assumption lets us conclude that every irreducible 𝔽_{2}[Aut_{0}(P, V)]–module occurs as a submodule in P_{V} H^{∗}(C_{P} (V)).

Now let S be an irreducible 𝔽_{2}[Out_{0}(P, V)]–module. Equivalently, S can be regarded as an irreducible 𝔽_{2}[Aut_{0}(P, V)]–module fixed by the subgroup W_{P} (V). From our remarks in the last paragraph, there exists an inclusion S ⊂ P_{V} H^{∗}(C_{P} (V)) of 𝔽_{2}[Aut_{0}(P, V)]–modules, and thus an inclusion

S = S^{W_{P} (V)} ⊂ P_{V} H^{∗}(C_{P} (V))^{W_{P} (V)}

of 𝔽_{2}[Out_{0}(P, V)]–modules.

9. Some 2–central groups satisfying Conjecture C

If P is a 2–central group, then C = Ω_{1}(P) < P is the unique maximal elementary subgroup and is central. In this case, we write Out_{0}(P) for Out_{0}(P, C).

Recall that P satisfies Conjecture C if every irreducible 𝔽_{2}[Out_{0}(P)]–module occurs as a submodule of the algebra of primitive P_{C}H^{∗}(P). In this section we verify this conjecture for many 2–central groups.

Our strategy is the following. First, we verify the conjecture for various indecomposable 2–central groups, including all of order up to 64. Second, we find various side hypotheses on a pair (P, Q) ensuring that if P and Q satisfy Conjecture C, so does P × Q.

All calculations have been done by hand, using information listed on the website [Ca] or the book [CTVZ]. For starters, we note that a 2–group P is 2–central exactly when rk(Z(P)) = rk(P), so 2–central groups can be easily identified. We occasionally use notation like ‘32#10’ to denote a 2–group: this would be the group of order 32 and numbered 10 on [Ca] or in [CTVZ].

9.1. Indecomposable 2–central groups of order dividing 64. Let P be 2– central. Recall that P_{C}H^{0}(P) = 𝔽_{2}, the trivial module, so P rather trivially satisfies Conjecture C whenever Out_{0}(P) is a 2–group. This is often the case.

Lemma 9.1. Let P be an indecomposable 2–central groups P of order dividing 64. Then Out_{0}(P) is a 2–group unless P is one of the three groups 8#5 (Q_{8}), 64#162, or 64#187 (G_{2} as in Example 2.4).

We sketch how one checks this. Firstly, here is a table of the indecomposable 2–central groups P having any nontrivial automorphisms of order prime to 2, and the order of Out(P) modulo its 2–Sylow subgroup.

\begin{matrix}P & 2′ part of |Out(P)| \\ 8#5 & 3 \\ 32#18 & 3 \\ 64#30 & 3 \\ 64#82 & 3 \\ 64#93 & 3 \\ 64#145 & 3 \\ 64#153 & 21 \\ 64#162 & 3 \\ 64#187 & 15 \end{matrix}

We note that 8#5 is Q_{8}, 64#153 is the 2–Sylow subgroup of the Suzuki group Sz(8), and 64#187 is the 2–Sylow subgroup of SU_{3}(𝔽_{4}) as in Example 2.4.

Now one checks that, except when P is 8#5 or 64#162, an automorphism of P of order 3 does not fix Ω_{1}(P), and similarly an automorphism of 64#153 of order 7 does not fix Ω_{1}(P).

Lemma 9.2. The three groups 8#5 (Q_{8}), 64#162, and 64#187 (G_{2}) all satisfy Conjecture C.

To see this, we look at each group in turn.

If P = Q_{8} with center C, then Out_{0}(P) is the symmetic group of order 6, and there is a 2 dimensional 𝔽_{2}[Out_{0}(P)] irreducible S. As 𝔽_{2}[Out_{0}(P)]–modules, S ≃ H^{1}(P/Φ(P)) = H^{1}(P/C) = P_{C}H^{1}(P), and we are done.

