THE COMPUTATION OF THE COHOMOLOGY RINGS OF ALL GROUPS OF ORDER 128

DAVID J. GREEN AND SIMON A. KING

Abstract. We describe the computation of the mod-2 cohomology rings of all 2328 groups of order 128. One consequence is that all groups of order less than 256 satisfy the strong form of Benson’s Regularity Conjecture.

1. Introduction

We have computed the mod-2 cohomology rings of all 2328 groups of order 128. The results are available in human-readable form at our website [21]. They may also be read into the computer for further inspection using our package [24] for the Sage [34] computer algebra system.

Several different methods have been used to compute the mod-p cohomology rings of p-groups. D. J. Rusin computed the mod-2 cohomology rings of all 51 groups of order 32 using the Eilenberg–Moore spectral sequence [33]. The approach most commonly used today is due to J. F. Carlson, who devised a way to compute cohomology rings completely, based on a suitably large initial segment of the minimal projective resolution [10]. In this way, Carlson successfully computed the cohomology of all 267 groups of order 64 [13, Appendix]. More recently, G. Ellis and P. Smith demonstrated that the Lyndon–Hochschild–Serre spectral sequence can also be used for automatic machine computation [16].

Being the most widely used computational technique, Carlson’s minimal resolution approach has been the subject of several improvements. D. J. Benson gave an improved test for completion [6], which led to the regularity conjectures discussed below. In this paper, we expose various improvements to Benson’s criterion.

For constructing the minimal projective resulution, we use the noncommutative Gröbner basis approach devised by the first author [19], which has since been used to construct the minimal resolution of the Sylow 2-subgroup of the Mathieu group M_{24} out to degree 13: the Poincaré series is

Date: 14 January 2010.

2000 Mathematics Subject Classification. Primary 20J06; Secondary 20-04, 20D15.

King was supported by DFG grant GR 1585/4-1 during most of this research. Green received travel assistance from the same source and from GR 1585/4-2.

1 + 4t + 12t^{2} + 25t^{3} + 49t^{4} + 85t^{5} + 143t^{6} + 222t^{7} + 336t^{8} + 485t^{9}

+ 691t^{10} + 953t^{11} + 1298t^{12} + 1726t^{13} + O(t^{14}) .

In this paper we describe how further work on the minimal resolutions approach allowed us to compute the mod-2 cohomology rings of all 2328 groups of order 128. Specifically we derive several results which improve the usablity of Benson’s test for completion, and the second author created a new and much more efficient package for cohomology computations.

In more detail, King rewrote large parts of Green’s cohomology program [19]. The new package makes extensive use of the computer algebra system Singular [22], and in order to make the various components of the program work well together we used Sage [34], developing extensive Cython code. Our Sage package is available online [24], and interested users can perform their own calculations in this system. In particular, the package contains the cohomology ring for each group of order 64, and the cohomology ring of each group of order 128 is automatically downloaded from a web repository when it is required. The cohomology rings and induced homomorphisms can then easily be transformed into objects of Singular.

Rusin got his list of groups from the book by Hall and Senior [23]. We took our list of groups from the Small Groups library [7], which is available in the systems GAP and Magma.

Using our package we computed the cohomology rings of all 2328 groups of order 128. We apply these calculations to study the statistical effectiveness of Duflot’s lower bound for the depth of the cohomology ring. We remark in Question 9.3 that this topic is worthy of further study. On inspecting the a-invariants of these cohomology rings we observe that amongst the 2595 groups of order 64 or 128, there are only fifteen groups where the a-invariants are not weakly increasing – see Table 2 on page 16. We also use our computations to verify the strong form of Benson’s regularity conjecture (our Conjecture 11.2) for all groups of order less than 256: see Theorem 11.4.

Structure of the paper. We start by recalling the notion of a filter-regular sequence in Section 2. In Section 3 we give a computationally effective variant (Theorem 3.1) of Benson’s completeness criterion. This variant uses two filterregular parameter systems. One parameter system has to lie in low degrees, but need not be explicitly constructed: we give an existence result in Section 4. The other parameter system has to be constructed, and the degrees should not be too high. In Section 5 we adapt Benson’s Dickson invariant construction to filter-regular parameters in order to obtain parameters in lower degrees. Section 6 provides a way to construct new parameters from old, whilst reducing the degrees. In Section 7 we discuss our implementation of these methods.

The last five sections are devoted to our computational findings. We discuss the computations for the groups of order 128 in Section 8. We consider depth and the Duflot bound in Section 9, introducing the concepts of group-theoretic defect and Duflot excess. The a-invariants are discussed in Section 10. The paper concludes with sections 11–13 on the strong form of Benson’s regularity conjecture.

2. Filter-regular sequences

We recall the key concepts from Benson’s paper [6].

Hypothesis 2.1. Let k be a field of characteristic p > 0; let A = ⨁_{n≥0} A_{n} be a graded commutative k-algebra which is connected and noetherian. Put differently, A_{0} = k and A is finitely generated.

Elements of A will always be understood to be homogeneous. If x ∈ A_{n} then we write |x| = n. Recall that the elements of positive degree form a maximal ideal, which we shall denote by A_{+}.

Definition. [6, §3] Let A be a ring satisfying Hypothesis 2.1.

a) A sequence of (homogeneous) elements h_{1}, …, h_{r} ∈ A_{+} is called filterregular if for each 1 ≤ i ≤ r the annihilator of h_{i} in the quotient ring A/(h_{1}, …, h_{i−1}) has bounded degree – or equivalently if it is a finite dimensional k-vector space.

b) A filter-regular system of parameters is a system of parameters which is simultaneously a filter-regular sequence. Note that in this case the quotient A/(h_{1}, …, h_{r}) also has bounded degree.

c) A filter-regular sequence of parameters h_{1}, …, h_{r} has type (d_{0}, d_{1}, …, d_{r}) ∈ ℤ^{r+1} if

d_{0} ≥ −1 ,  d_{i−1} − 1 ≤ d_{i} ≤ d_{i−1} for each 1 ≤ i ≤ r,

and the annihilator of h_{i+1} in A/(h_{1}, …, h_{i}) lies in degrees ≤ d_{i} +∑_{j=0}^{i} |h_{i}| for all 0 ≤ i ≤ r, where h_{r+1} = 0.

d) A system of parameters is strongly quasi-regular if it is filter-regular of type (0, −1, …, −r), and very strongly quasi-regular if it is filter-regular of type (−1, −2, …, −r, −r).

Lemma 2.2. Assume Hypothesis 2.1, and let h_{1}, …, h_{r} be a sequence of homogeneous elements in A_{+}. Let d be the Krull dimension of A. Then the following are equivalent:

a) h_{1}, …, h_{r} is a filter-regular system of parameters.

b) h_{1}, …, h_{r} is a filter-regular sequence, r ≤ d, and A/(h_{1}, …, h_{r}) is a finite k-vector space.

c) h_{1}, …, h_{r}, 0 is a filter-regular sequence and r ≤ d.

Remark. If A has bounded degree, then every sequence in A is filter-regular.

