THE MOD-2 COHOMOLOGY RING OF THE THIRD CONWAY GROUP IS COHEN–MACAULAY

SIMON A. KING, DAVID J. GREEN, AND GRAHAM ELLIS

Abstract. By explicit machine computation we obtain the mod 2 cohomology ring of the third Conway group Co_{3} and show that it is Cohen-Macaulay. Moreover, the maximal elementary abelian 2-subgroups detect the mod-2 cohomology of Co_{3}.

1. Introduction

There has been considerable work on the mod-2 cohomology rings of the finite simple groups. Every finite simple group of 2-rank at most three has Cohen– Macaulay mod-2 cohomology [3]. There are eight sporadic finite simple groups of 2-rank four. For six of these eight groups, the mod-2 cohomology ring has already been determined, at least as a module over a polynomial subalgebra [4, VIII.5]. In most cases it can be shown that the mod-2 cohomology ring is not Cohen–Macaulay. For instance, the Mathieu groups M_{22} and M_{23} each have maximal elementary abelian 2-subgroups of ranks 3 and 4, which means that the cohomology ring cannot be Cohen–Macaulay: see [4, p. 269].

The two outstanding cases are the two groups with the largest Sylow 2-sub-groups: the Higman–Sims group HS with size 2^{9} Sylow subgroup, and the third Conway group Co_{3} with size 2^{10} Sylow subgroup. For HS the cohomology ring of the Sylow 2-subgroup has already been determined [1].

In this paper we consider the third Conway group Co_{3}. It stands out for two reasons, one of which being that it has the largest Sylow 2-subgroup. The second reason requires a little explanation. The Mathieu group M_{12} has 2-rank three, and its mod-2 cohomology ring was determined by Adem, Maginnis and Milgram [2]. Milgram observed that 2-locally it looks as if M_{12} admits a faithful representation in the Lie group G_{2}, which is however impossible. Benson and Wilkerson made this observation more precise [11] by constructing a map of classifying spaces with good properties in mod-2 cohomology.

Benson then took a similar approach to Co_{3}. He showed that the 2-completed classifying space admits a map to the classifying space of DI(4). This map induces a monomorphism in mod-2 cohomology, and H^{∗}(Co_{3}, 𝔽_{2}) is finitely generated as a module over its image [5].

Date: 23 March 2010.

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

King was supported by Marie Curie grant MTKD-CT-2006-042685. Green received travel assistance from DFG grants GR 1585/4-1 and GR 1585/4-2.

Recall that DI(4) denotes the Dwyer–Wilkerson exotic finite loop space, whose mod-2 cohomology is the ring of rank four Dickson invariants [17]. So Benson’s result tells us that the rank four Dickson invariants form a homogeneous system of parameters for H^{∗}(Co_{3}, 𝔽_{2}) in degrees 8, 12, 14, 15. In [7], Benson asks whether the cohomology of Co_{3} is free as a module over these parameters. That is, he suggests that the mod-2 cohomology of Co_{3} might be Cohen–Macaulay. Certainly the Dickson invariants constitute a filter-regular system of parameters, by Benson’s [9, Thm 1.2].

By a mixture of machine computation and theoretical argument we obtain the following theorem, which answers Benson’s question in the affirmative:

Theorem 1.1. The mod-2 cohomology ring H^{∗}(Co_{3}, 𝔽_{2}) of the third Conway group Co_{3} has the following properties:

(1) As a commutative 𝔽_{2}-algebra, the minimal ring presentation of H^{∗}(Co_{3}, 𝔽_{2}) has 16 generators and 71 relations. A full presentation is given in Appendix A. The smallest generator degree is 3, the greatest generator degree is 15. The greatest degree of a relation is 33.

(2) The cohomology ring is Cohen–Macaulay, having Krull dimension 4 and depth 4.

(3) There are four conjugacy classes of maximal elementary abelian 2-subgroups of Co_{3}, each of which has rank 4. These maximal elementary abelian subgroups detect the cohomology ring.

(4) The Poincaré series of H^{∗}(Co_{3}, 𝔽_{2}) is of the form

P (t) = \frac{f(t)}{(1 − t^{8})(1 − t^{12})(1 − t^{14})(1 − t^{15})},

where f(t) ∈ ℤ[t] is the monic polynomial of degree 45 with the coefficients 1, 1, 1, 1, 2, 3, 3, 4, 4, 6, 7, 8, 9, 10, 10, 11, 13, 12, 14, 15, 13, 13, 15, 14, 12, 13, 11, 10, 10, 9, 8, 7, 6, 4, 4, 3, 3, 2, 1, 1, 1, 1. The denominator corresponds to choosing Benson’s Dickson invariants as the homogeneous system of parameters [5].

Remark 1.2. As noted below, we actually constructed Benson’s Dickson invariants within H^{∗}(Co_{3}, 𝔽_{2}) in order to obtain an explicit filter regular system of parameters.

Remark 1.3. The cohomology ring is Cohen–Macaulay, and has parameters in degrees 8, 12, 14 and 15. So by Benson–Carlson duality [10, Thm 1.1], the numerator f(t) in the Poincaré series has to be symmetric, in the sense that the coefficients remain the same when one reads them from back to front. Observe that this is indeed the case.

To perform the computation we first computed the mod-2 cohomology of the Sylow 2-subgroup using our package [23]. Then we computed the stable elements degree by degree, taking the same approach as in Holt’s 1985 paper [22]. We then used our variant [21] of Benson’s test for completion [9] to work out when to stop.

Structure of the paper. We discuss the group structure of Co_{3} and its Sylow 2subgroup in Sections 2 and 5. Then we spend three sections describing the stable elements method and its implementation: in §3 we present the abstract method and discuss how to reduce the number of stability checks; in §4 we describe the stability checks in a method suitable for machine computation; and in §6 we discuss the actual implementation of the stability checks. Finally in §7 we describe the construction of the filter-regular parameter systems required by Benson’s completeness criterion, and give the proof of Theorem 1.1.

2. The Sylow 2-subgroup

In the ATLAS [15] we read that the third Conway group Co_{3} is simple and admits a faithful permutation representation of degree 276. The Sylow 2-subgroups have order 2^{10}. An explicit construction of Co_{3} as a degree 276 permutation group may be found in R. Wilson’s Online ATLAS [25]. Using GAP [18] one easily constructs the Sylow 2-subgroup S.

Despite its size, we were able to compute H^{∗}(S, 𝔽_{2}) as a surprisingly routine application of our program [23]. The result may be viewed online [20]. As the centre Z(S) is cyclic, Duflot’s theorem [13, Thm 12.3.3] merely says that the depth should be at least one. In fact the depth is three, which is only one less than the Krull dimension. This led Dave Benson to reiterate to us his conjecture that H^{∗}(Co_{3}, 𝔽_{2}) could be Cohen–Macaulay.

3. Stable elements

The stable element method is a way to compute the cohomology of a finite group from the cohomology of a Sylow subgroup.

3.1. The basics. Let p be a prime, G a finite group and H ≤ G a subgroup, whose index is coprime to p. Restriction from G to H embeds H^{∗}(G, 𝔽_{p}) in H^{∗}(H, 𝔽_{p}) as the ring of stable elements. Recall that x ∈ H^{∗}(H, 𝔽_{p}) is stable if

∀g ∈ G Res_{H^{g}∩H}^{H}(x) = g^{∗} Res_{H∩^{g}H}^{H}(x) .

This method is best known in the case where H is a Sylow p-subgroup of G [14, XII §10], but the method of proof works equally well in the general case [8, Prop. 3.8.2].

Since inner automorphisms are trivial in cohomology [14, XII §8 (5)], the stability condition Res_{H^{g}∩H}^{H}(x) = g^{∗} Res_{H∩^{g}H}^{H}(x) only depends on the double coset HgH ∈ H \ G/H. So one can count stability conditions by counting double cosets. Here, g^{∗} = c_{g}^{∗}, for c_{g}(h) = ghg^{−1}.

3.2. Discarding double cosets. A priori there is one stability condition per double coset HgH. For some double cosets however, the associated stability condition is satisfied by every cohomology class x ∈ H^{∗}(H, 𝔽_{p}). If this behaviour can be recognised in advance, then the stability condition can be discarded. For example, the trivial double coset H1H can always be discarded.

Moreover, if H^{g} ∩ H has order coprime to p, then it has no cohomology in positive degrees, and so the stability condition for the double coset HgH can safely be ignored.

Generalizing Proposition 18 of [19] give us another, purely group-theoretic criterion for the redundancy of some stability conditions.

Lemma 3.1. Let G be a finite group and H ≤ G a subgroup with index coprime to p. Let g ∈ G, and denote by T a Sylow p-subgroup of H^{g} ∩ H. Suppose that the transfer map from T to G is identically zero. Then the stability condition associated to the double coset HgH is redundant.