If P is 64#162 with center C, the order of Out_{0}(P) has the form 3·2^{a}. It follows that there must be exactly one irreducible 𝔽_{2}[Out_{0}(P)]–module S on which an element of order 3 acts nontrivially. H^{1}(P/Φ(P)) = H^{1}(P/C) = P_{C}H^{1}(P) = H^{1}(P) is 4 dimensional, with basis {z, y, x, w} as in [CTVZ]. Examining the action of the generating automorphisms on H^{1}(P), one sees that the 2 dimensional subspace with basis {x + z, w} is an Out_{0}(P)–submodule acted on nontrivially by an automorphism of order 3. Thus this submodule must be isomorphic to S, and we are done.

If P is 64#187 with center C, the order of Out_{0}(P) has the form 5 · 2^{a}. It follows that there must be exactly one irreducible 𝔽_{2}[Out_{0}(P)]–module S on which an element of order 5 acts nontrivially. H^{1}(P/Φ(P)) = H^{1}(P/C) = P_{C}H^{1}(P) = H^{1}(P) is four dimensional, and is an irreducible 𝔽_{2}[ℤ/5]–module as seen in §6.3. Thus P_{C}H^{1}(P) ≃ S, and we are done.

In summary, in this subsection we have checked

Proposition 9.3. If P is an indecomposable 2–central subgroup of order dividing 64, with C = Ω_{1}(P), then |Out_{0}(P)| has the form 2^{a}, 3 · 2^{a}, or 5 · 2^{a}, and all irreducible 𝔽_{2}[Out_{0}(P)]–modules occur as submodules of P_{C}H^{0}(P) = 𝔽_{2} or P_{C}H^{1}(P) = H^{1}(P/C).

9.2. On Out_{0}(P × Q). The following Kunneth formula for primitives is easily verified.

Lemma 9.4. If C < P and D < Q are central elementary abelian subgroups, then

P_{C×D}H^{∗}(P × Q) = P_{C}H^{∗}(P) ⊗ P_{D}H^{∗}(Q).

Definition 9.5. Call a pair (P, Q) of 2–central groups good if the algebra homomorphism

𝔽_{2}[Out_{0}(P) × Out_{0}(Q)] → 𝔽_{2}[Out_{0}(P × Q)]

induces an isomorphism on their semisimple quotients.

In general, given two finite groups G and H, the irreducible 𝔽_{2}[G × H] modules will be direct summands of modules of the form S ⊗ T, where S is an irreducible 𝔽_{2}[G]–module, and T is an irreducible 𝔽_{2}[H]–module. Thus the lemma implies

Corollary 9.6. If a pair (P, Q) of 2–central groups is good, and both P and Q satisfy Conjecture C, so does P × Q.

Now we aim to prove that pairs of 2–central groups are frequently good. The following is our key observation.

Lemma 9.7. Let P and Q be 2–central. Given

α = ( \begin{matrix}α_{P} & α_{P,Q} \\ α_{Q,P} & α_{Q} \end{matrix} ) : P × Q → P × Q

in Aut_{0}(P × Q), the following conditions on the component homomorphisms must hold.

(a) Ω_{1}(P) ⊆ Ker(α_{Q,P}), and Ω_{1}(Q) ⊆ Ker(α_{P,Q}).

(b) Im(α_{Q,P}) ⊆ Z(Q), and Im(α_{P,Q}) ⊆ Z(P).

(c) α_{P} ∈ Aut_{0}(P), and α_{Q} ∈ Aut_{0}(Q).

Proof. As α must be the identity when restricted to Ω_{1}(P × Q) = Ω_{1}(P) × Ω_{1}(Q), condition (a) is clear. Furthermore, α_{P} and α_{Q} must be the identity when respectively restricted to Ω_{1}(P) and Ω_{1}(Q).

As α is a homomorphism, it follows that Im(α_{Q,P}) and Im(α_{Q}) must commute, and similarly Im(α_{P,Q}) and Im(α_{P}) must commute.

It follows that both (b) and (c) will follow if we can show that α_{P} and α_{Q} are isomorphisms.