Proof. Recalling what filter-regularity means for the term h_{r+1} = 0, one sees the equivalence of the last two statements. The first implies the second, since then r = d and A/(h_{1}, …, h_{r}) has Krull dimension zero. And the second implies the first, for A is a finite module for k[h_{1}, …, h_{r}], and so if h_{1}, …, h_{r} were algebraically dependent then the Krull dimension would have to be less than r ≤ d. □

Remark. Recall from Benson’s Theorem 4.5 that all filter-regular systems of parameters for A have the same type. By his Corollary 4.8, the cohomology ring H^{∗}(G, k) has a strongly quasi-regular system of parameters if and only if its Castelnuovo–Mumford regularity is zero.

3. A variant of Benson’s test for completion

Let G be a finite group and k a field of characteristic p dividing G. Benson’s test for completion [6, Theorem 10.1] makes two somewhat conflicting uses of a filter-regular system of parameters for H^{∗}(G, k). First one has to construct explicit filter-regular parameters, in order to determine the type. Secondly one has to combine the type information with the sum of the parameter degrees in order to obtain a degree bound. The problem is that one wants parameters in low degrees for the degree sum, whereas explicit parameter construction is easier in higher degrees.

These two conflicting uses could lead to computations taking too long to terminate. In this section we show (Theorem 3.1) that the conflict can be resolved by using two different systems of filter-regular parameters. Later in Proposition 4.2 we show that a non-constructive existence proof is sufficient for the parameters needed for the degree bound.

Following Benson [6, §10] we write τ_{N}H^{∗}(G, k) for the approximation to the graded commutative ring H^{∗}(G, k) obtained by taking all generators and relations in degree ≤ N. Recall that if K is an extension field of k then H^{∗}(G, K) ≅ K ⊗_{k} H^{∗}(G, k), and observe that therefore

τ_{N}H^{∗}(G, K) ≅ K ⊗_{k} τ_{N}H^{∗}(G, k) .

The following result is based on Theorem 10.1 of Benson’s paper [6]. Recall that the p-rank of G is the rank of the largest elementary abelian p-subgroup.

Theorem 3.1. Suppose that r = r_{p}(G) ≥ 2. Suppose that we have two homogeneous systems of elements ζ_{1}, …, ζ_{r} ∈ τ_{N}H^{∗}(G, k) and κ_{1}, …, κ_{r} ∈ τ_{N}H^{∗}(G, K) for some extension field K of k, with the following properties:

a) Each system is a filter-regular homogeneous system of parameters in its respective ring. Let (d_{0}, …, d_{r}) be the type of ζ_{1}, …, ζ_{r}.

b) The images of the ζ_{i} form a homogeneous system of parameters in H^{∗}(G, k).

c) Let n_{i} = |κ_{i}|, and let

α′ = max_{0≤i≤r−2}(d_{i} + i) .

Then n_{i} ≥ 2 for all i, and
(1)  N > max(α′, 0) + ∑_{j=1}^{r}(n_{j} − 1) .

Then the map τ_{N}H^{∗}(G, k) → H^{∗}(G, k) is an isomorphism.

Moreover, if a Sylow p-subgroup of G has centre of p-rank at least two, then in Eqn. (1) we only have to require ≥.

Proof. Since H^{∗}(G, K) ≅ K ⊗_{k} H^{∗}(G, k), it follows that the ζ_{i} are a filter-regular system of parameters of type (d_{0}, …, d_{r}) in τ_{N}H^{∗}(G, K) as well. By (i) ⇒ (iii) in Theorem 4.5 of [6], it follows that a_{𝔪}^{i}(τ_{N}H^{∗}(G, K)) ≤ d_{i} for each 0 ≤ i ≤ r. Hence α′ ≥ α for α as in Benson’s Theorem 10.1. Since the ζ_{i} are a homogeneous system of parameters for H^{∗}(G, K), so are the κ_{i}. Hence applying Benson’s Theorem 10.1 to the κ_{i} we deduce that τ_{N}H^{∗}(G, K) ≅ H^{∗}(G, K) and therefore τ_{N}H^{∗}(G, k) ≅ H^{∗}(G, k). The last part follows from the corrected version^{1} of Benson’s Remark 10.6(v). □

4. Filter-regular parameter systems and finite extension fields

As announced above, we now derive a non-constructive method for demonstrating the existence of a filter-regular system of parameters in low degrees.

Lemma 4.1. Suppose that A satisfies Hypothesis 2.1 and that additionally for some d ≥ 1 the algebra A is finitely generated as a module over A(d), the subalgebra generated by A_{d}. Then there is a finite extension L/k and an element x ∈ L ⊗_{k} A_{d} such that Ann_{A_{L}}(x) is a finite dimensional L-vector space.

Proof. We start by choosing an infinite field K that is an algebraic extension field of k. Let x_{1}, …, x_{n} be a k-basis of A_{d}. By assumption, A_{K} is finite over the subalgebra K[x_{1}, …, x_{n}]. One version of Noether normalisation now says that there are K-linear combinations y_{1}, …, y_{r} of the x_{i} such that A_{K} is finite over the polynomial subalgebra K[y_{1}, …, y_{r}]. Note that each y_{j} ∈ (A_{K})_{d}. Let Y ⊆ (A_{K})_{d} be the K-vector space with basis the y_{i}.

Denote by 𝔪_{K} the irrelevant ideal 𝔪_{K} = (A_{K})_{>0}. As A_{K} is noetherian we have Ass(A_{K}) = {𝔭_{1}, …, 𝔭_{m}}, a finite set. Suppose that Y ⊆ 𝔭_{i} for some i. If x ∈ 𝔪_{K} then x is integral over K[y_{1}, …, y_{r}] and therefore x^{s} lies in the ideal (Y) of A_{K} for some s ≥ 1. So x ∈ 𝔭_{i} and therefore 𝔭_{i} = 𝔪_{K}.

This means that if 𝔭_{i} ≠ 𝔪_{K} then 𝔭_{i} ∩ Y is a proper K-subspace of Y. As K is infinite, Y is not a union of finitely many proper subspaces. So there is at least one x ∈ Y such that x lies in no 𝔭_{i} except possibly 𝔪_{K}. Suppose a ∈ Ann_{A_{K}}(x), then x ∈ Ann_{A_{K}}(a), so there is a b ∈ A_{K} such that x ∈ Ann_{A_{K}}(ab) ∈ Ass A_{K}. It follows that Ann_{A_{K}}(ab) = 𝔪_{K} and therefore ab ∈ Ann_{A_{K}}(𝔪_{K}). Now, Ann_{A_{K}}(𝔪_{K}) is a finitely generated ideal in the noetherian ring A_{K} and therefore a finite dimensional K-vector space by virtue of its own definition. So there is a N ≥ 1 such that |c| ≤ N for every c ∈ Ann_{A_{K}}(𝔪_{K}). But then |a| ≤ N, and so Ann_{A_{K}}(x) is a finite dimensional K-vector space too.

1It is the centre of the Sylow p-subgroup which should have rank at least two.

Finally write x as a K-linear combination of the k-basis x_{1}, …, x_{n} of A_{d}. Let k ⊆ L ⊆ K be the intermediate field generated by the coefficients in this representation of x. Then L/k is a finite extension. □

Proposition 4.2. Suppose that A satisfies Hypothesis 2.1 and that additionally for some d ≥ 1 the algebra A is finitely generated as a module over A(d), the subalgebra generated by A_{d}. Then there is a finite extension K/k and a filterregular system of parameters x_{1}, …, x_{n} for A_{K} with |x_{i}| = d for every i.

