COCLASS AND COHOMOLOGY

JON F. CARLSON^{∗}

Dedicated to Claus Ringel for his sixtieth birthday

1. Introduction

This paper considers what the coclass classification of finite p-groups might say about the cohomology rings of groups. The classification was recently completed by Charles Leedham-Green and others (see [LG]). It says roughly that, with a finite number of exceptions, every p-group having coclass r is associated to one of a finite number of p-adic uniserial space groups. The nature of the association for 2-groups is given in detail in Theorem 5.2. For p-groups with p odd, the association is somewhat more complicated and we refer the reader to [LG] for details. The coclass of a p-group G is r = n − c where p^{n} is the order of G and c is the class of G, the length of the lower central series of G.

The model for the association is the situation with 2-groups of coclass 1. With exactly two exceptions, every 2 group of coclass 1 is one of the dihedral groups, semidihedral groups or (generalized) quaternion groups. Each of these is associated to the infinite dihedral group ⟨z, y|z^{2} = 1, zyz^{−1} = y^{−1}⟩ in the sense that it isomorphic either to a quotient of that group or to a central extension of a quotient of that group by a cyclic group of order 2. Thus, there is a unique family of 2-groups of coclass 1.

It has long been known that if k is a field of characteristic 2 and if G_{1} and G_{2} are dihedral 2-groups, then the cohomology rings H^{∗}(G_{1}, k) and H^{∗}(G_{2}, k) are isomorphic [CE]. Similarly, the cohomology rings of any two quaternion groups are isomorphic [CE] and the cohomology ring of any two semi-dihedral groups are isomorphic [EP]. Hence, among all of the groups of coclass 1 there is only a finite number of isomorphism types of cohomology rings. The main theorem of this paper shows that this is correct in general. That is, for any r there are only finitely many isomorphism classes of cohomology rings of 2-groups of coclass r with coefficients in a fixed field k of characteristic 2. We should note that H^{∗}(G, k) ≅ k ⊗_{𝔽_{2}} H^{∗}(G, 𝔽_{2}) is really defined over the prime field. Hence the choice of the field k is largely irrelevant in this consideration.

Date: November 17, 2004.

* Supported in part by a grant from NSF.

In fact, there is experimental evidence indicating that a lot more is true. The proof of the theorem shows only that the number is finite, without giving any sort of effective bounds. The proof is mostly a collection of counting arguments on the number of possible outcomes of certain spectral sequences. It does not actually say that any two groups have isomorphic cohomology rings. The computational evidence indicates that successive quotients of the uniserial space groups do indeed have isomorphic cohomology rings. The evidence is somewhat scant because of the limitations of the computational methods. However, there is enough to be intriguing.

For these reasons, this paper should be seen as a first step in an investigation of the isomorphisms between cohomology rings. The methods for calculating the rings are not yet sufficiently refined to determine what other theorems might be provable. For one thing, a similar theorem should be true for p-groups in the case that p is an odd prime. Also, similar results have been found in the studies [BP, W], which though much more specialized, give much stronger outcomes.

Following this introduction, the next two sections of the paper develop some technicalities of counting with spectral sequences. This is followed by an investigation of the cohomology ring of the quotients of a particularly important 2-adic space group. The proof of the main theorem is found in Section 5. The last section discusses the case of cohomology rings of p-groups for odd p as well as reviewing some of the experimental evidence.

Except as noted, all of the computer calculations of the last section were run using the MAGMA computer algebra package [BC]. Thanks are due to John Cannon and his group for their help in developing the computer programs. I also want to thank the Alexander von Humboldt Foundation and the Max Planck Institute for Mathematics in Bonn for their support during a time when part of this paper was written. Finally, I would like to thank Mike Newman and especially Eamonn O’Brien for several helpful conversations.