We are assuming that α is invertible. Thus there exists

β = ( \begin{matrix}β_{P} & β_{P,Q} \\ β_{Q,P} & β_{Q} \end{matrix} ) : P × Q → P × Q

such that β ∘ α : P × Q → P × Q is the identity. In particular,

9.8. The composite P ⟶^{α_{P} ,α_{Q,P}} P × Q ⟶^{β_{P} ·β_{P,Q}} P is the identity.

We will show that β_{P} ∘ α_{P} : P → P is epic. Let γ = β_{P,Q} ∘ α_{Q,P} : P → P. Then 9.8 says

x = β_{P} (α_{P} (x))γ(x) for all x ∈ P.

By condition (a), Ω_{1}(P) ⊆ Ker(γ), and so the order of γ(x) will be strictly less than the order of x for all x ≠ e ∈ P. Thus we can prove that β_{P} ∘ α_{P} is epic by induction on the order of x, as follows. Given x ∈ P, there exists z such that β_{P} (α_{P} (z)) = γ(x). But then

β_{P} (α_{P} (xz)) = β_{P} (α_{P} (x))β_{P} (α_{P} (z)) = β_{P} (α_{P} (x))γ(x) = x.
□

Theorem 9.9. A pair (P, Q) of 2–central groups is good if every homomorphism P/Ω_{1}(P) → Z(Q) has image contained in Φ(Q).

Proof. Let Ãut_{0}(P) denote the image of Aut_{0}(P) → Aut(P/Φ(P)). Proposition 6.4 implies that 𝔽_{2}[Aut_{0}(P)] → 𝔽_{2}[Ãut_{0}(P)] induces an isomorphism on semisimple quotients, and similarly with P replaced by Q and P × Q.

Using parts (a) and (b) of the lemma, the hypothesis that every homomorphism P/Ω_{1}(P) → Z(Q) has image contained in Φ(Q) implies that for all α ∈ Aut_{0}(P ×Q), the component α_{Q,P} : P → Q induces the zero map P/Φ(P) → Q/Φ(Q). Again using the lemma, it follows that

Ãut_{0}(P × Q) = ( \begin{matrix}Ãut_{0}(P) & B \\ 0 & Ãut_{0}(Q) \end{matrix} ) ,

where B is a quotient of Hom(Q/Ω_{1}(Q), Z(P)), and is thus a 2–group.

The inclusion Ãut_{0}(P) × Ãut_{0}(Q) → Ãut_{0}(P × Q) thus has a retract with kernel a 2–group, and so induces an isomorphism on the semisimple quotients of the associated 𝔽_{2} group rings. □

Corollary 9.10. A pair (P, Q) of 2-central groups is good if any of the following conditions hold.

(a) Z(Q) ⊆ Φ(Q).

(b) P is elementary abelian.

(c) P/[P, P]Ω_{1}(P) is elementary abelian, and Q has no ℤ/2 summands.

(d) Q = ℤ/2^{t} and P has exponent dividing 2^{t}.

In summary, we have shown

Proposition 9.11. If P and Q satisfy any of the conditions in the last corollary, and both satisfy Conjecture C, then P × Q also satisfies Conjecture C.

9.3. Many 2–central groups satisfy Conjecture C. When one does a census of indecomposable noncyclic 2-central groups, one finds 1 of order 8, 2 of order 16, 9 of order 32, and 41 of order 64.^{7} Condition (a) of Corollary 9.10 holds for all but one of these groups.

Proposition 9.12. Let Q be an indecomposable 2–central group of order dividing 64. Then Z(Q) ⊆ Φ(Q) unless Q is cyclic or 64#30. The group 64#30 has exponent 4, and thus satisfies the condition for P listed in (c) of Corollary 9.10.

Checking this is made easier by the observation that a 2-central group Q has no ℤ/2 summands exactly when Ω_{1}(Q) ⊆ Φ(Q). Thus Z(Q) ⊆ Φ(Q) whenever Z(Q) is elementary abelian. This is the case for all but 11 of the 53 indecomposable noncyclic 2-central groups in question.

We can now easily prove Theorem 2.12, which we restate here.

7Group 32#25 is mistakenly listed at indecomposable in both [Ca] and [CTVZ]. It is isomorphic to ℤ/2 × Q_{16}.