In particular if there is a p-group W ≠ 1 such that T × W ≤ G, then the stability condition associated to HgH is redundant.

Proof. First part: Note that we are not actually claiming that the stability condition is always satisfied. Inspecting the proof of the stable elements method in [8, Prop. 3.8.2], we observe that it actually uses a weaker condition: that each stability condition is satisfied after transfer from H^{g} ∩ H back to G. So if this transfer map is zero, then the stability condition is redundant. But transfer from H^{g} ∩ H factors through transfer from T to G, since transfer from T to H^{g} ∩ H is a split surjection in mod-p cohomology.

Second part: Transfer from T to G factors through transfer from T to T × W, which is zero: for restriction from T ×W to T is a split surjection, and restriction followed by transfer is multiplication by |W |, which is zero modulo p. □

Remark 3.2. An easy application of this result: If H^{g} ∩H has order coprime to p, then transfer back to H is zero, since H^{∗}(H^{g} ∩ H, 𝔽_{p}) is zero in positive degrees. Hence this stability condition may be discarded.

Example 3.3. As noted in §5.2, every maximal elementary abelian 2-subgroup of Co_{3} has rank four. So if the Sylow p-subgroup of H^{g} ∩H is elementary abelian of rank ≤ 3, then this redundancy criterion applies with W = C_{2}. This method proves to be very effective: see Remark 5.2.

Not all methods for eliminating stability conditions are group-theoretic. To perform the stability test for HgH we first construct induced homomorphisms Res_{H^{g}∩H}^{H} and g^{∗} Res_{H∩^{g}H}^{H} as ring homomorphisms, meaning that we determine the images of the ring generators. If each ring generator has the same image under each map, then we can discard this stability condition. Similarly, we can discard it if the pair of maps turns out to coincide with a pair that has already been processed. This too saves an appreciable amount of effort, for it is the next step which is the most time-intensive. This next step consists of working out the matrices of the two linear maps from H^{n}(H^{g} ∩ H, 𝔽_{p}) to H^{n}(H, 𝔽_{p}) degree by degree.

4. Computing stable elements

We now address the question of how to perform the stability tests on a computer. First though we must consider how to represent

• The mod-p cohomology ring of an arbitrary finite group; and

• The homomorphism between two such cohomology rings induced by a group homomorphism.

4.1. Representing cohomology rings of arbitrary finite groups. Let G be a finite group and S ≤ G a Sylow p-subgroup. We may assume that the cohomology ring of S is already known. More precisely, we may assume that we already know the cohomology of a p-group S̄ with the same isomorphism type as S. In order to make use of this pre-existing computation we need to choose an isomorphism f: S̄ → S. Once determined, the cohomology of G can then be stored by recording the map f together with the image R_{G,f} under f^{∗}: H^{∗}(S, 𝔽_{p}) → H^{∗}(S̄, 𝔽_{p}) of the ring of stable elements in H^{∗}(S, 𝔽_{p}). That is,

R_{G,f} = f^{∗} Res_{S}^{G} H^{∗}(G, 𝔽_{p}) ⊆ H^{∗}(S̄, 𝔽_{p}) .

4.2. Representing induced homomorphisms. Suppose that ϕ: G_{1} → G_{2} is a group homomorphism; and that for i = 1, 2 we calculated H^{∗}(G_{i}, 𝔽_{p}) using the Sylow p-subgroup S_{i} and the isomorphism f_{i}: S̄_{i} → S_{i}. We want to calculate the composition

R_{G_{2},f_{2}} ⟶^{≅} H^{∗}(G_{2}, 𝔽_{p}) ⟶^{ϕ^{∗}} H^{∗}(G_{1}, 𝔽_{p}) ⟶^{≅} R_{G_{1},f_{1}} .

As ϕ(S_{1}) is a p-subgroup of G_{2}, we may pick g ∈ G_{2} such that ϕ′(S_{1}) ≤ S_{2}, where ϕ′ = c_{g} ∘ ϕ. Since c_{g} is an inner automorphism of G_{2}, we have that ϕ′^{∗} = ϕ^{∗}: H^{∗}(G_{2}, 𝔽_{p}) → H^{∗}(G_{1}, 𝔽_{p}). Let ϕ̄: S̄_{1} → S̄_{2} be the group homomorphism ϕ̄ = f_{2}^{−1} ∘ c_{g} ∘ ϕ ∘ f_{1}. Then ϕ̄^{∗} maps R_{G_{2},f_{2}} ⊆ H^{∗}(S̄_{2}, 𝔽_{p}) to R_{G_{1},f_{1}} ⊆ H^{∗}(S̄_{1}, 𝔽_{p}) in the desired way. And since ϕ̄: S̄_{1} → S̄_{2} is a group homomorphism, we may easily compute ϕ̄^{∗}.

4.3. Performing the stability tests. As before, let H ≤ G be a subgroup whose index is coprime to p. We assume that the cohomology of H is already known. That is, we have chosen a Sylow p-subgroup S ≤ H and an isomorphism f: S̄ → S, and we already know the subring R_{H,f} ⊆ H^{∗}(S̄, 𝔽_{p}) given by R_{H,f} = f^{∗} Res_{S}^{H} H^{∗}(H, 𝔽_{p}). The stability test for the double coset HgH involves taking the equalizer of two maps ϕ_{1}^{∗}, ϕ_{2}^{∗}: H^{∗}(H, 𝔽_{p}) → H^{∗}(H^{g} ∩ H, 𝔽_{p}), induced by two different group homomorphisms ϕ_{1}, ϕ_{2} : H^{g} ∩ H → H: specifically

ϕ_{1}(h) = h  ϕ_{2}(h) = ghg^{−1} .

Remark 4.1. With the exception of Lemma 3.1, our approach is essentially the same as that of Holt [22], who was also the first to use a tower of intermediate subgroups, as described below. One very minor difference is that Holt chooses his double coset representatives with great care at the outset, whereas we take the first ones we find and in effect correct them at a later date.

Remark 4.2. The third author has implemented the stable elements method in his HAP system, and with Dutour Sikirić he used it to compute the integral homology of the Mathieu group M_{24} out to degree four [16].

Typically the cohomology of H^{g} ∩ H will not already be known, but the cohomology of its Sylow p-subgroup, T say, will be. There are now two ways in which we could proceed.

• We use stability checks to determine the cohomology of H^{g} ∩ H, and then construct ϕ_{1}^{∗}, ϕ_{2}^{∗} as above.

• Since restriction from H^{g} ∩ H to T is injective, we instead construct ψ_{1}^{∗}, ψ_{2}^{∗}: H^{∗}(H, 𝔽_{p}) → H^{∗}(T, 𝔽_{p}) for ψ_{i} = ϕ_{i}|_{T}. This avoids the need to determine H^{∗}(H^{g} ∩ H, 𝔽_{p}), which may well not be needed again.

It is perhaps surprising that in practice the first method turned out to be the more efficient. The reason seems to be that for fixed n, the dimension of the 𝔽_{p}-vector space H^{n}(H^{g} ∩ H, 𝔽_{p}) is often considerably smaller than the dimension of H^{n}(T, 𝔽_{p}). This reduces the size of the matrices which represent the two maps, and it is the size of these matrices which seems to have the greatest influence on the speed of the computations.

4.4. Using an intermediate subgroup. Let S be a Sylow p-subgroup of the finite group G, and let H be an intermediate group, in the sense that S ≤ H ≤ G. Using the methods of this section, there are two ways to compute H^{∗}(G, 𝔽_{p}) from H^{∗}(S, 𝔽_{p}):

• Direct computation of the G-stable elements in H^{∗}(S, 𝔽_{p}); or

• Computation of the H-stable elements in H^{∗}(S, 𝔽_{p}), followed by compu-

tation of the G-stable elements in H^{∗}(H, 𝔽_{p}). Typically the second method greatly reduces the number of stability checks that need to be considered. As the trivial double coset can be discarded, the first method involves |S \ G/S| − 1 stability checks, whereas the second involves |S \ H/S| + |H \ G/H| − 2.

Example 4.3. Let G be the Higman–Sims sporadic finite group G = HS. Then |S| = 2^{9}. With H = N_{G}(Z(S)) we have

|S \ G/S| = 270  |S \ H/S| = 4  |H \ G/H| = 9 .

Therefore the first method involves 269 stability checks, whereas the second only involves a total of most 11. As noted in §3.2, the number of stability conditions that actually matter may well be somewhat smaller in both cases.