Proof. Induction over the Krull dimension of A. If dim(A) = 0 then dim_{k}(A) < ∞ and we are done, with n = 0. If dim(A) ≥ 1 then by Lemma 4.1 there is a finite extension L/k and an x_{1} ∈ (A_{L})_{d} such that Ann_{A_{L}}(x_{1}) is finite dimensional. This means that dim(A_{L}/(x_{1})) < dim(A). So we have the inductive step. □

5. The weak rank-restriction condition

Benson’s test for completion requires us to construct an explicit filter-regular system of parameters, in order to determine the common type of all such systems in H^{∗}(G, k). He shows in [6, Corollary 9.8] that certain Chern classes of the regular representation of the Sylow p-subgroup of G provide one such explicit parameter system. These Chern classes restrict to each elementary abelian subgroup as (powers of) the appropriate Dickson invariants. Given information about restriction to subrings it is a straightforward task to compute classes with the appropriate restriction patterns. However the degrees involved can be inconveniently large, even for our variant (Theorem 3.1) of Benson’s test.

Restricting to the case of a p-group, we shall now see (Lemma 5.2) that filterregular parameters in smaller degrees can be obtained using the Dickson invariants for a complement of the centre.

Definition. (c.f. [6, §8]) Let G be a p-group with p-rk(G) = K. Set C to be Ω_{1}(Z(G)), the greatest central elementary abelian subgroup of G. Homogeneous elements ζ_{1}, …, ζ_{K} ∈ H^{∗}(G) satisfy the weak rank restriction condition if for each elementary abelian subgroup V ≥ C of G the following holds, where s = p-rk(V):

The restrictions of ζ_{1}, …, ζ_{s} to V form a homogeneous system of parameters for H^{∗}(V); and the restrictions of ζ_{s+1}, …, ζ_{K} are zero.

Lemma 5.1. If ζ_{1}, …, ζ_{K} ∈ H^{∗}(G) satisfy the weak rank restriction condition then they constitute a filter-regular system of parameters.

Proof. By a well known theorem of Quillen (see e.g. Evens’ book [17]), ζ_{1}, …, ζ_{K} is a homogeneous system of parameters for H^{∗}(G). The proof of Theorem 9.6 of [6] applies just as well to parameters satisfying the weak rank restriction condition, because if E is an arbitrary rank i elementary abelian subgroup of G, then setting V = ⟨C, E⟩ one has V ≥ C and C_{G}(V) = C_{G}(E), yet the rank of V is at least as large as the rank of E. So the restrictions of ζ_{1}, …, ζ_{i} to H^{∗}(C_{G}(E)) do form a regular sequence, by the same argument based on the Broto–Henn approach [8] to Duflot’s theorem. □

Lemma 5.2. Let G be a p-group with p-rk(G) = K and p-rk(Z(G)) = r. Let C = Ω_{1}(Z(G)). Suppose that homogeneous elements ζ_{1}, …, ζ_{K} ∈ H^{∗}(G) satisfy the following conditions:

a) The restrictions of ζ_{1}, …, ζ_{r} to H^{∗}(C) form a regular sequence there; and b) For each rank r+s elementary abelian subgroup V ≥ C of G the restrictions

of ζ_{r+s+1}, …, ζ_{K} to V are zero, and for 1 ≤ i ≤ s the restrictions of ζ_{r+i} to V is a power of the ith Dickson invariant in H^{∗}(V/C).

Then ζ_{1}, …, ζ_{K} ∈ H^{∗}(G) is a filter-regular system of parameters for H^{∗}(G). Moreover such systems of parameters exist.

Remark. By the ith Dickson invariant we mean the one which restricts nontrivially to dimension i subspaces, but has zero restriction to smaller subspaces. That is, if i < j then the ith Dickson invariant lies in lower degree than the jth Dickson invariant.

Proof. Such classes clearly satisfy the weak rank restriction condition. The existence of ζ_{1}, …, ζ_{r} already follows from Evens’ theorem that the cohomology ring of an arbitrary subgroup H ≤ G is a finitely generated module over the image of restriction from G. For the ζ_{r+i}: these are given by restrictions to each elementary abelian subgroup, and these restrictions satisfy the compatibility conditions that one expects from genuine restrictions, c.f. Quillen’s work on the spectrum of a cohomology ring [31]. This means that – on raising these defining restrictions by sufficiently high pth powers – the ζ_{r+i} do indeed exist. □

Remark. The point is that Lemma 5.2 is a recipe for constructing a filter-regular system of parameters. It follows from work of Kuhn [25] that the first r parameters ζ_{1}, …, ζ_{r} may be chosen from amongst the ring generators of H^{∗}(G). We refer to such generators as Duflot regular.

An additional saving follows from the fact that if ζ_{1}, …, ζ_{K} is a system of parameters and ζ_{1}, …, ζ_{K−1} is filter-regular, then the whole system is automatically filter-regular. This means that one can replace the ζ_{K} of Lemma 5.2 by any element that completes a system of parameters.

6. New filter-regular seqences from old

As we observed in the introduction to §5, we should where possible choose a filter-regular system of parameters in low degrees, as this also makes the determination of the type more straightforward. Used together, the factorization method (Lemma 6.1) and the nilpotent alteration method (Lemma 6.4) can be very effective at reducing the degrees of a filter-regular parameter system.

Lemma 6.1. Suppose that A is a ring satisfying Hypothesis 2.1. Suppose that p_{1}, …, p_{r} is a filter-regular sequence consisting of homogeneous elements of A_{+}. Suppose further that f_{1}, …, f_{r} is another sequence of homogeneous elements in A_{+} such that f_{i} | p_{i} for each i. Then f_{1}, …, f_{r} is filter-regular too.

In particular, if p_{1}, …, p_{r} is a filter-regular system of parameters, then so is f_{1}, …, f_{r}.

Proof. Last part: p_{1}, …, p_{r} is a filter-regular system of parameters if and only if its length is the Krull dimension and p_{1}, …, p_{r}, 0 is a filter-regular sequence.

Main part: By induction on r it suffices to prove that f_{1}, p_{2}, …, p_{r} is filterregular. If f_{1} = p_{1} then we are done, so we may assume that p_{1} = f_{1}g_{1} with f_{1}, g_{1} ∈ A_{+}. We show that f_{1}, p_{2}, …, p_{r} is filter-regular by induction on r. If f_{1}x = 0 then p_{1}x = 0, so x is in bounded degree and the case r = 1 is done. Now suppose that r ≥ 1. Observe that we may assume that both f_{1}, p_{2}, …, p_{r−1} and g_{1}, p_{2}, …, p_{r−1} are filter-regular.

Suppose that p_{r}x ∈ (f_{1}, p_{2}, …, p_{r−1}) and that x is in large degree. Then p_{r}g_{1}x ∈ (p_{1}, p_{2}, …, p_{r−1}) and g_{1}x is in large degree, so by filter-regularity of p_{1}, …, p_{r} we have

g_{1}x ∈ (p_{1}, …, p_{r−1}) .

Hence for some w ∈ A we have g_{1}(x − f_{1}w) ∈ (p_{2}, …, p_{r−1}). Recall that g_{1}, p_{2}, …, p_{r−1} is filter regular: so as x is in large degree, Lemma 6.2 below says that x − f_{1}w ∈ (p_{2}, …, p_{r−1}). Hence x ∈ (f_{1}, p_{2}, …, p_{r−1}), as required. □

Lemma 6.2. Suppose that the sequence p_{1}, …, p_{r} of homogeneous elements in A_{+} is filter-regular. Suppose that p_{1}x ∈ (p_{2}, …, p_{r}) and that x lies in sufficiently high degree. Then x ∈ (p_{2}, …, p_{r}).