Theorem 9.13. Conjecture C is true for all 2–central groups that can be written as the product of groups of order dividing 64.

Proof. By Proposition 9.3, all indecomposable 2–central groups of order dividing 64 satisfy Conjecture C. Now suppose a 2–central group P is a product P = A×B ×C where A is abelian with no ℤ/2 summands, B is a product of noncyclic indecomposable summands of order dividing 64, and C is elementary abelian. Repeatedly using condition (d) of Corollary 9.10, one deduces that A satisfies Conjecture C.^{8} Repeatedly using conditions (a) and (c), one then deduces that A × B satisfies Conjecture C, and finally that P = A × B × C does, thanks to condition (b). □

10. A discussion of Example 2.14

In this section we discuss in detail Example 2.14, and use this to illustrate Theorem 2.8 and its proof. So let P be the group 64#108: all of our information about this group is available on [Ca].

10.1. Subgroup structure. The commutator subgroup Z = [P, P] has order 2. The center C is elementary abelian of rank 2, and C = Φ(P), so Z < C and P/C is elementary abelian of rank 4. There is a unique maximal elementary abelian group V of rank 3, and its centralizer K = C_{P} (V) has order 32, so that W_{P} (V) = G/K ≃ ℤ/2. More precisely, the 2–central group K is isomorphic to (ℤ/2)^{2} × Q_{8}, with Q_{8} embedded so that V ∩ Q_{8} = Z.^{9}

We have the following picture of A^{C}(P):

(10.1)  [figure: C; V; ℤ/2]

and from this it is already clear that H^{∗}(P) = H^{∗}(V)^{ℤ/2}.

10.2. H_{1}(P) as an 𝔽_{2}[Out(P)]–module. The group Out(P) has order 768 = 3·2^{8}. From this, one can deduce that 𝔽_{2}[Out(P)] has precisely two irreducible modules, the trivial module ‘1’, and another one which we will call ‘S’ of dimension at least two. Furthermore, S occurs as a composition factor in an 𝔽_{2}[Out(P)]–module M if and only if any element of order 3 in Out(P) acts nontrivially on M.

Now we consider H_{1}(P) = P/C as an 𝔽_{2}[Out(P)]–module. It is nontrivial by Proposition 6.4, thus S occurs as a composition factor. Notice that each of the subgroups C,V, and K are characteristic, and recall that Inn(P) acts trivially on H_{1}(P). Thus there is a composition series of 𝔽_{2}[Out(P)]–modules:

V/C ⊂ K/C ⊂ P/C,

with composition factors V/C, K/V, and P/K. By dimension considerations, V/C and P/K are copies of the trivial module, and thus K/V must be S, which is thus two dimensional.

10.3. H^{∗}(V) as an 𝔽_{2}[Aut(P)]–module. An automorphism of P order 3 fixes V.^{10} Thus, as an Aut(P)–module, H^{∗}(V) has only trivial composition factors.

8Indeed Aut_{0}(A) is a 2–group, as is well known.

9In terms of the generators g_{1}, …, g_{6} on the website, Z = ⟨g_{6}⟩, C = ⟨g_{3}, g_{6}⟩, V = ⟨g_{2}, g_{3}, g_{6}⟩, Q_{8} = ⟨g_{4}, g_{5}, g_{6}⟩, and K = ⟨g_{2}, g_{3}, g_{4}, g_{5}, g_{6}⟩. K is maximal subgroup #11.

10See automorphism #7 on the Carlson website.

10.4. P_{V} H^{∗}(K) and P_{C}H^{∗}(K). We have maps of unstable algebras equipped with with Aut(P) action:

(10.2)  P_{V} H^{∗}(K) ↪ P_{C}H^{∗}(K) ⟵^{j^{∗}} P_{C}H^{∗}(P),

where j : K → P is the inclusion. In degree 1, this reads

Hom(K/V, 𝔽_{2}) ↪ Hom(K/C, 𝔽_{2}) ← Hom(P/C, 𝔽_{2}),

and so we see that j^{∗} is onto in degree 1, and that P_{V} H^{1}(K) is a copy of S.