2. Ungrading the spectral sequence

In this section we investigate the extent to which a ring may differ from a graded version of itself. The main interest here is in the difference between a cohomology ring H^{∗}(G, k) of a group G and the graded version which is the E_{∞} page of a spectral sequence

E_{2}^{r,s} = H^{r}(G/H, H^{s}(H, k)) ⇒ H^{∗}(G, k)

of the group extension, where H is a normal subgroup of G. We write the theorem in more general terms. In the above situation it tells us that given the ring E_{∞}^{∗,∗}, there is only a finite number of ways of creating H^{∗}(G, k).

Suppose that R = ⊕_{i≥0}R^{i} is a graded-commutative, finitely generated k-algebra over a finite field k. Assume that R^{0} = k. Then each R^{i} is finite dimensional over k. Assume that R has a filtration F^{∗} such that F^{j}(R^{i}) = 0 whenever j > i. Assume also that the filtration is multiplicative in the sense that F^{i}(R)F^{j}(R) ⊆ F^{i+j}(R) for all i and j.

Let S = ∑_{i,j≥0} S^{i,j} be the doubly graded algebra obtained from R by letting

S^{i,j} = F^{i}(R^{i+j})/F^{i+1}(R^{i+j}).

The multiplication on S is given by the following formula. If x ∈ F^{i}(R^{m}) and y ∈ F^{j}(R^{n}), then

(x + F^{i+1}(R^{m}))(y + F^{j+1}(R^{n})) = xy + F^{i+j+1}(R^{m+n}).

Theorem 2.1. Let R be a finitely generated, graded-commutative k-algebra as above and let S be the doubly graded ring defined by a filtration F^{∗} on R. Assume that the field k is finite. Then the structure of R is determined by the structure of S within a finite number of possibilities.

Proof. Suppose that s_{1}, …, s_{t} is a set of homogeneous generators for the ring S. For each i, there is a pair of non-negative integers (j_{i}, k_{i}) such that s_{i} ∈ S^{j_{i},k_{i}}. For each i, let r_{i} ∈ R^{j_{i}+k_{i}} be any element such that r_{i} + F^{j_{i}}(R) = s_{i}. We claim first that r_{1}, …, r_{t} generate R.

For suppose that r ∈ F^{i}(R^{n}), r ∉ F^{i+1}(R^{n}). Then we know that r + F^{i+1} = f(s_{1}, …, s_{t}) for some polynomial f (in graded commuting variables). Then we have that r − f(r_{1}, … r_{t}) ∈ F^{i+1}(R^{n}). Now we continue along this line until we get that r − g(r_{1}, …, r_{t}) = 0 for some polynomial g. This establishes the claim.

Let F = k⟨α_{1}, …, α_{t}⟩ be a free graded-commutative algebra such that we have a graded homomorphism ψ : F ⟶ S. Note that this requires that the degree of α_{i} be the same as that of s_{i} for all i. The variables α_{1}, …, α_{t} satisfy the relations α_{i}α_{j} = (−1)^{deg(α_{i})deg(α_{j})}α_{j}α_{i}, and these are essentially the only relations on the variables. Of course, if p = 2, then F is a polynomial ring. In any case, every element of F is a sum of monomials α_{1}^{u_{1}}α_{2}^{u_{2}} … α_{t}^{u_{t}}, where if p > 2, the exponent u_{i} is either 0 or 1 when the degree of α_{i} is odd (since α_{i}^{2} = 0 in such a case). Moreover, we can regard F as a doubly graded ring and ψ as map of doubly graded rings by assigning to each α_{i} the double degree (j_{i}, k_{i}).