As we shall now see in the case of Co_{3}, one can reduce the number of stability checks yet further by working one’s way up a tower of intermediate subgroups S = G_{0} ≤ G_{1} ≤ G_{2} ≤ ⋯ ≤ G_{n} = G.

5. Some group theory for Co_{3}

5.1. A tower of subgroups. The Sylow 2-subgroup has 484,680 double cosets in Co_{3}. It is therefore essential that we find a convenient tower of subgroups, in order to obtain a manageable total number of stability checks.

The ATLAS [15] tells us that there are two conjugacy classes of elements of order 4 in Co_{3}. These may be distinguished by the sizes of their centralizers: an element of type 4A has centralizer of size 23,040, whereas only 1,536 elements of Co_{3} centralize an element of type 4B.

Lemma 5.1. Let S be a Sylow 2-subgroup of G = Co_{3}. Then

(1) The centre Z(S) and the second centre Z_{2}(S) have isomorphism types

Z(S) ≅ C_{2}  Z_{2}(S) ≅ C_{4} × C_{2} .

(2) Z(S) is the Frattini subgroup of Z_{2}(S), and therefore of each copy of C_{4} in Z_{2}(S).

(3) There is a unique subgroup U ≤ Z_{2}(S) which is generated by an element of type 4A.

(4) N_{G}(Z_{2}(S)) ≤ N_{G}(U) ≤ N_{G}(Z(S)).

Proof. The first two statements are easily checked in GAP [18] using the degree 276 permutation representation from the Online ATLAS [25]. For the third statement one inspects the four elements of order 4 in Z_{2}(S) ≅ C_{4} × C_{2} and finds that two are of type 4A, and two of type 4B. As the centralizer sizes differ, the two type 4A elements generate the same cyclic subgroup.

The last part now follows, for Z(S) is a characteristic subgroup of U, and no other subgroup of Z_{2}(S) is conjugate to U in G = Co_{3}. □

Consider therefore the tower of subgroups S = G_{0} ≤ G_{1} ≤ G_{2} ≤ G_{3} ≤ G_{4} = Co_{3}, given by

G_{1} = N_{G}(Z_{2}(S))  G_{2} = N_{G}(U)  G_{3} = N_{G}(Z(S)) .

Note from the ATLAS [15] that G_{3} is a maximal subgroup of Co_{3}. The index of G_{i−1} in G_{i} and the number of double cosets is given in the following table:

\begin{matrix}i & |G_{i}: G_{i−1}| & |G_{i−1} \ G_{i}/G_{i−1}| \\ 1 & 3 & 2 \\ 2 & 15 & 3 \\ 3 & 63 & 3 \\ 4 & 170, 775 & 7 \end{matrix}

As there is one stability condition per double coset, and as the trivial double coset can be discarded, working up this enables us to determine the cohomology of Co_{3} by applying a total of 1 + 2 + 2 + 6 = 11 stability conditions. In practice, even some of these 11 conditions turned out to be redundant, so that eventually we only needed to consider a total of 6 stability conditions. Remark 5.2. When computing H^{∗}(Co_{3}, 𝔽_{2}) from H^{∗}(G_{3}, 𝔽_{2}) it turns out that all but two of the stability conditions may be discarded. After we have discarded the trivial one, there are six double cosets left to consider. In one case H^{g} ∩ H has the cyclic group of order 2 as its Sylow subgroup, and in three cases the Sylow subgroup is elementary abelian of order 4. So by Example 3.3 we may discard these four double cosets, with only two left over. In fact, for these four double cosets, the two maps in the associated stability condition coincide, i.e., these stability conditions are not only redundant but trivial. Remark 5.3. The depth of H^{∗}(G_{i}, 𝔽_{2}) is weakly increasing in i: see [6, Thm 2.1], and note that although the statement calls for a Sylow p-subgroup, the proof only requires the index to be coprime to p.

As noted in §2, H^{∗}(G_{0}, 𝔽_{2}) already has depth 3. It turns out that H^{∗}(G_{1}, 𝔽_{2}) has depth 4. As each G_{i} has odd index in Co_{3} and therefore has 2-rank 4, it follows that the depth cannot increase any more, and that H^{∗}(G_{i}, 𝔽_{2}) is Cohen– Macaulay for all i ≥ 1. 5.2. The maximal elementary abelian 2-subgroups. There are two conjugacy classes of order 2 elements in Co_{3}: class 2A elements have size 2,903,040 centralizer and class 2B elements have size 190,080 centralizer. A GAP computation shows that there are four conjugacy classes of maximal elementary abelian 2-subgroups of Co_{3}. Each has rank 4, and they are distinguished from each other by the number of 2A elements they contain. In ATLAS notation:

V_{1} = 2A_{1}B_{14}  V_{2} = 2A_{3}B_{12}  V_{3} = 2A_{7}B_{8}  V_{4} = 2A^{4} .

For each 1 ≤ r ≤ 4 there is a subgroup 2A^{r} ≤ V_{r} which contains all the 2A elements. Lemma 5.4. In Theorem 1.1, (3) follows from (2). Proof. We have just verified the group-theoretic statements in Theorem 1.1 (3). As H^{∗}(Co_{3}, 𝔽_{2}) is Cohen–Macaulay of dimension 4, Carlson’s [12, Thm 2.3] (repeated as [13, Thm 12.5.2]) states that the centralizers of the rank four elementary abelian subgroups detect the cohomology ring. Using GAP one checks that each of these rank four elementary abelians is self-centralizing in Co_{3}. It follows that these four rank four elementary abelian subgroups detect the mod-2 cohomology of Co_{3}. □

6. Computing the stable elements degree by degree

Let p be a prime number, G a finite group of p-rank r, and H < G a subgroup that contains a Sylow p-subgroup S < G. For d ≥ 0 we follow Benson and denote by τ_{d}H^{∗}(G, 𝔽_{p}) the 𝔽_{p}-algebra generated by the indecomposable elements of H^{∗}(G, 𝔽_{p}) in degree ≤ d and subject to those relations which hold in H^{∗}(G, 𝔽_{p}) in degrees ≤ d.

We assume that we know H^{∗}(H, 𝔽_{p}), and we want to compute H^{∗}(G, 𝔽_{p}), the subring of stable elements. There are two possible approaches. The first approach respects the fact that the maps which we are comparing with each other are ring homomorphisms. For each pair of maps one directly computes their equalizer, a subring. Then one takes the intersection of these subrings, again directly. The problem is that it is not clear how compute such equalizers directly, and intersecting subrings is a challenge as well. One way to salvage this approach would be to compute parameters for H^{∗}(G, 𝔽_{p}) by brute force: for this resettles the stability checks in the category of noetherian modules, which is better suited to the computation of equalizers and intersections.

The second approach is to work degree by degree, which reduces each stability check to determining the null space of a matrix: the matrix being the difference between two linear maps. This approach is much easier to implement, but as it ignores so much algebraic structure it cannot of itself decide in which degree it is safe to stop. For this reason we use (our variant of) Benson’s completeness condition, just as we do for p-groups.

So assume that we know the approximation τ_{d−1}H^{∗}(G; 𝔽_{p}) out to and including degree d − 1, and how each generator of τ_{d−1}H^{∗}(G; 𝔽_{p}) corresponds to a stable element of H^{∗}(H; 𝔽_{p}).

After constructing the stable subspace U_{d} < H^{d}(H; 𝔽_{p}), it is easy to extend the ring structure of τ_{d−1}H^{∗}(G; 𝔽_{p}) out to degree d. First, we determine the degree d standard monomials of τ_{d−1}H^{∗}(G; 𝔽_{p}), and express them as elements of U_{d}. Linear dependencies between these elements of U_{d} correspond to new relations in degree d between the generators of H^{∗}(G, 𝔽_{p}) in degree < d. The subspace of U_{d} spanned by the standard monomials is the decomposable subspace of H^{d}(G, 𝔽_{p}). If it is not the whole of U_{d}, then a basis for a complement yields new generators in degree d for the cohomology of G. Adding the new relations and generators to τ_{d−1}H^{∗}(G; 𝔽_{p}), we obtain the approximation τ_{d}H^{∗}(G; 𝔽_{p}) out to degree d of H^{∗}(G; 𝔽_{p}).

In order to ensure termination of this algorithm, one needs a method to prove that there will be no further relations or generators in degree > d. We use a modification of Benson’s completeness criterion [9]. Benson’s criterion depends on the filter degree type and the degrees of a filter regular HSOP — the smaller the degrees, the better the criterion.

7. The construction of Filter Regular Parameters — application of Benson’s criterion

How does one construct elements of τ_{d}H^{∗}(G; 𝔽_{p}) that are guaranteed to correspond to a filter regular HSOP for H^{∗}(G; 𝔽_{p})? According to Benson [9], it is possible to find elements d_{1}, ..., d_{r} that simultaneously restrict to (powers of) Dickson elements in the maximal p-elementary abelian subgroups of G.