The maps of pairs (Q_{8}, Z) → ((ℤ/2)^{2} × Q_{8}, (ℤ/2)^{2} × Z) = (K, V) induces an isomorphism of algebras:

P_{V} H^{∗}(K) ≃ P_{Z}H^{∗}(Q_{8}).

The algebra P_{Z}H^{∗}(Q_{8}) is familiar: the calculation of H^{∗}(Q_{8}) using the Serre spectral sequence associated to Z → Q_{8} → Q_{8}/Z reveals that P_{Z}H^{∗}(Q_{8}) = Im {H^{∗}(Q_{8}/Z) → H^{∗}(Q_{8})} = B^{∗}, where B^{∗} is the Poincaré duality algebra 𝔽_{2}[x, w]/(x^{2} + xw + w^{2}, x^{2}w + xw^{2}), where x and w both have degree 1. As an 𝔽_{2}[Aut(P)]–module, P_{V} H^{∗}(K) = B^{∗} is given by

B^{d} = \begin{cases}1 & if d = 3 \\ S & if d = 2 \\ S & if d = 1 \\ 1 & if d = 0. \end{cases}

From this we learn that P_{C}H^{∗}(K) ≃ B^{∗}[y] where y is also in degree 1, and thus is generated by elements in degree 1. It follows that j^{∗} : P_{C}H^{∗}(P) → P_{C}H^{∗}(K) is onto, and then that Inn(P) acts trivially on both P_{V} H^{∗}(K) and P_{C}H^{∗}(K).

10.5. R̄_{d}H^{∗}(P). Applying the formula for R̄_{d}H^{∗}(P) in Proposition 7.1 to our group, we see that, for all d, there is a pullback diagram of graded 𝔽_{2}[Out(P)]– modules:

\xymatrix{R̄_{d}H^{∗}(P) \ar[d] \ar[r] & H^{∗}(C) ⊗ P_{C}H^{d}(P) \ar[d]^{1⊗j^{∗}} \\ (H^{∗}(V) ⊗ P_{V} H^{d}(K))^{Inn(P)} \ar[r] & H^{∗}(C) ⊗ P_{C}H^{d}(K).}

By our comments above, this simplifies: for all d, there is a pullback diagram of unstable 𝔽_{2}[Out(P)]–modules:

\xymatrix{R̄_{d}H^{∗}(P) \ar[d] \ar[r] & H^{∗}(C) ⊗ P_{C}H^{∗}(P) \ar[d]^{1⊗j^{∗}} \\ H^{∗}(V)^{ℤ/2} ⊗ P_{V} H^{∗}(K) \ar[r] & H^{∗}(C) ⊗ P_{C}H^{∗}(K).}

As j^{∗} is onto, we conclude

Proposition 10.1. For all d, there is a short exact sequence of unstable 𝔽_{2}[Out(P)]– modules 0 → H^{∗}(C) ⊗ ker j^{d} → R̄_{d}H^{∗}(P) → H^{∗}(V)^{ℤ/2} ⊗ P_{V} H^{d}(K) → 0, where H^{∗}(C) and H^{∗}(V)^{ℤ/2} have only trivial composition factors, while P_{V} H^{∗}(K) ≃ B^{∗}.

We note that in the notation of §7, Ess^{∗}(C) = ker j^{∗} and Ess^{∗}(V) = P_{V} H^{∗}(K). From Proposition 10.1, one can already place the 3’s occurring in the table in

Example 2.14. The 2’s will be determined by knowing the composition factors of the ideal ker j^{∗} ⊂ P_{C}H^{∗}(P). This we discuss next.

10.6. P_{C}H^{∗}(P) and the ideal ker j^{∗}. Obviously, ker j^{0} = 0, and we also know that ker j^{1} is one dimensional. Using more detailed information about H^{∗}(P), we outline how all of the composition factors of ker j^{∗} can be determined.