Because F is a free object in the category of graded commutative k-algebras, there is a homomorphism θ : F ⟶ R given by θ(α_{i}) = r_{i} for all i. In particular, note that the degrees match up and this is a graded homomorphism. From the above argument, we have that θ is surjective. Let I denote the kernel of ψ, and suppose that f_{1}, …, f_{n} is a minimal set of doubly homogeneous generators for I. For each i, let (a_{i}, b_{i}) be the double degree of f_{i} = f_{i}(α_{1}, …, α_{t}). It follows that f_{i}(r_{1}, …, r_{t}) ∈ F^{a_{i}+1}(R^{a_{i}+b_{i}}). Hence there exist f_{i,1}, …, f_{i,b_{i}} in F having homogeneous (double) degrees (a_{i} + 1, b_{i} − 1), …, (a_{i} + b_{i}, 0), respectively, such that

f_{i}(r_{1}, …, r_{t}) + ∑_{k=1}^{b_{i}} f_{i,k}(r_{1}, …, r_{t}) = 0.

Hence, if g_{i} = f_{i} + ∑ f_{i,k}, then g_{i} is in the kernel K of θ. Let J be the ideal of F generated by g_{1}, …, g_{n}. Clearly J ⊆ K. We claim that J = K.

Suppose that h = h(α_{1}, …, α_{t}) is a homogeneous polynomial of degree m that is contained in K, the kernel of θ. Write h = h_{ℓ} + h_{ℓ+1} + ⋯ + h_{m}, where h_{j} is of homogeneous double degree (j, m − j) and ℓ is some integer between 0 and m with h_{ℓ} ≠ 0. Then

h_{ℓ}(s_{1}, …, s_{t}) = h(r_{1}, …, r_{t}) + F^{ℓ+1}(R^{m}) = 0.

We must have that h_{ℓ} ∈ I and hence that h_{ℓ} = ∑_{k=1}^{n} β_{k}f_{k}, where β_{k} is homogeneous of double degree (ℓ − a_{k}, m − ℓ − b_{k}). Then we get that h − ∑_{k} β_{k}g_{k} is in K and moreover, h − ∑_{k} β_{k}g_{k} is a sum of doubly homogeneous polynomials in double degrees (u, m − u) where u ≥ ℓ. Now perform this operation again with h replaced by h′ = h − ∑_{k} β_{k}g_{k}. After a finite number of steps we have h ∈ J as desired.

So we have shown that R ≅ F/J. The proof of the theorem is a consequence of the fact that J = (g_{1}, …, g_{n}) and for each i, g_{i} = f_{i} + u_{i} where u_{i} has the same total degree as g_{i} but higher double degree. That is, R differs from S only in the choices of the elements u_{1}, …, u_{n}, and there are only finitely many possibilities for each such choice. We leave it as an exercise to the reader to show that the number of possibilities does not depend on our choice of the generators. □

Some of the steps in the proof of the theorem can be summarized in the following results which are useful to us later in the paper. The results are corollaries to the proof and we do not repeat the arguments.

Corollary 2.2. Suppose that we have an extension 1 ⟶ H ⟶ G ⟶ Q ⟶ 1 of finite groups. Assume that the structure of the E_{∞} page of the spectral sequence of the extension is known either as a ring or as a finitely generated module over some homogeneous set of parameters in E_{∞}^{∗,0} ∪ E_{∞}^{0,∗}. Then the structure of H^{∗}(G, k) as a ring is determined up to a finite number of possibilities.

Corollary 2.3. Suppose that the structure of H^{∗}(G, k) is known as a finitely generated module over some homogeneous finitely generated subring. Then the ring structure of H^{∗}(G, k) is determined up to a finite number of possibilities.

3. The counting theorems

The purpose of this section is to prove some theorems on the number of cohomology rings that can be built from an extension. Another way of stating the proposition below is that if G is a p-group with a normal subgroup H, the cohomology ring H^{∗}(G, k) is determined by H^{∗}(H, k) and H^{∗}(G/H, k) within a finite number of possibilities provided the action of G/H on H is trivial and it is known that the spectral sequence

E_{2}^{r,s} = H^{r}(G/H, H^{s}(H, k)) ⇒ H^{r+s}(G, k)

collapses after some bounded number of pages. In particular, we need not know anything about the groups H and G/H, only about their cohomology rings and the action of G/H on H. We suspect that the second condition in the proposition is not necessary.