The Dickson elements are of rather high degree. Therefore, it is reasonable to invest some energy into finding smaller parameters. In [21], we suggested two methods that may transform a given filter regular HSOP d_{1}, ..., d_{r} into another filter regular HSOP of smaller degrees:

(1) Filter regularity of the last parameter is automatic. Hence, d_{r} can be replaced by any other parameter (this is easy to construct).

(2) If d_{i} can be decomposed as p · q for elements p, q of positive degree, then one can replace d_{i} by p (see Lemma 6.1 of [21]). We refer to this as the factorisation method.

In Lemma 5.2 of [21], we suggest a further improvement, that is guaranteed to work in the case of p-groups: It is possible to choose the cohomology generators such that they contain a Duflot regular sequence, i.e., elements c_{1}, ..., c_{z} that restrict to a regular HSOP of Ω_{1}(Z(S)). For sufficiently large d and if G = S, one can find elements c_{z+1}, ..., z_{r} of τ_{d}H^{∗}(G; 𝔽_{p}) that simultaneously restrict to powers of Dickson elements in a complement of Ω_{1}(Z(S)) in the maximal elementary abelian subgroups of G. We refer to them as restricted Dickson elements for G. Then, c_{1}, ..., c_{r} corresponds to a filter regular HSOP of H^{∗}(G; 𝔽_{p}), and the degrees of c_{1}, ..., c_{r} are smaller than the degrees of d_{1}, ..., d_{r}.

If G is not a prime power group, two problems arise. First, it is not guaranteed that one can choose the generators such that they contain a Duflot regular sequence. But this is not much of a problem: It is easy to construct a Duflot regular sequence c_{1}, ..., c_{z} of minimal degree, even when it is not formed by generators. Secondly, it is, in general, impossible to find restricted Dickson elements for G. The reason is that the centre of S is, in general, not invariant under inner automorphisms of G.

Therefore, we proceed as follows. Assume that τ_{d}H^{∗}(G; 𝔽_{p}) is known, and that d is large enough, such that we also know a Duflot regular sequence c_{1}, ..., c_{z}.

(1) Try to construct d_{1}, ..., d_{r}; if successfull, store the result.

(2) If d is large enough, test whether the restricted Dickson elements for S are stable, thus, give rise to elements c_{z+1}, ..., c_{r} of τ_{d}H^{∗}(G; 𝔽_{p}); if successful for all or all but the last element, then store the result.

(3) If it was possible to construct c_{z+1}, ..., c_{r−1}, but impossible to find c_{r}, then try to find an element p ∈ τ_{d}H^{∗}(G; 𝔽_{p}) that corresponds to a parameter of the quotient ring (τ_{d}H^{∗}(G; 𝔽_{p})) /⟨c_{1}, ..., c_{r−1}⟩. Note that this step must be repeated in higher degrees, should it turn out that τ_{d}H^{∗}(G; 𝔽_{p}) is not isomorphic to H^{∗}(G; 𝔽_{p}).

Lemma 7.1. If d is large enough that τ_{d}H^{∗}(G; 𝔽_{p}) is isomorphic to H^{∗}(G; 𝔽_{p}), then at least one of the sequences d_{1}, ..., d_{r} or c_{1}, ..., c_{r} or c_{1}, ..., c_{r−1}, p can be constructed, and each of them provides a filter regular HSOP of τ_{d}H^{∗}(G; 𝔽_{p}) that is mapped to a HSOP for H^{∗}(G; 𝔽_{p}).

Proof. By [9], d_{1}, ..., d_{r} can be constructed, provides a filter regular HSOP, and is mapped to a HSOP for H^{∗}(G; 𝔽_{p}).

If it is possible to construct c_{z+1}, ..., c_{r−1}, then by construction their restrictions to the elementary abelian subgroups of S coincide with those of the restricted Dickson elements for S. It thus follows from Lemma 5.1 of [21] that c_{1}, ..., c_{r−1} is a filter regular sequence for τ_{d}H^{∗}(G; 𝔽_{p}). Hence, the quotient ring (τ_{d}H^{∗}(G; 𝔽_{p})) /⟨c_{1}, ..., c_{r−1}⟩ is of dimension one. It is thus certainly possible to find a last parameter p, and c_{1}, ..., c_{r−1}, p is a filter regular HSOP for τ_{d}H^{∗}(G; 𝔽_{p}).

Again by Lemma 5.1 of [21], it follows that c_{1}, ..., c_{r} is a filter regular HSOP as well, should it be possible to find c_{r}.

Since there exist parameters d_{1}, ..., d_{r} for τ_{d}H^{∗}(G; 𝔽_{p}) that yield parameters for the cohomology ring and since c_{1}, ..., c_{r−1}, p respectively c_{1}, ..., c_{r} is a HSOP for τ_{d}H^{∗}(G; 𝔽_{p}), it follows that c_{1}, ..., c_{r−1}, p respectively c_{1}, ..., c_{r} are mapped to parameters for the cohomology ring as well. □

Hence, if d is large enough, then we have at least one filter regular HSOP for τ_{d}H^{∗}(G; 𝔽_{p}), and it may be further simplified by the factorisation method. Let P be the one among these filter regular HSOPs with the smallest degrees. We can now compute the filter degree type of P. It is still possible that the degrees of P are not small enough, i.e., that Benson’s criterion applied to P would not ensure termination in degree d.

Theorem 3.1 in combination with Proposition 4.2 of [21] may help to improve the degree bound further. Theorem 3.1 of [21] provides a completeness criterion for τ_{d}H^{∗}(G; 𝔽_{p}), expressed in terms of (an upper bound for) the regularity of τ_{d}H^{∗}(G; 𝔽_{p}), together with the degrees of filter regular parameters p_{1}, ..., p_{r} for the degree-d approximation of the cohomology ring of G with coefficients in a finite field extension of 𝔽_{p}. The upper bound for the regularity of τ_{d}H^{∗}(G; 𝔽_{p}) can be computed from the filter regular HSOP for τ_{d}H^{∗}(G; 𝔽_{p}) obtained from Lemma 7.1 above. The existence of the p_{1}, ..., p_{r} and their degrees come out of Proposition 4.2 of [21]. We refer to this as the existence proof method.

Examples of the completeness criteria. The above methods may be illustrated with the different “stories” of the subgroup tower. The 2-rank of Co_{3} is four.

Thus, the Dickson elements are of degree 8, 12, 14 and 15, for any subgroup in the subgroup tower.

Since the centre of G_{0} = S is of rank one, Benson’s criterion applied to parameters in degrees 8, 12, 14 and 15 only yields the degree bound 46 for the cohomology of G_{0}. However, we find a Duflot regular sequence formed by one generator in degree 8, and the restricted Dickson elements can be found in degrees 4, 6 and 7. This yields the degree bound 22 for H^{∗}(G_{0}; 𝔽_{2}). But this still is not good enough — it would take months to compute a minimal projective resolution out to that degree. By the existence proof method, we can show that the last two parameters can be replaced by some elements of degree two, after suitably extending the coefficient field. This ensures termination right after finding the last relation, in degree 14.

It turns out that both the mod-2 cohomology rings of G_{1}, G_{2} and G_{3} have a Duflot regular sequence formed by a generator in degree 8, and the first two restricted Dickson elements for G_{0} turn out to be stable with respect to G_{1}, G_{2} and G_{3}. Hence, we found the first three parameters in degrees 8, 4 and 6. A last parameter for G_{1} and G_{2} can be found in degree one, which means that the computation terminates right after finding the last relation, in degree 16.

A last parameter for the mod-2 cohomology of G_{3} can only be found in degree 7. We already know that the depth of H^{∗}(G_{1}; 𝔽_{2}) is four, which is a lower bound for the depth of H^{∗}(G_{3}; 𝔽_{2}). Hence, using Remark 10.6(v) of [9], parameters in degrees 8, 4, 6 and 7 yield the degree bound 21. We were not able to improve it using the existence proof method. Anyway, the last relation was already found in degree 18.

The restricted Dickson elements for G_{0} are not stable in G_{4} = Co_{3}. Hence, we need to work with the original Dickson elements in degrees 8, 12, 14, 15. Again by Remark 10.6(v) of [9], Benson’s criterion can be very slightly improved and yields the degree bound 45. It turns out that this can not be further improved by the existence proof method. Fortunately, H^{∗}(Co_{3}; 𝔽_{2}) < H^{∗}(G_{3}; 𝔽_{2}) is defined by only two stability conditions. Hence, it is feasible to carry the computation out to degree 45, even though the last relation was only found in degree 33.