H^{∗}(P) is generated by classes z, y, x, w, v, u, t of degrees 1,1,1,1,2,5,8 respectively, and the kernel of the restriction H^{∗}(P) → H^{∗}(K) is identified as the ideal (z).^{11} Furthermore, this ideal is a free module on 𝔽_{2}[v, t] on an explicit basis of 24 monomials in the generators of degree 1.

All the degree 1 generators are H^{∗}(C)–primitive, while v and t restrict to algebraically independent elements of H^{∗}(C). It follows that ker j^{∗} = P_{C}H^{∗}(P) ∩ (z) is precisely the 𝔽_{2}–span of this basis of mononomials.

This allows us to conclude the following.

Firstly P_{C}H^{∗}(P) ⊂ H^{∗}(P) is precisely the subalgebra generated by the classes in degree 1. Explicitly, P_{C}H^{∗}(P) is

A^{∗} = 𝔽_{2}[z, y, x, w]/(z^{2}, zy + x^{2} + xw + w^{2}, zy^{2} + x^{2}w + xw^{2}, y^{2}w^{3} + yw^{4} + w^{5}).

For our purposes, this can be simplified as follows. From the second and third relations, one can deduce that x^{3} = z(yx + y^{2}) and w^{3} = z(yw + y^{2}). Using these and the first relation, one deduces that y^{2}w^{3} + yw^{4} + w^{5} = zy^{4}. So we have

A^{∗} = 𝔽_{2}[z, y, x, w]/(z^{2}, zy^{4}, zy + x^{2} + xw + w^{2}, zy^{2} + x^{2}w + xw^{2}).

Secondly, ker j^{∗} = Σ(A^{∗}/Ann(z)), as unstable 𝔽_{2}[Out(P)]–modules. Since z^{2} = zy^{4} = 0, A^{∗}/Ann(z) is a quotient of A^{∗}/(z, y^{4}) = 𝔽_{2}[y, x, w]/(y^{4}, x^{2}+xw+w^{2}, x^{2}w+ xw^{2}) = B^{∗}[y]/(y^{4}). Inspection of the degrees of the monomial basis shows that A^{∗}/Ann(z) has dimension 1,3,5,6,5,3,1 in degrees 0,1,2,3,4,5,6. This agrees with B^{∗}[y]/(y^{4}), and we can conclude that Ann(z) = (z, y^{4}), and then

Proposition 10.2. ker j^{∗} ≃ Σ(B^{∗}[y]/(y^{4})) as unstable 𝔽_{2}[Out(P)]–modules.

References

[Alp] J. L. Alperin, Local Representation Theory, Cambridge Studies in Advanced Math. 11, Cambridge U. Press, 1986.

[Asch] M. Aschbacher, Finite Group Theory, 2nd edition, Cambridge Studies in Advanced Math. 10, Cambridge U. Press, 2000.

[BoZ] D. Bourguiba and S. Zarati, Depth and the Steenrod algebra. With an appendix by J. Lannes. Invent. Math. 128 (1997), 589–602.

[BrH] C.Broto and H.-W. Henn, Some remarks on central elementary abelian p–subgroups and cohomology of classifying spaces, Quart. J. Math. 44 (1993), 155–163.

[BrZ1] C.Broto and S.Zarati, Nil–localization of unstable algebras over the Steenrod algebra, Math. Zeit. 199 (1988), 525–537.

[BrZ2] C.Broto and S.Zarati, On sub–A_{p}^{∗}–algebras of H^{∗}(V), Springer L. N. Math. 1509 (1992), 35–49.

[Ca] Jon Carlson, Mod 2 cohomology of 2 groups, MAGMA computer computations on the website http://www.math.uga.edu/∼lvalero/cohointro.html.

11See the description of the depth essential cohomology on the website.

[CTVZ] J.F.Carlson, L.Townsley, L.Valeri-Elizondo,and M. Zhang, Cohomology rings of finite groups. With an appendix, Calculations of cohomology rings of groups of order dividing 64, by Carlson, Valeri-Elizondo and Zhang. Algebras and Applications 3, Kluwer, Dordrecht, 2003.

[Cr] M.C.Crabb, Dickson-Mui invariants, Bull. London Math. Soc. 37 (2005), 846–856.