Proposition 3.1. Suppose that S and T are finitely generated k-algebras and that b is a positive integer. Then there are only finitely many k-algebras R such that R ≅ H^{∗}(G, k) where G is a finite p-group satisfying the following the properties:

(1) G has a normal subgroup H such that H^{∗}(H, k) ≅ S and H^{∗}(G/H, k) ≅ T,

(2) G/H has trivial action on H, and (3) the spectral sequence of the group extension

1 ⟶ H ⟶ G ⟶ G/H ⟶ 1

stops (collapses) on or before page b (i.e. E_{b}^{∗,∗} = E_{∞}^{∗,∗}).

Proof. Suppose that G and H satisfy the conditions of the theorem. Because G/H has trivial action on H, E_{2}^{r,s} ≅ H^{r}(G/H, k) ⊗ H^{s}(H, k), and moreover, the isomorphism is an isomorphism of rings. Now choose a complete homogeneous set of parameters γ_{1}, …, γ_{u} for H^{∗}(H, k). That is, γ_{1}, …, γ_{u} generate a polynomial subring over which H^{∗}(H, k) is a finitely generated module. For convenience of notation we identify H^{∗}(H, k) with the subalgebra H^{∗}(H, k) ⊗ 1 ⊆ E_{2}^{∗,∗}.

For an element γ of even degree s in H^{∗}(H, k), suppose that γ survives to the E_{n} page of the spectral sequence. If d_{n}(γ) = μ ∈ E_{n}^{n,s−n+1}, then because d_{n} is a derivation we have d_{n}(γ^{p}) = p · d_{n}(γ)γ^{p−1} = pμγ^{p−1} = 0. Because the spectral sequence stops at the E_{b} page, we must have that for all i, ζ_{i} = γ_{i}^{p^{b}} is a universal cycle. That is, ζ_{i} is in the kernel of every one of the differentials d_{j} for j ≥ 2.

Let ζ_{u+1}, …, ζ_{v} be a homogeneous set of parameters for H^{∗}(G/H, k) which we identify with the subalgebra E_{2}^{∗,0} of E_{2}^{∗,∗}. Then E_{2}^{∗,∗} is a finitely generated module over the polynomial subalgebra W generated by ζ_{1}, …, ζ_{v}. Next we notice inductively that, because d_{j}(ζ_{i}) = 0, each d_{j} is W -module homomorphism. Therefore, each E_{j+1}^{∗,∗} is a finitely generated W -module. The generators can be chosen to be homogeneous. The key point is that if α_{1}, …, α_{q} is a set of generators for E_{j}^{∗,∗} as a W -module, then the homomorphism d_{j} is determined entirely by the choice of the images d_{j}(α_{1}), …, d_{j}(α_{q}). For each i there is only a finite number of choices of the image d_{j}(α_{i}) simply because E_{j}^{ℓ_{1}+j,ℓ_{2}−j+1} is a finite set, (with (ℓ_{1}, ℓ_{2}) being the degree of α_{i}). Consequently, for each j ≥ 2 there is only a finite number of possibilities for the map d_{j} and for the structure of E_{j+1}^{∗,∗} as both a W -module and as a ring. Because d_{j} = 0 for j ≥ b, there are only finitely many possibilities for E_{∞}^{∗,∗}.

The fact that there are only finitely many possibilities for R follows from Theorem 2.1. □

Next we consider the case that H has bounded order. It is sufficient to consider the situation in which H ≅ C_{p} is cyclic of order p and apply induction. What we need is the following.

Lemma 3.2. Suppose that H ≅ C_{p} is contained in an abelian subgroup A such that |G : A| = p^{n}. Then the spectral sequence E_{2}^{r,s} = H^{r}(G/H, H^{s}(C_{p}, k)) ⇒ H^{r+s}(G, k) stops at the p^{n} + 1 page.