Proof of Theorem 1.1. We computed the mod-2 cohomology ring of the Sylow subgroup using our package [23]. We then used the stable elements method to work up the tower of subgroups in §5.1. As described in Sections 6 and 7, for each successive group in the tower we computed the stable elements in low degrees, and then constructed the filter-regular parameter system needed to detect completion by Benson’s method. Whilst applying Benson’s method we obtained the depth as a by-product. From G_{1} onward, each group in the tower has Cohen–Macaulay mod-2 cohomology. Thus we established (1), (2) and (3). By Lemma 5.4, we deduce that (3) holds too. □

Appendix A. A Minimal Ring Presentation

When we name generators of a cohomology ring, we use the following conventions. If the generator is nilpotent, its name starts with the letter “a”.^{1} If it is not nilpotent, but the restriction to the greatest elementary abelian subgroup in the centre of a Sylow subgroup is nilpotent, then its name starts with the letter “b”. Otherwise, its restriction to the greatest elementary abelian subgroup in the centre of a Sylow subgroup is not nilpotent; we call such element a Duflot element, and let its name start with the letter “c”. The first index gives the degree of the generator. The second index is just for distinguishing generators of the same degree.

A minimal generating set for H^{∗}(Co_{3}; 𝔽_{2}) is given by

b_{4,0}, b_{6,1}, b_{8,1}, c_{8,3}, b_{12,1}, b_{12,7}, b_{14,1}, b_{3,0}, b_{5,0}, b_{7,0}, b_{7,1}, b_{9,0}, b_{11,5}, b_{13,1}, b_{13,7}, b_{15,13}.

The following polynomials form a minimal generating set of the relation ideal:

(1) b_{5,0}^{2} + b_{3,0}b_{7,0} + b_{4,0}b_{6,1}

(2) b_{3,0}^{2}b_{5,0} + b_{8,1}b_{3,0} + b_{4,0}b_{7,1}

(3) b_{3,0}b_{9,0} + b_{4,0}b_{8,1} + b_{4,0}^{3}

(4) b_{5,0}b_{7,0} + b_{3,0}^{4} + b_{6,1}b_{3,0}^{2} + b_{6,1}^{2} + b_{4,0}^{3}

(5) b_{5,0}b_{7,1} + b_{4,0}b_{8,1} + b_{4,0}^{3}

(6) b_{6,1}b_{7,1} + b_{4,0}b_{9,0} + b_{4,0}b_{6,1}b_{3,0} + b_{4,0}^{2}b_{5,0}

(7) b_{3,0}^{2}b_{7,0} + b_{8,1}b_{5,0} + b_{4,0}b_{9,0} + b_{4,0}b_{6,1}b_{3,0}

(8) b_{3,0}^{2}b_{7,1} + b_{4,0}b_{9,0} + b_{4,0}b_{3,0}^{3} + b_{4,0}^{2}b_{5,0}

(9) b_{6,1}b_{3,0}b_{5,0} + b_{6,1}b_{8,1} + b_{4,0}b_{3,0}b_{7,1} + b_{4,0}^{2}b_{3,0}^{2} + b_{4,0}^{2}b_{6,1}

(10) b_{5,0}b_{9,0} + b_{4,0}b_{3,0}b_{7,1} + b_{4,0}b_{3,0}b_{7,0} + b_{4,0}^{2}b_{3,0}^{2} + b_{4,0}^{2}b_{6,1}

(11) b_{7,0}b_{7,1} + b_{4,0}b_{3,0}b_{7,0}

(12) b_{6,1}b_{9,0} + b_{4,0}b_{6,1}b_{5,0} + b_{4,0}^{2}b_{7,1} + b_{4,0}^{3}b_{3,0}

(13) b_{12,7}b_{3,0} + b_{4,0}b_{11,5} + b_{4,0}c_{8,3}b_{3,0}

(14) b_{3,0}^{5} + b_{8,1}b_{7,0} + b_{6,1}b_{3,0}^{3} + b_{6,1}^{2}b_{3,0} + b_{4,0}^{2}b_{7,0} + b_{4,0}^{3}b_{3,0}

(15) b_{3,0}b_{13,1} + b_{8,1}b_{3,0}b_{5,0} + b_{8,1}^{2} + b_{4,0}^{2}b_{8,1}

(16) b_{3,0}b_{13,7} + b_{4,0}b_{12,7} + c_{8,3}b_{3,0}b_{5,0}

(17) b_{5,0}b_{11,5} + b_{4,0}b_{12,7} + c_{8,3}b_{3,0}b_{5,0}

(18) b_{7,0}b_{9,0} + b_{4,0}b_{3,0}^{4} + b_{4,0}b_{6,1}b_{3,0}^{2} + b_{4,0}b_{6,1}^{2} + b_{4,0}^{4}

(19) b_{7,1}b_{9,0} + b_{8,1}b_{3,0}b_{5,0} + b_{8,1}^{2} + b_{4,0}^{2}b_{3,0}b_{5,0} + b_{4,0}^{2}b_{8,1}

(20) b_{6,1}b_{11,5} + b_{4,0}b_{13,7} + b_{6,1}c_{8,3}b_{3,0} + b_{4,0}c_{8,3}b_{5,0}

(21) b_{8,1}b_{9,0} + b_{4,0}b_{13,1} + b_{4,0}b_{8,1}b_{5,0}

(22) b_{12,7}b_{5,0} + b_{4,0}b_{13,7} + b_{4,0}c_{8,3}b_{5,0}

(23) b_{3,0}^{2}b_{11,5} + b_{4,0}b_{13,7} + c_{8,3}b_{3,0}^{3} + b_{4,0}c_{8,3}b_{5,0}

(24) b_{3,0}b_{7,1}^{2} + b_{4,0}b_{13,1} + b_{4,0}^{2}b_{3,0}^{3}

(25) b_{6,1}b_{12,7} + b_{4,0}b_{3,0}b_{11,5} + b_{4,0}c_{8,3}b_{3,0}^{2}

(26) b_{3,0}b_{15,13} + b_{6,1}b_{12,1} + b_{4,0}b_{7,1}^{2} + b_{4,0}b_{7,0}^{2} + b_{4,0}b_{14,1} + b_{4,0}^{3}b_{3,0}^{2} + c_{8,3}b_{3,0}b_{7,0} (27) b_{5,0}b_{13,1} + b_{4,0}b_{7,1}^{2} + b_{4,0}^{3}b_{3,0}^{2}

1There is no nilpotent generator for H^{∗}(Co_{3}; 𝔽_{2})

(28) b_{5,0}b_{13,7} + b_{4,0}b_{3,0}b_{11,5} + c_{8,3}b_{3,0}b_{7,0} + b_{4,0}c_{8,3}b_{3,0}^{2} + b_{4,0}b_{6,1}c_{8,3}

(29) b_{7,0}b_{11,5} + c_{8,3}b_{3,0}b_{7,0}

(30) b_{9,0}^{2} + b_{4,0}b_{7,1}^{2} + b_{4,0}^{2}b_{3,0}b_{7,0} + b_{4,0}^{3}b_{3,0}^{2} + b_{4,0}^{3}b_{6,1}

(31) b_{6,1}b_{13,1} + b_{4,0}b_{8,1}b_{7,1} + b_{4,0}^{2}b_{8,1}b_{3,0}

(32) b_{6,1}b_{13,7} + b_{4,0}^{2}b_{11,5} + b_{6,1}c_{8,3}b_{5,0} + b_{4,0}^{2}c_{8,3}b_{3,0}

(33) b_{12,1}b_{7,0}

(34) b_{12,7}b_{7,0}

(35) b_{12,7}b_{7,1} + b_{8,1}b_{11,5} + b_{4,0}^{2}b_{11,5} + b_{8,1}c_{8,3}b_{3,0} + b_{4,0}^{2}c_{8,3}b_{3,0}

(36) b_{14,1}b_{5,0} +b_{8,1}^{2}b_{3,0} +b_{6,1}b_{8,1}b_{5,0} +b_{6,1}^{2}b_{7,0} +b_{4,0}b_{15,13} +b_{4,0}b_{12,1}b_{3,0} +b_{4,0}b_{6,1}b_{3,0}^{3} + b_{4,0}b_{6,1}^{2}b_{3,0} + b_{4,0}^{2}b_{6,1}b_{5,0} + b_{4,0}^{3}b_{7,1} + b_{4,0}^{3}b_{7,0} + b_{4,0}c_{8,3}b_{7,0}