Proof. It is as if we had a regular sequence. Induction on r. Clear for r = 1. Now suppose r ≥ 2 and

p_{1}x = ∑_{i=2}^{r} p_{i}w_{i} .

Then since w_{r} is in sufficiently large degree, there are v_{1}, …, v_{r−1} ∈ A with w_{r} = ∑_{i=1}^{r−1} p_{i}v_{i}. Then p_{1}(x − p_{r}v_{1}) ∈ (p_{2}, …, p_{r−1}), and we are done by induction. □

Lemma 6.3. Let A be a ring satisfying Hypothesis 2.1. Suppose that p_{1}, …, p_{r} is a sequence of homogeneous elements in A_{+}, and that m_{1}, …, m_{r} ≥ 1 are integers. Then: p_{1}, …, p_{r} is a filter-regular sequence if and only if p_{1}^{m_{1}}, …, p_{r}^{m_{r}} is.

Proof. Lemma 6.1 gives us one implication, so we just need to show that the sequence p_{1}^{m_{1}}, …, p_{r}^{m_{r}} is filter-regular if p_{1}, …, p_{r} is. By induction on r it will suffice to show that the sequence p_{1}^{m}, p_{2}, …, p_{r} is filter regular for any m ≥ 1. So suppose that x ∈ A_{+} has sufficiently large degree and satisfies p_{r}x ∈ (p_{1}^{m}, p_{2}, …, p_{r−1}). We need to show that x ∈ (p_{1}^{m}, p_{2}, …, p_{r−1}). We proceed by induction on m, the case m = 1 being known by assumption. Now assume that m ≥ 2.

Since p_{r}x ∈ (p_{1}^{m}, p_{2}, …, p_{r−1}) there is y ∈ A with

p_{r}x ∈ p_{1}^{m}y + (p_{2}, …, p_{r−1}) .

Since p_{r}x ∈ (p_{1}^{m}, p_{2}, …, p_{r−1}) ⊆ (p_{1}, …, p_{r−1}) and x has sufficiently large degree, we have x ∈ (p_{1}, …, p_{r−1}) by filter-regularity of p_{1}, …, p_{r}. Hence there is z ∈ A with

x ∈ p_{1}z + (p_{2}, …, p_{r−1}) .

Hence

p_{1}(p_{r}z − p_{1}^{m−1}y) ∈ (p_{2}, …, p_{r−1}) .

Since p_{1}, …, p_{r−1} is filter-regular and y has sufficiently large degree, we deduce from Lemma 6.2 that p_{r}z − p_{1}^{m−1}y ∈ (p_{2}, …, p_{r−1}) and hence

p_{r}z ∈ (p_{1}^{m−1}, p_{2}, …, p_{r−1}) .

As p_{1}^{m−1}, p_{2}, …, p_{r} is filter-regular by the inductive hypothesis, and as y and therefore z lies in sufficiently high degree, we deduce that z ∈ (p_{1}^{m−1}, p_{2}, …, p_{r−1}). Combining this with x ∈ p_{1}z + (p_{2}, …, p_{r−1}), we see that x ∈ (p_{1}^{m}, p_{2}, …, p_{r−1}).

□

Lemma 6.4. Let A be a ring satisfying Hypothesis 2.1. Suppose that f_{1}, …, f_{r} and g_{1}, …, g_{r} are two sequences of homogeneous elements in A_{+} with the property that each g_{i} only differs from f_{i} by a nilpotent element. Then f_{1}, …, f_{r} is a filter regular sequence if and only if g_{1}, …, g_{r} is.

Proof. Since g_{i} − f_{i} is nilpotent, there is an N_{i} ≥ 1 such that (g_{i} − f_{i})^{N_{i}} = 0. Choose e_{i} ≥ 1 such that p^{e_{i}} ≥ N_{i}, then g_{i}^{m_{i}} = f_{i}^{m_{i}} for m_{i} = p^{e_{i}}. Now apply Lemma 6.3. □

Remark 6.5. The preceding lemma had two benefits for our computations. Firstly, it means that for the purpose of finding filter-regular parameters we may quotient out the nilpotent generators. This reduces the computational complexity. Secondly, doing this often makes it easier to find a factorization of the parameters and so reduce the degrees by applying Lemma 6.1.

7. Notes on the Implementation

In order to succeed with the computation of the modular cohomology rings of all groups of order 128, much care had to be taken on the implementation of the algorithms, in addition to the theoretical progress provided by Proposition 4.2, Lemma 5.2, and Lemma 6.1.

The first author already had C-programs and GAP [18] functions for the computation of modular group cohomology ([19]), which was able to compute the cohomology algebras for all groups of order 64 and of some interesting groups of order 128, but it would not have sufficed to handle all groups of order 128. However, the strength of these programs is the computation of minimal projective resolutions. So apart from some minor improvements we preserved that part of the old programs.

The computation of the ring structure involves fairly standard commutative algebra techniques, namely the computation of Gröbner bases in the context of graded commutativity. First tests indicated that available specialised software was much more efficient than the self made implementation of the first author. So, we henceforth used specialised software, namely Singular [22].

Of course, the computation of the ring structure also involves the computation of many cup products. We compute cup products by means of the Yoneda product. The computation of the cup product is not easy, particularly in high degrees. If x, y and z are previously found cohomology generators and we need to know their cup product, then usually y · z has been computed before. Our implementation takes care that previous intermediate results are cached for later usage.

Caching is a standard task in programming, and so is memory management. We did not want to bother with it and thus wanted to chose a programming language that provides all that out of the box. Moreover, in order to make use of the old C-routines for the computation of minimal projective resolutions, we wanted to be able to link against C-libraries. Also for performance reason, our programming language had to be a compiled (not interpreted) language.

This programming language should come along with software specialised in computational group theory and in commutative algebra (including graded commutative rings). And ideally, it should all be free open source software, so that it is more easy for us to make our results publically available.

It turned out that the Sage project [34] perfectly matched our needs. It is free open source software. On the one hand, it provides a common interface for and contains a distribution of several independent computer algebra systems. These include Singular [22], providing very efficient Gröbner basis computations in commutative and non-commutative rings, and GAP, which is specialised on groups and was used in the old programs of the first author anyway. On the other hand, Sage is based on Cython [3], which is a compiled version of Python. It makes linking against C-libraries easy, and it yielded a huge speed-up compared with interpreted code.

Our project eventually resulted in an optional Sage package, called the p-Group Cohomology Package [24]. Its constituent components are as follows:

• A modified version of the Aachen C-MeatAxe [32] for linear algebra over finite fields

• The first author’s C-routines for the computation of minimal resolutions • GAP functions for computing elementary abelian subgroups, dealing with groups from the SmallGroups library [7] and the Atlas of finite groups [14] • Singular functions that compute a Gröbner basis of the relation ideal, detect new relations, detect filter-regular systems of parameters and compute their filter degree type, and (optionally) construct simultaneous lifts of the Dickson elements

• Cython modules written by the second author, providing methods for homological algebra computations (cup product, restriction to maximal elementary abelian subgroups, induced homomorphisms, etc)

The package also provides general Massey products and restricted Massey powers. The latest version features Persistent Group Cohomology, introduced by Graham Ellis and the second author [15].