[D] L.E.Dickson, A fundamental system of invariants of the general modular linear group with a solution of the form problem, T..A.M.S. 12 (1911), 75–98.

[DS] T. Diethelm and U. Stammbach, On the module structure of the mod p cohomology of a p-group, Arch. Math. 43 (1984), 488–492.

[D] J. Duflot, Depth and equivariant cohomology, Comm. Math. Helv. 56 (1981), 627–637.

[Gor] D. Gorenstein, Finite Groups, 2nd edition, Chelsea Publishing, New York, 1980.

[Gr] D.J.Green, The essential ideal in group cohomology does not square to zero, J. Pure Appl. Algebra 193 (2004), no. 1-3, 129–139.

[HK] J.C.Harris and N.J.Kuhn, Stable decompositions of classifying spaces of finite abelian pgroups, Math. Proc. Camb. Phil. Soc. 103(1988), 427–449.

[H] H.-W. Henn, Finiteness properties of injective resolutions of certain unstable modules over the Steenrod algebra and applications, Math. Ann. 291 (1991), 191-203.

[HLS1] H.-W. Henn, J. Lannes, and L. Schwartz, The categories of unstable modules and unstable algebras modulo nilpotent objects, Amer. J. Math. 115 (1993), 1053-1106.

[HLS2] H.–W.Henn, J.Lannes, and L.Schwartz, Localizations of unstable A-modules and equivariant mod p cohomology, Math. Ann. 301 (1995), 23–68.

[HP] H.–W.Henn and S.Priddy, p–nilpotence, classifying space indecomposability, and other properties of almost all finite groups, Comm. Math. Helv. 69(1994), 335–350.

[Hi] R. Higginbottom, Ph.D. thesis, University of Virginia, 2005.

[K1] N.J.Kuhn, Character rings in algebraic topology, Advances in Homotopy Theory (Cortona 1988), L. M. S. Lectures Notes 139(1989), 111–126.

[K2] N.J.Kuhn, Generic representations of the finite general linear groups and the Steenrod algebra:I, Amer. J. Math. 116(1994), 327–360.

[K3] N.J.Kuhn, On topologically realizing modules over the Steenrod algebra, Ann. Math. 141(1995), 321–347.

[K4] N.J.Kuhn, Cohomology primitives associated to central extensions, Oberwolfach Reports 2 (2005), 2383–2386.

[K5] N.J.Kuhn, Cohomology primitives associated to central extensions, in preparation.

[LZ] J.Lannes, and S.Zarati, Sur les U–injectifs, Ann. Sci. Ec. Norm. Sup. 19 (1986), 303–333.

[MP] J.Martino and S.Priddy, On the dimension theory of dominant summands, Adams Memorial Symposium on Algebraic Topology, 1 (Manchester, 1990), L.M.S. Lect. Note Ser. 175 (1992), 281–292.

[Mat] H.Matsumura, Commutative algebra, 2nd edition, Math. Lect. Note. Series, Benjamin, 1980.

[N] G. Nishida, Stable homotopy type of classifying spaces of finite groups, Algebraic and topological theories (Kinosaki 1984), Kinokuniya, Tokyo, 1986, 391–404.

[Q1] D. Quillen, The spectrum of an equivariant cohomology ring I, Ann. Math. 94 (1971), 549-572.

[Q2] D. Quillen, The spectrum of an equivariant cohomology ring II, Ann. Math. 94 (1971), 573-602.

[S1] L. Schwartz, La filtration nilpotente de la catégorie U et la cohomologie des espaces de lacets , Algebraic Topology – Rational Homotopy (Louvain la Neuve 1986), S. L. N. M. 1318 (1988), 208-218.

[S2] L. Schwartz, Modules over the Steenrod algebra and Sullivan’s fixed point conjecture, Chicago Lectures in Math., U. Chicago Press, 1994.

[Sy] P. Symonds, The action of automorphisms on the cohomology of a p-group, Math. Proc. Camb. Phil. Soc. 127 (1999), 495–496.

Department of Mathematics, University of Virginia, Charlottesville, VA 22904

   E-mail address: njk4x@virginia.edu