(37) b_{14,1}b_{3,0}^{2} + b_{12,1}b_{3,0}b_{5,0} + b_{8,1}b_{3,0}^{4} + b_{6,1}b_{7,0}^{2} + b_{6,1}b_{14,1} + b_{6,1}b_{8,1}b_{3,0}^{2} + b_{6,1}^{2}b_{8,1} + b_{4,0}b_{6,1}b_{3,0}b_{7,0} + b_{4,0}^{2}b_{3,0}^{4} + b_{4,0}^{2}b_{12,1} + b_{4,0}^{2}b_{6,1}b_{3,0}^{2} + b_{4,0}^{2}b_{6,1}^{2} + b_{4,0}^{3}b_{8,1} + b_{4,0}^{5}

(38) b_{5,0}b_{15,13} +b_{12,1}b_{3,0}b_{5,0} +b_{6,1}b_{7,0}^{2} +b_{6,1}b_{14,1} +b_{4,0}b_{8,1}b_{3,0}b_{5,0} +b_{4,0}b_{8,1}^{2} +b_{4,0}^{3}b_{8,1} + c_{8,3}b_{3,0}^{4} + b_{6,1}c_{8,3}b_{3,0}^{2} + b_{6,1}^{2}c_{8,3} + b_{4,0}^{3}c_{8,3}

(39) b_{7,0}b_{13,1}

(40) b_{7,0}b_{13,7} + c_{8,3}b_{3,0}^{4} + b_{6,1}c_{8,3}b_{3,0}^{2} + b_{6,1}^{2}c_{8,3} + b_{4,0}^{3}c_{8,3}

(41) b_{7,1}b_{13,7} + b_{8,1}b_{12,7} + b_{4,0}^{2}b_{12,7} + b_{4,0}b_{8,1}c_{8,3} + b_{4,0}^{3}c_{8,3}

(42) b_{9,0}b_{11,5} + b_{8,1}b_{12,7} + b_{4,0}^{2}b_{12,7} + b_{4,0}b_{8,1}c_{8,3} + b_{4,0}^{3}c_{8,3}

(43) b_{12,1}b_{3,0}^{3} + b_{6,1}b_{15,13} + b_{6,1}b_{12,1}b_{3,0} + b_{4,0}b_{14,1}b_{3,0} + b_{4,0}b_{12,1}b_{5,0} + b_{4,0}b_{8,1}b_{3,0}^{3} + b_{4,0}b_{6,1}b_{8,1}b_{3,0} + b_{4,0}b_{6,1}^{2}b_{5,0} + b_{4,0}^{2}b_{13,1} + b_{4,0}^{2}b_{6,1}b_{7,0} + b_{4,0}^{3}b_{3,0}^{3} + b_{4,0}^{3}b_{6,1}b_{3,0} + b_{4,0}^{4}b_{5,0} + b_{6,1}c_{8,3}b_{7,0}

(44) b_{12,7}b_{9,0} + b_{8,1}b_{13,7} + b_{4,0}^{2}b_{13,7} + b_{8,1}c_{8,3}b_{5,0} + b_{4,0}^{2}c_{8,3}b_{5,0}

(45) b_{3,0}b_{7,1}b_{11,5} + b_{8,1}b_{13,7} + b_{4,0}^{2}b_{13,7} + b_{8,1}c_{8,3}b_{5,0} + b_{4,0}c_{8,3}b_{9,0} + b_{4,0}c_{8,3}b_{3,0}^{3}

(46) b_{7,0}^{3} + b_{14,1}b_{7,0}

(47) b_{7,1}^{3} + b_{8,1}b_{13,1} + b_{4,0}^{2}b_{13,1} + b_{4,0}^{3}b_{9,0} + b_{4,0}^{3}b_{3,0}^{3} + b_{4,0}^{4}b_{5,0}

(48) b_{7,0}b_{15,13} + c_{8,3}b_{7,0}^{2}

(49) b_{7,1}b_{15,13} + b_{12,1}b_{3,0}b_{7,1} + b_{8,1}b_{7,1}^{2} + b_{8,1}b_{14,1} + b_{8,1}^{2}b_{3,0}^{2} + b_{6,1}b_{8,1}^{2} + b_{6,1}^{2}b_{3,0}b_{7,0} + b_{4,0}b_{6,1}b_{3,0}^{4}+b_{4,0}b_{6,1}^{2}b_{3,0}^{2}+b_{4,0}^{2}b_{7,1}^{2}+b_{4,0}^{2}b_{14,1}+b_{4,0}^{2}b_{8,1}b_{3,0}^{2}+b_{4,0}^{3}b_{3,0}b_{7,0}+b_{4,0}^{4}b_{6,1}+ b_{4,0}c_{8,3}b_{3,0}b_{7,0}

(50) b_{9,0}b_{13,1} + b_{8,1}b_{7,1}^{2} + b_{4,0}^{2}b_{7,1}^{2} + b_{4,0}^{2}b_{8,1}b_{3,0}^{2} + b_{4,0}^{4}b_{3,0}^{2}

(51) b_{9,0}b_{13,7} + b_{4,0}b_{7,1}b_{11,5} + b_{4,0}c_{8,3}b_{3,0}b_{7,0} + b_{4,0}^{2}c_{8,3}b_{3,0}^{2} + b_{4,0}^{2}b_{6,1}c_{8,3}

(52) b_{11,5}^{2} + b_{12,1}b_{3,0}b_{7,1} + b_{8,1}b_{7,1}^{2} + b_{8,1}b_{14,1} + b_{8,1}^{2}b_{3,0}^{2} + b_{6,1}b_{8,1}^{2} + b_{6,1}^{2}b_{3,0}b_{7,0} + b_{4,0}b_{6,1}b_{3,0}^{4} + b_{4,0}b_{6,1}b_{12,1} + b_{4,0}b_{6,1}^{2}b_{3,0}^{2} + b_{4,0}^{2}b_{7,0}^{2} + b_{4,0}^{2}b_{8,1}b_{3,0}^{2} + b_{4,0}^{3}b_{3,0}b_{7,0} + b_{4,0}^{4}b_{3,0}^{2} + b_{4,0}^{4}b_{6,1} + c_{8,3}b_{7,1}^{2} + b_{4,0}^{2}c_{8,3}b_{3,0}^{2} + c_{8,3}^{2}b_{3,0}^{2}

(53) b_{12,7}b_{11,5} + b_{8,1}b_{15,13} + b_{6,1}b_{12,1}b_{5,0} + b_{4,0}^{2}b_{12,1}b_{3,0} + b_{8,1}c_{8,3}b_{7,1} + b_{8,1}c_{8,3}b_{7,0} + b_{4,0}c_{8,3}b_{11,5} + b_{4,0}b_{8,1}c_{8,3}b_{3,0} + b_{4,0}c_{8,3}^{2}b_{3,0}

(54) b_{14,1}b_{9,0} + b_{8,1}b_{15,13} + b_{8,1}^{2}b_{7,1} + b_{6,1}b_{12,1}b_{5,0} + b_{4,0}b_{12,1}b_{7,1} + b_{4,0}b_{6,1}b_{8,1}b_{5,0} + b_{4,0}b_{6,1}^{2}b_{7,0} + b_{4,0}^{2}b_{15,13} + b_{4,0}^{2}b_{12,1}b_{3,0} + b_{4,0}^{2}b_{6,1}b_{3,0}^{3} + b_{4,0}^{2}b_{6,1}^{2}b_{3,0} + b_{4,0}^{3}b_{6,1}b_{5,0} + b_{4,0}^{4}b_{7,1} + b_{4,0}^{4}b_{7,0} + b_{8,1}c_{8,3}b_{7,0} + b_{4,0}^{2}c_{8,3}b_{7,0} (55) b_{14,1}b_{3,0}b_{7,1} + b_{12,7}^{2} + b_{4,0}b_{7,1}b_{13,1} + b_{4,0}b_{8,1}b_{3,0}^{4} + b_{4,0}b_{8,1}b_{12,1} + b_{4,0}b_{6,1}b_{7,0}^{2} +

b_{4,0}b_{6,1}b_{14,1}+b_{4,0}b_{6,1}b_{8,1}b_{3,0}^{2}+b_{4,0}b_{6,1}^{2}b_{8,1}+b_{4,0}^{2}b_{8,1}b_{3,0}b_{5,0}+b_{4,0}^{2}b_{8,1}^{2}+b_{4,0}^{2}b_{6,1}b_{3,0}b_{7,0}+ b_{4,0}^{3}b_{3,0}^{4} + b_{4,0}^{3}b_{6,1}b_{3,0}^{2} + b_{4,0}^{3}b_{6,1}^{2} + b_{4,0}^{6} + b_{8,1}c_{8,3}b_{3,0}b_{5,0} + b_{8,1}^{2}c_{8,3} + b_{4,0}^{2}b_{8,1}c_{8,3}