Remark 7.1. Surprisingly, the computationally most challenging groups of order 128 were not the ones with extremal structural properties presented in Section 8. Instead, the computations that took the longest were those for groups number 2298 and 2300 (about one week each) and for group number 2327, the extraspecial group 2_{−}^{1+6} (about 11 days). The cohomology of this extraspecial group has of course been known for a long time [30].

Effect of the theoretical improvements. Much of our success is due to the improvements of Benson’s completeness criterion. Without these improvements, the computation would have been far beyond the realm of possibilities.

In the first version of our program, we already used Lemma 5.2 (Dickson elements away from the centre) and Lemma 6.1 (factorisation of parameters). By these improvements, the computation was feasible, taking very roughly 10 months on an AMD Opteron Processor 270 (2 GHz).

When we adopted Proposition 4.2 as well, the computation time dropped to roughly two months, albeit on a different machine, namely a Sun X4450 (2.66 GHz). This already shows the strength of the improvements.

Remark 7.2. The effect of the theoretical improvements can also be demonstrated with the Sylow 2-subgroups of the third Conway group and of the Higman-Sims group. These groups are of p-rank 4. Hence, the unrestricted Dickson elements live in degree 8,12,14 and 15. The centre is of p-rank 1, thus, Benson’s criterion would only apply in degree 46. However, by using the Duflot regular generators and Dickson elements restricted away from the centre, one obtains parameters in degrees 8, 4, 6 and 7, thus reducing the degree bound to 22. But using Proposition 4.2, one can show that the last two parameters can be replaced by elements in degree 2. Hence the computation can be stopped in degree 14, when the last relations were found.

Remark 7.3. A closer inspection shows that the improved Benson criterion also works perfectly for 1779 groups of order 128. That is, the criterion already applies in the degree where the last generator or relation is found. In the remaining case there is a delay before our improved Benson criterion recognises that the ring presentation is complete. This delay is one degree for 341 groups, two degrees for 168 groups, and three degrees for 39 groups of order 128. There is one group with a delay of four degrees: it is group number 2320, which is the direct product of the dihedral group of order 8 and a rank 4 elementary abelian group. However this particular computation is not very difficult: the presentation is complete in degree 2, but we can only prove this in degree 6.

8. Experimental Results

With our package, we computed a minimal presentation, the depth, the Poincaré series and the a-invariants of the modular cohomology rings of

• all 2-groups of order at most 128 and some further 2-groups of order up to 1024, including the Sylow 2-subgroups of the Higman-Sims group and the third Conway group

• all but six 3-groups of order up to 243 and some further groups of order 729

• all 5-groups up to order 625 Our results on the groups of order 64 agree with the results obtained by Carlson [13, Appendix] and by the old programs of the first author [19]. We can also confirm the computation of Adem, Carlson, Karagueuzian and Milgram [1] of the cohomology of the Sylow 2-subgroup of the Higman-Sims group.

The latest version of our package can compute the modular cohomology of all groups of order 64 in about 20 minutes on a Mac Pro desktop machine running at 2.6 Ghz with 8 GB RAM (Darwin Kernel Version 10.0.0) in 64 bit mode.

We obtained the first complete modular cohomology computation of the groups of order 128. Rather than spending a lot of paper, we refer the reader to the ring presentations provided in the internet [21].

The first complete computation in summer 2008 comprised a total of very roughly 10 months computation time. At that time, we only used Lemmas 6.1 and 5.2 to improve Benson’s criterion. After adopting the existence result for filterregular homogeneous systems of parameters in low degrees (Proposition 4.2), the computation time dropped to a total of about 2 months, on a Sun X4450 machine with 2.66 GHz.

Another new result is the cohomology ring of the Sylow 2-subgroup of the third Conway group. Actually we started with two different group presentations and obtained consistent results. The ring is of depth 3. Its Poincaré series is

\frac{t^{9} + t^{8} + 2t^{7} + 3t^{6} + t^{5} + 3t^{4} + 2t^{3} + 3t^{2} + t + 1}{t^{13} − 3t^{12} + 4t^{11} − 4t^{10} + 3t^{9} − t^{8} − t^{5} + 3t^{4} − 4t^{3} + 4t^{2} − 3t + 1}

The a-invariants are −∞, −∞, −∞, −6, −4.

A minimal algebra presentation of the cohomology ring would fill about 20 pages, thus we don’t include it here. Two minimal algebra presentations can be found on our web pages [21].

Extremal cases. We conclude this section by highlighting some extremal cases amongst the cohomology rings of the groups of order 128.

Remark 8.1. Number of ring generators There are 2328 groups of order 128. In all but 6 cases, the number of ring generators for the mod-2 cohomology ring is less than 34. There are three groups with 34 ring generators, and one group each with 36 and with 39 ring generators. The extremal case is group number 836, whose cohomology ring has 65 minimal generators. This group is the Sylow 2-subgroup of one double cover of Sz(8). by less than 65 elements.

Remark 8.2. Number of relations in cohomology ring Group number 836 is the group whose cohomology ring has the most generators, and it also has the largest number of minimal relations: 1859. Then there is one group each where the number of minimal generators is 626, 526, 486 or 461. Each remaining cohomology ring has less than 440 relations.

Remark 8.3. Maximum ring generator degree In 2301 cases, the classes in degree ≤ 11 generate the cohomology ring. There is one group where the maximal degree in a minimal generating set of the cohomology ring is 13 and three groups where it is 14. In 22 cases, the last cohomology generator is found in degree 16; it is not surprising that there are so many, because Duflot regular generators can only be found in prime power degrees. But for the cohomology ring of group number 562, degree 16 is not enough: there is one generator in degree 17. We have not been able to identify this group.

Remark 8.4. Maximum degree of a minimal relation Group number 562 has a ring generator in degree 17, so it is not surprising that its cohomology ring has a minimal relation in degree 34. There is one group where the last relation is in degree 32, three where it is in degree 30, and four groups where it is in degree 28. In all other cases the last minimal relation occurs in degree 26 at most.

9. Depth and the Duflot bound

There are several results relating the group structure of a finite group G to the commutative algebra of its cohomology ring H^{∗}(G) = H^{∗}(G, k) with coefficients in a field of characteristic p. For the Krull dimension and depth we have the following inequalities, where S denotes a Sylow p-subgroup of G:

(2) p-rk(Z(S)) ≤ depth H^{∗}(S) ≤ depth H^{∗}(G) ≤ dim H^{∗}(G) = p-rk(G) .

See Evens’ book [17] for proofs of the first inequality (Duflot’s theorem) and the last one (due to Quillen). The second inequality is Theorem 2.1 of Benson’s paper [4] and “must be well known”. The third is automatic for finitely generated connected k-algebras. Note that the dimension and depth only depend on G, p: not on k. These inequalities motivate the following definitions.

Definition. Let G, p, k, S be as above. The group-theoretic defect gtD_{p}(G), and the Cohen–Macaulay defect δ_{p}(G) are defined by

gtD(G) = p-rk(G) − p-rk(Z(S)) δ(G) = dim H^{∗}(G, k) − depth H^{∗}(G, k)

The Duflot excess e(G) = e_{p}(G) is defined by

e_{p}(G) = depth H^{∗}(G, k) − p-rk(Z(S)) .