Proof. There exists an element η ∈ H^{2}(A, k) such that the restriction from A to H of η is not zero. This is because A is abelian and we know the cohomology of abelian p-groups. So let ζ = N orm_{A}^{G}(η) where N orm_{A}^{G} is the Evens Norm Map. The usual Mackey formula for norms (see [CT], Theorem 6.3.5.3) tells us that res_{G,H}(ζ) ≠ 0. The restriction map on cohomology from G to H is the edge homomorphism on the spectral sequence. So the image of the restriction res_{G,H} : H^{2p^{n}}(G, k) ⟶ H^{2p^{n}}(H, k) is isomorphic to E_{∞}^{0,2p^{n}}. Let ζ′ ∈ E_{∞}^{0,2p^{n}} be an element representing ζ.

Now if t ≥ 2p^{n} + 2, and if μ ∈ E_{t}^{r,s} with s = a(2p^{n}) + b, then μ := (ζ′)^{a}μ′ for some μ′ ∈ E_{t}^{r,b}. It follows that d_{t}(μ) = (ζ′)^{a}d_{t}(μ′) = 0. That is, ζ′ is regular on E_{t}^{∗,∗} and ζ is regular on H^{∗}(G, k). Therefore, d_{t} = 0 for t ≥ 2p^{n} + 2 as asserted. □

Theorem 3.3. Let f and n be positive integers. Suppose that

1 ⟶ H ⟶ G ⟶ Q ⟶ 1

is an extension of finite p-groups such that (1) |H| ≤ n, and

(2) G has an abelian subgroup A such that H ⊆ A and |G : A| ≤ f. Then the ring H^{∗}(G, k) is determined up to a finite number of possibilities by the ring H^{∗}(Q, k).

Proof. Consider first the case that H ≅ C_{p}. In that case, the conditions of Proposition 3.1 are satisfied and the cohomology ring H^{∗}(G, k) is determined by that of H^{∗}(Q, k) and H^{∗}(C_{p}, k) up to a finite number of possibilities. As the cohomology ring H^{∗}(C_{p}, k) is known, we say that H^{∗}(G, k) is determined by H^{∗}(Q, k) up to a finite number of possibilities. Now the proof of the theorem follows from the fact that we can form G from Q by taking a sequence of at most n extensions by C_{p}. For each extension there are only finitely many possibilities for the cohomology ring H^{∗}(G, k), given the cohomology ring of Q. □

The next theorem approaches the extension from another direction, with the cyclic factor on top.

Theorem 3.4. Suppose that G is an extension 1 ⟶ H ⟶ G ⟶ C_{p} ⟶ 1 where H is a finite p-group. Suppose we know that the spectral sequence of the extension stops at page m for some fixed m. Then H^{∗}(G, k) is determined by H^{∗}(H, k) up to a finite number of possibilities.

Proof. First note that there is only a finite number of possible actions of C_{p} on H^{∗}(H, k) and so it does not matter if we know what that action is. Suppose that we choose a homogeneous set of parameters γ_{1}, …, γ_{t} for the ring of invariants H^{∗}(H, k)^{C_{p}} of H^{∗}(H, k) under the action of C_{p}. That is, γ_{1}, …, γ_{t} are elements such that the subring that they generate is a polynomial ring and the ring of invariants is finitely generated as a module over the subring. Let ζ_{i} = γ_{i}^{p}. Then ζ_{i} is in the image of the restriction of H^{∗}(G, k) to H^{∗}(H, k), by a norm argument (see Theorem 6.3.5 of [CT]). Also, ζ_{1}, …, ζ_{t} is a homogeneous set of parameters for H^{∗}(H, k). Let S be the subring of E_{2}^{∗,∗} generated by the classes of ζ_{1}, …, ζ_{t} in E_{2}^{0,∗} and by μ where μ is the class of the inflation of the generator in H^{2}(C_{p}, k) to E_{2}^{2,0}. Notice that each of the differentials d_{n} is zero on the classes of ζ_{1}, …, ζ_{t} and μ on the n^{th} page of the spectral sequence.