(56) b_{9,0}b_{15,13} + b_{12,7}^{2} + b_{4,0}b_{12,1}b_{3,0}b_{5,0} + b_{4,0}b_{6,1}b_{7,0}^{2} + b_{4,0}b_{6,1}b_{14,1} + b_{4,0}^{2}b_{8,1}b_{3,0}b_{5,0} + b_{4,0}^{2}b_{8,1}^{2} + b_{4,0}^{4}b_{8,1} + b_{8,1}c_{8,3}b_{3,0}b_{5,0} + b_{8,1}^{2}c_{8,3} + b_{4,0}c_{8,3}b_{3,0}^{4} + b_{4,0}b_{6,1}c_{8,3}b_{3,0}^{2} + b_{4,0}b_{6,1}^{2}c_{8,3} + b_{4,0}^{2}b_{8,1}c_{8,3} + b_{4,0}^{4}c_{8,3}

(57) b_{11,5}b_{13,7} + b_{12,7}^{2} + c_{8,3}^{2}b_{3,0}b_{5,0}

(58) b_{12,7}b_{13,7}+b_{4,0}b_{14,1}b_{7,1}+b_{4,0}b_{12,1}b_{9,0}+b_{4,0}b_{8,1}b_{13,1}+b_{4,0}^{2}b_{14,1}b_{3,0}+b_{4,0}^{2}b_{12,1}b_{5,0}+ b_{4,0}c_{8,3}b_{13,7} + b_{4,0}c_{8,3}b_{13,1} + b_{4,0}c_{8,3}^{2}b_{5,0}

(59) b_{7,1}^{2}b_{11,5} + b_{12,7}b_{13,1} + b_{4,0}^{3}b_{13,7} + b_{4,0}c_{8,3}b_{13,1} + b_{4,0}^{2}c_{8,3}b_{3,0}^{3} + b_{4,0}^{3}c_{8,3}b_{5,0}

(60) b_{11,5}b_{15,13} +b_{12,7}b_{14,1} +b_{12,1}b_{3,0}b_{11,5} +b_{8,1}b_{7,1}b_{11,5} +b_{4,0}^{2}b_{7,1}b_{11,5} +b_{4,0}^{3}b_{3,0}b_{11,5} + c_{8,3}b_{12,1}b_{3,0}^{2}+b_{6,1}c_{8,3}b_{12,1}+b_{4,0}c_{8,3}b_{7,0}^{2}+b_{4,0}c_{8,3}b_{14,1}+b_{4,0}b_{8,1}c_{8,3}b_{3,0}^{2}+b_{4,0}^{2}c_{8,3}b_{3,0}b_{7,1}+ b_{4,0}^{3}c_{8,3}b_{3,0}^{2} + c_{8,3}^{2}b_{3,0}b_{7,0}

(61) b_{13,1}^{2}+b_{12,7}b_{14,1}+b_{12,1}b_{7,1}^{2}+b_{12,1}b_{3,0}b_{11,5}+b_{8,1}b_{7,1}b_{11,5}+b_{4,0}^{2}b_{7,1}b_{11,5}+b_{4,0}^{2}b_{12,1}b_{3,0}^{2}+ b_{4,0}^{3}b_{3,0}b_{11,5} + c_{8,3}b_{12,1}b_{3,0}^{2} + b_{4,0}c_{8,3}b_{7,1}^{2} + b_{4,0}b_{8,1}c_{8,3}b_{3,0}^{2} + b_{4,0}^{2}c_{8,3}b_{3,0}b_{7,1}

(62) b_{13,1}b_{13,7}+b_{8,1}b_{7,1}b_{11,5}+b_{4,0}^{2}b_{7,1}b_{11,5}+b_{4,0}^{3}b_{3,0}b_{11,5}+b_{4,0}b_{8,1}c_{8,3}b_{3,0}^{2}+b_{4,0}^{2}c_{8,3}b_{3,0}b_{7,1}+ b_{4,0}^{3}c_{8,3}b_{3,0}^{2}

(63) b_{13,7}^{2} + b_{4,0}b_{12,1}b_{3,0}b_{7,1} + b_{4,0}b_{8,1}b_{7,1}^{2} + b_{4,0}b_{8,1}b_{14,1} + b_{4,0}b_{8,1}^{2}b_{3,0}^{2} + b_{4,0}b_{6,1}b_{8,1}^{2} + b_{4,0}b_{6,1}^{2}b_{3,0}b_{7,0} + b_{4,0}^{2}b_{6,1}b_{3,0}^{4} + b_{4,0}^{2}b_{6,1}b_{12,1} + b_{4,0}^{2}b_{6,1}^{2}b_{3,0}^{2} + b_{4,0}^{3}b_{7,0}^{2} + b_{4,0}^{3}b_{8,1}b_{3,0}^{2} + b_{4,0}^{4}b_{3,0}b_{7,0}+b_{4,0}^{5}b_{3,0}^{2}+b_{4,0}^{5}b_{6,1}+b_{4,0}c_{8,3}b_{7,1}^{2}+b_{4,0}^{3}c_{8,3}b_{3,0}^{2}+c_{8,3}^{2}b_{3,0}b_{7,0}+b_{4,0}b_{6,1}c_{8,3}^{2}

(64) b_{14,1}b_{13,7} + b_{12,7}b_{15,13} + b_{8,1}^{2}b_{11,5} + b_{4,0}b_{12,1}b_{11,5} + b_{6,1}b_{8,1}c_{8,3}b_{5,0} + b_{6,1}^{2}c_{8,3}b_{7,0} + b_{4,0}c_{8,3}b_{15,13} + b_{4,0}b_{6,1}c_{8,3}b_{3,0}^{3} + b_{4,0}b_{6,1}^{2}c_{8,3}b_{3,0} + b_{4,0}^{2}b_{6,1}c_{8,3}b_{5,0} + b_{4,0}^{3}c_{8,3}b_{7,1} + b_{4,0}^{3}c_{8,3}b_{7,0} + b_{4,0}c_{8,3}^{2}b_{7,0}

(65) b_{7,1}^{2}b_{13,1}+b_{12,7}b_{15,13}+b_{8,1}b_{12,1}b_{7,1}+b_{4,0}b_{6,1}b_{14,1}b_{3,0}+b_{4,0}b_{6,1}b_{12,1}b_{5,0}+b_{4,0}b_{6,1}b_{8,1}b_{3,0}^{3}+ b_{4,0}b_{6,1}^{2}b_{8,1}b_{3,0}+b_{4,0}b_{6,1}^{3}b_{5,0}+b_{4,0}^{2}b_{12,1}b_{7,1}+b_{4,0}^{2}b_{6,1}^{2}b_{7,0}+b_{4,0}^{3}b_{15,13}+b_{4,0}^{3}b_{12,1}b_{3,0}+ b_{4,0}^{3}b_{6,1}b_{3,0}^{3} + b_{4,0}^{3}b_{6,1}^{2}b_{3,0} + b_{4,0}^{4}b_{6,1}b_{5,0} + b_{4,0}^{3}c_{8,3}b_{7,0}

(66) b_{14,1}b_{3,0}b_{11,5}+b_{8,1}b_{7,1}b_{13,1}+b_{8,1}^{2}b_{12,1}+b_{4,0}b_{11,5}b_{13,1}+b_{4,0}b_{12,1}b_{12,7}+b_{4,0}b_{6,1}b_{12,1}b_{3,0}^{2}+ b_{4,0}^{2}b_{7,1}b_{13,1} + b_{4,0}^{3}b_{8,1}b_{3,0}b_{5,0} + b_{4,0}^{3}b_{8,1}^{2} + b_{4,0}^{4}b_{12,1} + b_{4,0}^{5}b_{8,1} + c_{8,3}b_{12,1}b_{3,0}b_{5,0} + b_{8,1}c_{8,3}b_{3,0}^{4}+b_{6,1}c_{8,3}b_{7,0}^{2}+b_{6,1}c_{8,3}b_{14,1}+b_{6,1}b_{8,1}c_{8,3}b_{3,0}^{2}+b_{6,1}^{2}b_{8,1}c_{8,3}+b_{4,0}b_{8,1}c_{8,3}b_{3,0}b_{5,0}+ b_{4,0}b_{8,1}^{2}c_{8,3} + b_{4,0}b_{6,1}c_{8,3}b_{3,0}b_{7,0} + b_{4,0}^{2}c_{8,3}b_{3,0}^{4} + b_{4,0}^{2}c_{8,3}b_{12,1} + b_{4,0}^{2}b_{6,1}c_{8,3}b_{3,0}^{2} + b_{4,0}^{2}b_{6,1}^{2}c_{8,3} + b_{4,0}^{5}c_{8,3}