It follows from Eqn. (2) that

(3) 0 ≤ δ(G) ≤ gtD(G)  gtD(G) = gtD(S)  δ(G) ≤ δ(S) .

and that

(4) 0 ≤ e(G) ≤ gtD(G)  δ(G) + e(G) = gtD(G)  e(G) ≥ e(S) .

The term Cohen–Macaulay defect (sometimes deficiency) is already in use among workers in the field.

Question 9.1. How (for large values of n) are the p-groups of order p^{n} distributed on the graph with δ(G) on the x-axis and e(G) on the y-axis?

Examples. Every abelian group has (δ, e) = (0, 0). So does every p-central group, such as the Sylow 2-subgroup of SU_{3}(4).

Quillen showed that the extraspecial 2-group G = 2_{+}^{1+2n} has Cohen–Macaulay cohomology [30]. So this group has (δ, e) = (0, n).

For the extraspecial p-group G = p_{+}^{1+2n} of odd exponent p, Minh proved [28] that (δ, e) = (n, 0). There is one exception: 3_{+}^{1+2} has (δ, e) = (0, 1), as its cohomology ring is Cohen–Macaulay [27].

One good way to produce groups with small e(G)/δ(G) ratio is by iterating the wreath product construction. By passing from H to H ≀ C_{p} one multiplies the p-rank by p but increases the depth by one only [12].

Table 1 lists the number of groups of various orders with each possible value of (δ, e). For |G| = 64 this information can be read off from the tables in [5, Appendix]. The other numbers come from our calculations.

Remarks 9.2. The unique group of order 64 with (δ, e) = (1, 2) is group number 138 (Hall–Senior number 259). The unique group of order 64 with (δ, e) = (2, 1) is group number 32 (Hall–Senior number 250).

The 14 groups of order 128 with (δ, e) = (1, 4) are the groups of interest for Conjecture 11.2, the strong form of Benson’s conjecture. They are listed in Proposition 13.1, and also in Table 4 on page 21. The unique group of order 128 with (δ, e) = (0, 3) is the extraspecial group 2_{+}^{1+6}, which is group number 2326.

The unique group of order 81 with (δ, e) = (1, 1) is the wreath product C_{3} ≀ C_{3} which occurs as the Sylow 3-subgroup of S_{9}. The two groups of order 625 with (δ, e) = (2, 0) are numbers 7 and 8; group number 7 being the Sylow 5-subgroup of the Conway group Co_{1}.

\begin{matrix}\begin{aligned}e & 3 & 0 \\ 2 & 5 & 1 \\ 1 & 45 & 22 & 1 \\ 0 & 69 & 103 & 21 & 0 \\ & 0 & 1 & 2 & 3 \\ N = 64 & & δ \end{aligned} & \begin{aligned}e & 3 & 1 \\ 2 & 27 & 10 \\ 1 & 256 & 220 & 32 \\ 0 & 353 & 1123 & 292 & 14 \\ & 0 & 1 & 2 & 3 \\ N = 128 & & δ \end{aligned} \\ \begin{aligned}e & 2 & 0 \\ 1 & 2 & 1 \\ 0 & 6 & 6 & 0 \\ & 0 & 1 & 2 \\ N = 3^{4} & & δ \end{aligned} & \begin{aligned}e & 2 & 0 \\ 1 & 0 & 0 \\ 0 & 6 & 7 & 2 \\ & 0 & 1 & 2 \\ N = 5^{4} & & δ \end{aligned} \end{matrix}

Table 1. Distribution of the groups of order N by defect δ and excess e

Question 9.3. Is the Duflot bound sharp for most groups of a given order p^{n}? That is, do most groups of order p^{n} lie on the bottom row e = 0 of the δ/e-graph? If so, is this effect more pronounced for large primes p?

10. The a-invariants

Let k be a field and R a connected finitely presented graded commutative kalgebra. Let M be a finitely generated graded R-module, and 𝔪 the ideal in R of all elements in positive degree. The a-invariants of M are defined by

a_{𝔪}^{i}(M) = max{m | H_{𝔪}^{i,m}(M) ≠ 0} ,

with a_{𝔪}^{i}(M) = −∞ if H_{𝔪}^{i}(M) = 0. One can then take

Reg(M) = max_{i≥0}{a_{𝔪}^{i}(M) + i}

as the definition of the Castelnuovo–Mumford regularity of M.

In order to calculate the a-invariants, one uses Lemma 4.3 of [6], which is based on methods of N. V. Trung. The lemma says that if ζ ∈ R^{n} is such that Ann_{M}(ζ) consists entirely of 𝔪-torsion, then

(5)  a_{𝔪}^{i+1}(M) + n ≤ a_{𝔪}^{i}(M/ζM) ≤ max(a_{𝔪}^{i}(M), a_{𝔪}^{i+1}(M) + n) .

Replacing ζ by a suitable power if necessary, one can arrange for a_{𝔪}^{i+1}(M) + n ≥ a_{𝔪}^{i}(M) and therefore a_{𝔪}^{i+1}(M) = a_{𝔪}^{i}(M/ζM) − n. So the a-invariants of H^{∗}(G) may be computed recursively. To start the recursion we need a_{𝔪}^{0}(M), which is the 𝔪-torsion: so if ζ ∈ R^{n} is such that Ann_{M}(ζ) is finite dimensional, then a_{𝔪}^{0}(M) is the top dimension of Ann_{M}(ζ).

|G| Group a_{𝔪}^{0} a_{𝔪}^{1} a_{𝔪}^{2} a_{𝔪}^{3} a_{𝔪}^{4} a_{𝔪}^{5}  Note
64 0242 −∞ −∞ −3 −5 −4  Sylow in L_{3}(4)
128 0391 −∞ −∞ −3 −5 −4
0741 −∞ −4 −5 −3
0749 −∞ −3 −7 −3

0836 −∞ −4 −5 −4 −4 Sylow in a particular 2.Sz(8)

0931 −∞ −∞ −3 −4 −4  Sylow in M_{22}
0934 −∞ −∞ −3 −5 −4  Sylow in J_{2}
1411 −∞ −∞ −∞ −4 −6 −5
1931 −∞ −∞ −3 −5 −4
2005 −∞ −∞ −3 −5 −4
2191 −∞ −∞ −4 −6 −4
2258 −∞ −∞ −∞ −4 −6 −5  Sylow in 2 × L_{3}(4)
2261 −∞ −∞ −3 −5 −4
2272 −∞ −∞ −3 −5 −4
2300 −∞ −∞ −3 −4 −4

Table 2. The fifteen groups of orders 64 and 128 whose ainvariants are not weakly increasng.

Hence by starting from a filter-regular system of parameters and raising some of the parameters to higher powers if necessary, one may compute the a-invariants of H^{∗}(G) by just computing kernels.

Benson computed the a-invariants of several particularly interesting groups of order 64 in [5, Appendix]. Using our cohomology package we computed these invariants for all groups of orders 64 and 128. Interestingly, amongst the 2595 groups of these orders there are only 15 cases where the a-invariants are not weakly increasing. These are listed in Table 2.

Cohomology rings with defect 3 have not been computed before. Table 3 lists the a-invariants of H^{∗}(G) for the 14 groups of order 128 with δ(G) = 3. Note that group number 836 occurs in both tables.

Remark 10.1. The a-invariants of the Sylow 2-subgroup of M_{22} (group 931 of order 128) are particularly interesting. Recall from [6, Theorem 4.5(iii)] that

Group a_{𝔪}^{0} a_{𝔪}^{1} a_{𝔪}^{2} a_{𝔪}^{3} a_{𝔪}^{4} a_{𝔪}^{5}
036 −∞ −∞ −5 −5 −5 −5
048 −∞ −∞ −5 −5 −5 −5
052 −∞ −4 −4 −4 −4
194 −∞ −∞ −6 −6 −5 −5
515 −∞ −∞ −5 −5 −5 −5
551 −∞ −∞ −6 −5 −5 −5
560 −∞ −6 −5 −4 −4
561 −∞ −5 −4 −4 −4
761 −∞ −∞ −5 −5 −5 −5
780 −∞ −6 −4 −4 −4
801 −∞ −4 −4 −4 −4
813 −∞ −4 −4 −4 −4
823 −∞ −4 −4 −4 −4
836 −∞ −4 −5 −4 −4

Table 3. The a-invariants of the fourteen groups of order 128 with δ(G) = 3.

a_{𝔪}^{i} ≤ d_{i} if there is a filter-regular system of parameters of type (d_{0}, …, d_{K}). This group has 2-rank four, and the type is (−1, −2, −3, −4, −4). Hence equality a_{𝔪}^{i} = d_{i} holds for i = 2, 3, 4.

Remark 10.2. For a cohomology computation one chooses parameters in low degrees. So it is perhaps surprising that amongst the 2328 groups of order 128 there were only 18 cases where nontrivial powers of the chosen parameters were actually necessary in order to compute the a-invariants.

The most extreme case was group number 2258, which is the Sylow 2-subgroup of C_{2} × L_{3}(4). The a-invariants are −∞, −∞, −∞, −4, −6, −5. We found a filter-regular system of parameters in degrees 1, 4, 4, 2, 2; the annihilators had top degrees −∞, −∞, −∞, 5, 5, 8. This led to problems with the calculation of a_{𝔪}^{3} and a_{𝔪}^{4}. We had to raise the third parameter to the fourth power, and square the fourth parameter.

11. The strong form of the regularity conjecture

We now consider two conjectures of D. J. Benson [6] about the commutative algebra of group cohomology rings.

Theorem 11.1 (Symonds [35]: Benson’s Regularity Conjecture). Let G be a finite group, p a prime number and k a field of characteristic p. The cohomology ring H^{∗}(G, k) has Castelnuovo–Mumford regularity zero. Equivalently, the cohomology ring H^{∗}(G, k) always contains a strongly quasi-regular system of parameters.

Conjecture 11.2 (Benson’s Regularity Conjecture, strong form). Let G be a finite group, p a prime number and k a field of characteristic p. The cohomology ring H^{∗}(G, k) always contains a very strongly quasi-regular system of parameters.

Remark. Theorem 11.1 is Conjecture 1.1 of Benson’s paper [6], which is now a theorem of P. Symonds [35]. The equivalent formulation is from [6, Corollary 4.8]. Kuhn has shown that Symonds’ result has applications to the study of central essential cohomology [25]. The stronger Conjecture 11.2 remains open. It originated as a remark on page 175 of Benson’s paper [6].

Theorem 11.3 (Benson). Let G be a finite group, p a prime number and k a field of characteristic p.

If the Cohen–Macaulay defect of G satisfies δ(G) ≤ 2 then Conjecture 11.2 holds for G. Every group of order 64 satisfies δ(G) ≤ 2.

Proof. The main reference is Benson’s paper [6]. The first statement is Theorem 1.5 of [6]; the second one was observed by Carlson [11, 13], who computed the cohomology ring of every group of order 64. The reader may find the tabulated data in [5, Appendix] useful. □

Remark. A weaker version of the δ(G) = 2 case of Theorem 11.3 was earlier proved by Okuyama and Sasaki. It is a shame that their paper [29] was delayed in publication: we know that it had completed the refereeing process by the end of April 2001, but it had been superseded by the time it finally appeared in 2004.

Our calculations demonstrate the following:

Theorem 11.4. Conjecture 11.2 holds for every group of order less than 256.

Remark. This result predates the computation of the cohomology rings of all groups of order 128 – see [20].

Proof. By Theorem 11.3 a counterexample has to have δ(G) ≥ 3. By Proposition 12.1 the only groups of order less than 256 satisfying δ(G) ≥ 3 have order 128 and satisfy δ(G) = 3. By Proposition 13.1 there are fourteen groups of order 128 with δ(G) = 3, and each of these satisfies the conjecture. □

Remark 11.5. To test Conjecture 11.2 further we need to find more high defect groups.

There are 24 groups of order 3^{6} with gtD(G) = 3. The presence of essential classes in low degrees demonstrates that at least three of these groups have δ(G) = 3. These groups are numbers 35, 56 and 67 in the Small Groups Library. There are essential classes in degrees 4, 2 and 4 respectively. Recall from Carlson’s paper [9] that the presence of essential classes means that depth H^{∗}(G, k) = p-rkZ(G) and therefore δ(G) = gtD(G).

Group number 299 of order 256 has gtD(G) = 4. The presence of an essential class in H^{3}(G) means that δ(G) = 4 too.

12. Reduction to the case |G| = 128

Proposition 12.1. Let G be a group of order less than 256. Then δ(G) ≤ 3; and if δ(G) = 3 then |G| = 128.

Proof. Let S be a Sylow p-subgroup of G. In view of the inequality δ(G) ≤ δ(S) and the restriction |G| < 256 it suffices to consider the case where G itself is a p-group.

So suppose G is a p-group with δ(G) ≥ 3. It follows that gtD(G) ≥ 3. By Lemma 12.2 below it follows that p = 2, that δ(G) = 3, and that either |G| = 64 or |G| = 128. But Carlson’s computations (see Theorem 11.3) above) show that δ(G) ≤ 2 if |G| = 64. □

Lemma 12.2. Let G be a finite group and p a prime number.

a) If gtD(G) ≥ 3 then p^{5} divides the order of G. If p = 2 or p = 3 then p^{6} must divide |G|.

b) If gtD(G) ≥ 4 then p^{6} divides the order of G. If p = 2 or p = 3 then p^{7} must divide |G|.

c) If p = 2 and gtD(G) ≥ 4 then |G| is divisible by 256.

Proof. a): By Eqn. (3) we have δ(G) ≤ gtD(G). It is apparent from the definition that a finite group and its Sylow p-subgroups have the same group-theoretic defect. So it suffices to consider the case where G is a p-group and gtD(G) ≥ 3.

Every nontrivial p-group has a centre of rank at least one. So a p-group with gtD ≥ 3 must have a subgroup which is elementary abelian of rank 4. It must be nonabelian too, so the order must be at least p^{5}.

Suppose that there is such a group of order p^{5}. Then the centre is cyclic of order p, and there is an elementary abelian subgroup V of order p^{4}. This V is a maximal subgroup of a p-group and therefore normal. Let a ∈ G lie outside V. Then G = ⟨a, V ⟩ and the conjugation action of a on V must be nontrivial of order p. So the minimal polynomial of the action divides X^{p} − 1 = (X − 1)^{p}. This means that the action has a Jordan normal form with sole eigenvalue 1. The eigenvectors in each Jordan block belong to the centre of G. So as the centre is cyclic there can only be one Jordan block, of size 4. But for p = 2 the size 3 Jordan block does not square to the identity, so there can be no blocks of size 3 or higher. Similarly there can be no size 4 block for p = 3, since it does not cube to the identity. So for p = 2, 3 there must be more than one Jordan block.

b): analogous.

c): We need to show that if |G| = 128 then gtD(G) ≥ 4 cannot happen. Such a group would need to contain an elementary abelian 2-group V satisfying the equation p-rk(V) = 4 + p-rk(Z(G)). The rank of the centre must be at least one, and if it is three or more then V = G and therefore G is abelian, a contradiction. If the centre has rank two, then V has index two in G and we are in the same situation as in a), except now we want two Jordan blocks. But we need at least three, since V has rank six and only Jordan blocks of size one or two are allowed.