(67) b_{13,1}b_{15,13} + b_{14,1}b_{7,1}^{2} + b_{8,1}b_{7,1}b_{13,1} + b_{8,1}^{2}b_{12,1} + b_{4,0}b_{6,1}b_{12,1}b_{3,0}^{2} + b_{4,0}^{2}b_{7,1}b_{13,1} + b_{4,0}^{2}b_{12,1}b_{3,0}b_{5,0}+b_{4,0}^{2}b_{8,1}b_{3,0}^{4}+b_{4,0}^{2}b_{6,1}b_{7,0}^{2}+b_{4,0}^{2}b_{6,1}b_{14,1}+b_{4,0}^{2}b_{6,1}b_{8,1}b_{3,0}^{2}+b_{4,0}^{2}b_{6,1}^{2}b_{8,1}+ b_{4,0}^{3}b_{8,1}b_{3,0}b_{5,0} + b_{4,0}^{3}b_{8,1}^{2} + b_{4,0}^{3}b_{6,1}b_{3,0}b_{7,0} + b_{4,0}^{4}b_{3,0}^{4} + b_{4,0}^{4}b_{6,1}b_{3,0}^{2} + b_{4,0}^{4}b_{6,1}^{2} + b_{4,0}^{7}

(68) b_{13,7}b_{15,13}+b_{8,1}b_{7,1}b_{13,1}+b_{8,1}^{2}b_{12,1}+b_{4,0}b_{6,1}b_{12,1}b_{3,0}^{2}+b_{4,0}^{2}b_{7,1}b_{13,1}+b_{4,0}^{3}b_{8,1}b_{3,0}b_{5,0}+ b_{4,0}^{3}b_{8,1}^{2} + b_{4,0}^{4}b_{12,1} + b_{4,0}^{5}b_{8,1} + c_{8,3}b_{12,1}b_{3,0}b_{5,0} + b_{6,1}c_{8,3}b_{7,0}^{2} + b_{6,1}c_{8,3}b_{14,1} + b_{4,0}b_{8,1}c_{8,3}b_{3,0}b_{5,0} + b_{4,0}b_{8,1}^{2}c_{8,3} + b_{4,0}^{3}b_{8,1}c_{8,3} + c_{8,3}^{2}b_{3,0}^{4} + b_{6,1}c_{8,3}^{2}b_{3,0}^{2} + b_{6,1}^{2}c_{8,3}^{2} + b_{4,0}^{3}c_{8,3}^{2}

(69) b_{7,1}b_{11,5}b_{13,1}+b_{14,1}^{2}b_{3,0}+b_{8,1}b_{12,1}b_{11,5}+b_{8,1}^{3}b_{7,0}+b_{6,1}b_{8,1}^{2}b_{3,0}^{3}+b_{6,1}^{4}b_{7,0}+b_{4,0}b_{12,7}b_{15,13}+ b_{4,0}b_{12,1}^{2}b_{3,0} + b_{4,0}b_{8,1}b_{12,1}b_{7,1} + b_{4,0}b_{8,1}^{2}b_{11,5} + b_{4,0}b_{6,1}b_{14,1}b_{7,0} + b_{4,0}b_{6,1}b_{8,1}^{2}b_{5,0} + b_{4,0}b_{6,1}^{2}b_{8,1}b_{7,0}+b_{4,0}b_{6,1}^{3}b_{3,0}^{3}+b_{4,0}b_{6,1}^{4}b_{3,0}+b_{4,0}^{2}b_{8,1}^{2}b_{7,0}+b_{4,0}^{2}b_{6,1}b_{14,1}b_{3,0}+b_{4,0}^{2}b_{6,1}b_{12,1}b_{5,0}+ b_{4,0}^{2}b_{6,1}b_{8,1}b_{3,0}^{3}+b_{4,0}^{2}b_{6,1}^{3}b_{5,0}+b_{4,0}^{3}b_{12,1}b_{7,1}+b_{4,0}^{3}b_{8,1}^{2}b_{3,0}+b_{4,0}^{4}b_{15,13}+b_{4,0}^{4}b_{12,1}b_{3,0}+ b_{4,0}^{4}b_{8,1}b_{7,0} + b_{4,0}^{4}b_{6,1}b_{3,0}^{3} + b_{4,0}^{4}b_{6,1}^{2}b_{3,0} + b_{4,0}^{6}b_{7,1} + b_{8,1}c_{8,3}b_{15,13} + b_{8,1}^{2}c_{8,3}b_{7,1} + b_{6,1}c_{8,3}b_{14,1}b_{3,0}+b_{6,1}b_{8,1}c_{8,3}b_{3,0}^{3}+b_{6,1}^{2}b_{8,1}c_{8,3}b_{3,0}+b_{6,1}^{3}c_{8,3}b_{5,0}+b_{4,0}c_{8,3}b_{12,1}b_{7,1}+ b_{4,0}b_{6,1}^{2}c_{8,3}b_{7,0}+b_{4,0}^{2}c_{8,3}b_{15,13}+b_{4,0}^{2}b_{6,1}c_{8,3}b_{3,0}^{3}+b_{4,0}^{2}b_{6,1}^{2}c_{8,3}b_{3,0}+b_{4,0}^{3}b_{6,1}c_{8,3}b_{5,0}+ b_{4,0}^{4}c_{8,3}b_{7,0} + b_{8,1}c_{8,3}^{2}b_{7,0} + b_{4,0}^{2}c_{8,3}^{2}b_{7,0}

(70) b_{8,1}b_{11,5}b_{13,1}+b_{8,1}b_{12,1}b_{12,7}+b_{4,0}b_{14,1}b_{7,0}^{2}+b_{4,0}b_{14,1}^{2}+b_{4,0}b_{8,1}b_{7,1}b_{13,1}+b_{4,0}^{2}b_{12,1}^{2}+ b_{4,0}^{3}b_{7,1}b_{13,1} + b_{4,0}^{4}b_{8,1}b_{3,0}b_{5,0} + b_{4,0}^{4}b_{8,1}^{2} + b_{4,0}^{6}b_{8,1} + c_{8,3}b_{12,7}^{2} + b_{4,0}c_{8,3}b_{7,1}b_{13,1} + b_{4,0}^{2}b_{8,1}c_{8,3}b_{3,0}b_{5,0}+b_{4,0}^{2}b_{8,1}^{2}c_{8,3}+b_{4,0}^{4}b_{8,1}c_{8,3}+b_{8,1}c_{8,3}^{2}b_{3,0}b_{5,0}+b_{8,1}^{2}c_{8,3}^{2}+b_{4,0}^{2}b_{8,1}c_{8,3}^{2}

(71) b_{8,1}b_{12,7}b_{13,1} + b_{8,1}b_{12,1}b_{13,7} + b_{4,0}b_{14,1}b_{15,13} + b_{4,0}b_{12,1}b_{14,1}b_{3,0} + b_{4,0}b_{12,1}^{2}b_{5,0} + b_{4,0}b_{8,1}b_{14,1}b_{7,1} + b_{4,0}^{2}b_{14,1}b_{11,5} + b_{4,0}^{2}b_{12,7}b_{13,1} + b_{4,0}^{2}b_{12,1}b_{13,7} + b_{4,0}^{2}b_{12,1}b_{13,1} + b_{4,0}^{2}b_{8,1}^{2}b_{3,0}^{3} + b_{4,0}^{2}b_{6,1}b_{8,1}^{2}b_{3,0} + b_{4,0}^{2}b_{6,1}^{2}b_{8,1}b_{5,0} + b_{4,0}^{3}b_{14,1}b_{7,1} + b_{4,0}^{3}b_{6,1}b_{15,13} + b_{4,0}^{3}b_{6,1}b_{8,1}b_{7,0} +b_{4,0}^{4}b_{12,1}b_{5,0} +b_{4,0}^{4}b_{6,1}^{2}b_{5,0} +b_{4,0}^{5}b_{13,1} +b_{4,0}^{5}b_{8,1}b_{5,0} +b_{4,0}^{5}b_{6,1}b_{7,0} + b_{4,0}^{6}b_{3,0}^{3}+b_{4,0}^{6}b_{6,1}b_{3,0}+b_{4,0}^{7}b_{5,0}+b_{4,0}c_{8,3}b_{14,1}b_{7,1}+b_{4,0}c_{8,3}b_{14,1}b_{7,0}+b_{4,0}b_{8,1}c_{8,3}b_{13,1}+ b_{4,0}b_{6,1}c_{8,3}b_{12,1}b_{3,0} + b_{4,0}^{3}b_{6,1}c_{8,3}b_{7,0}

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Mathematics Department, National University of Ireland, Galway, Ireland

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

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

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

Mathematics Department, National University of Ireland, Galway, Ireland

E-mail address: graham.ellis@nuigalway.ie