If the centre has rank one, then V has rank five and we may pick a subgroup H with V ≤ H ≤ G and [G : H] = [H : V] = 2. If H is elementary abelian then we are back in the immediately preceding case of an order two action on a rank six elementary abelian. If H is not elementary abelian then the usual Jordan block considerations mean that C has rank at least 3, where C is the unique largest central elementary abelian of H. By applying the Jordan block considerations to the conjugation action of G/H on C, we see that Z(G) must have rank at least two, a contradiction. □

Remark 12.3. From Table 1 we see that there are in fact only two groups of order 64 with gtD = 3: see Remarks 9.2. We also see that there are 57 groups of order 128 with gtD = 3, but that only 14 of these groups have δ(G) = 3.

Remark 12.4. For p ≥ 5 let G be the following semidirect product group of order p^{5}: there is a rank four elementary abelian on the bottom and a cyclic group of order p on top. The conjugation action is a size 4 Jordan block. This group has gtD(G) = 3, since

(\begin{matrix}1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 0 & 0 & 0 & 1 \end{matrix})^{n} = (\begin{matrix}1 & \binom{n}{1} & \binom{n}{2} & \binom{n}{3} \\ 0 & 1 & \binom{n}{1} & \binom{n}{2} \\ 0 & 0 & 1 & \binom{n}{1} \\ 0 & 0 & 0 & 1 \end{matrix}) .

13. The groups of order 128

Let G be a group of order 128. By Lemma 12.2c) we have gtD(G) ≤ 3.

Proposition 13.1. Only 57 out of the 2328 groups of order 128 satisfy gtD(G) = 3. Of these 57 groups, 43 satisfy δ(G) ≤ 2. The remaining 14 groups satisfy δ(G) = 3. Each of these 14 groups of order 128 with δ(G) = 3 satisfies Conjecture 11.2.

According to the numbering of the Small Groups Library [7] these fourteen groups are: numbers 36, 48, 52, 194, 515, 551, 560, 561, 761, 780, 801, 813, 823 and 836.

gp K d r δ  gp K d r δ
\underline{36} 5 2 2 3  850 5 3 2 2
\underline{48} 5 2 2 3  852 4 2 1 2
\underline{52} 4 1 1 3  853 4 2 1 2
\underline{194} 5 2 2 3  854 4 2 1 2
513 5 3 2 2  859 4 2 1 2
\underline{515} 5 2 2 3  860 4 2 1 2
527 4 2 1 2  866 4 2 1 2
\underline{551} 5 2 2 3  928 4 3 1 1
\underline{560} 4 1 1 3  929 4 2 1 2
\underline{561} 4 1 1 3  931 4 2 1 2
621 5 3 2 2  932 4 2 1 2
623 4 2 1 2  934 4 2 1 2
630 5 3 2 2  1578 6 4 3 2
635 4 2 1 2  1615 4 3 1 1
636 4 2 1 2  1620 4 2 1 2
642 4 2 1 2  1735 5 3 2 2
643 4 2 1 2  1751 4 2 1 2
645 4 3 1 1  1753 4 3 1 1
646 4 2 1 2  1755 5 4 2 1
740 4 2 1 2  1757 4 3 1 1
742 4 2 1 2  1758 4 3 1 1
753 5 3 2 2  1759 4 3 1 1
\underline{761} 5 2 2 3  1800 4 2 1 2
764 4 2 1 2  2216 5 4 2 1
\underline{780} 4 1 1 3  2222 5 3 2 2
\underline{801} 4 1 1 3  2264 5 3 2 2
\underline{813} 4 1 1 3  2317 4 3 1 1
\underline{823} 4 1 1 3  2326 4 4 1 0
\underline{836} 4 1 1 3

Table 4. For each group of order 128 with gtD(G) = 3, we give its number in the Small Groups library, the Krull dimension K and depth d of H^{∗}(G), the rank r = K − 3 of Z(G) and the Cohen– Macaulay defect δ = K − d. Underlined entries have δ = 3. Notation based on that of [5, Appendix].

Proof. By machine computation. Inspecting the Small Groups library, one sees that there are 57 groups with gtD(G) = 3.

These 57 groups are listed in Table 4. The cohomology rings of these 57 groups were computed using the authors’ cohomology package [24]. These cohomology rings, as the cohomology rings of all other groups of order 128, may be viewed online [21], and read into the computer using our package. As discussed above, Benson’s test for completion involves constructing a filter-regular system of parameters and determining its type. Hence the computation of the Cohen– Macaulay defect and the testing of Conjecture 11.2 are an integral part of our computations.

The value of δ(G) for each of the 57 groups is given in Table 4. The fourteen groups listed in the statement of the proposition are indeed the only ones with δ(G) = 3. The computations showed that these 14 groups do satisfy the conjecture. □

Remark. The “biggest” cohomology ring among the above examples belongs to group number 836 of order 128. It has 65 generators and 1859 generators.

Remark. Some of these groups have been studied before. Groups 928 and 1578 are wreath products: D_{8}≀2 and 2^{3}≀2 respectively. By the Carlson–Henn result [12] one has δ(D_{8}≀2) = 1 and δ(2^{3}≀2) = 2. Groups 850 of order 128 is a direct product of the form G = H × 2, where H is group number 32 of order 64; and the same applies to group 1755 of order 128 and group 138 of order 64. It is immediate that δ(G) = δ(H) for such groups, so Carlson’s work guarantees δ(G) ≤ 2 for both these groups of order 128.

Group number 2326 is the extraspecial group 2_{+}^{1+6}; Quillen showed that its cohomology ring is Cohen–Macaulay [30]. Group number 931 is the Sylow 2subgroup of the Mathieu groups M_{22} and M_{23}; its cohomology was studied by Adem and Milgram [2]. Group number 934 is the Sylow 2-subgroup of the Janko group J_{2}; its cohomology ring was calculated by Maginnis [26].

We are not aware of any previous cohomological investigations concerning the other two groups that we can name. One of these is group 932, the Sylow 2subgroup of G_{2}(3) : 2. The other is group number 836, the Sylow 2-subgroup of one double cover of the Suzuki group Sz(8). This group (number 836) turned out to have the most complicated cohomology ring in the study.

Acknowledgements

The idea for Theorem 3.1 arose during a discussion with Dave Benson.

The work that we are presenting here was supported by the German Science Foundation (DFG), project numbers GR 1585/4-1 and -2.

We are grateful to William Stein for giving us the opportunity to work on the sage.math computer, which is supported by National Science Foundation Grant No. DMS-0821725.

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Mathematical Institute, University of Jena, D-07737 Jena, Germany

E-mail address: david.green@uni-jena.de

Mathematics Department, National University of Galway, Galway, Ireland

E-mail address: simon.king@nuigalway.ie