The ring E_{2}^{∗,∗} is finitely generated as a module over S. Moreover, since H^{s}(H, k) is periodic of period at most 2 as a module over H^{∗}(C_{p}, k), the generators of E_{2}^{∗,∗} can be assumed to lie in E_{2}^{0,∗} ∪ E_{2}^{1,∗}. The differentials in the spectral sequence are all S-module homomorphisms. It should be clear that the module E_{2}^{j,∗} = H^{j}(C_{p}, H^{∗}(H, k)), as a module over the ring generated by ζ_{1}, …, ζ_{t}, is determined by the action of C_{p} on H^{∗}(H, k) and is independent of the actual group H.

Now suppose that α_{1}, …, α_{w} is a set of homogeneous generators for E_{2}^{∗,∗} as an S-module. For each j, there is only a finite number of choices for d_{2}(α_{i}). Hence E_{3}^{∗,∗}, which is the homology of E_{2}^{∗,∗}, is determined up to a finite number of choices by E_{2}^{∗,∗}. Likewise, E_{4}^{∗,∗} is determined up to a finite number of choices by E_{3}^{∗,∗}. Continuing in this way we get that E_{n}^{∗,∗} is determined up to a finite number of choices by H^{∗}(H, k). However, E_{n}^{∗,∗} = E_{∞}^{∗,∗}. The proof of the theorem is completed by an application of Theorem 2.1. □

Finally, we are interested in extensions from another angle, where we know the cohomology of the extension and we want to construct the cohomology of the subgroup. Once again we are not assuming that we know anything about the group G beyond its cohomology ring.

Theorem 3.5. Let n be a positive integer. Suppose that S is a finitely generated k-algebra. Then there are only finitely many k algebras R with the property that R ≅ H^{∗}(H, k) for H a subgroup of a p-group G with H^{∗}(G, k) ≅ S and |G : H| ≤ p^{n}.

Proof. By an induction we can assume that n = 1 and that G/H ≅ C_{p} is a cyclic group of order p. In any case, H^{∗}(H, k) is finitely generated as a module over H^{∗}(G, k), and in fact it is isomorphic to the module H^{∗}(G, k) ≅ H^{∗}(G, k_{H}^{↑G}) by the Eckmann-Shapiro Lemma. The induced module k_{H}^{↑G} has a unique sequence of submodules

0 = M_{0} ⊆ M_{1} ⊆ ⋯ ⊆ M_{p} = k_{H}^{↑G}

such that for each i, M_{i} has dimension i. Hence we have a collection of exact sequences 0 ⟶ M_{i−1} ⟶ M_{i} ⟶ k ⟶ 0, and long exact sequences

… ⟶ H^{r}(G, M_{i−1}) ⟶ H^{r}(G, M_{i}) ⟶ H^{r}(G, k) ⟶^{δ} H^{r+1}(G, M_{i−1}) ⟶ …

where the maps commute with multiplication by elements of H^{∗}(G, k). Now assume by induction that there is only a finite number of possibilities for H^{∗}(G, M_{i−1}) as an H^{∗}(G, k)-module. This is certainly the case for i = 2. There are only finitely many possibilities for the map δ. We have an exact sequence of H^{∗}(G, k)-modules

0 ⟶ Coker(δ) ⟶ H^{∗}(G, M_{i}) ⟶ Ker(δ) ⟶ 0.

It remains only to prove that there is only a finite number of possibilities for H^{∗}(G, M_{i}). For this we need only show that Ext_{H^{∗}(G,k)}(Ker(δ), Coker(δ)) is finite. However, if

… ⟶ P_{2} ⟶ P_{1} ⟶ P_{0} ⟶ Ker(δ) ⟶ 0

is a projective resolution of Ker(δ), then because H^{∗}(G, k) is noetherian, we can assume that P_{1} is finitely generated. Moreover, any cocycle P_{1} ⟶ Coker(δ) must be a graded map. Hence there are only finitely many choices for the image of each of the generators of P_{1}. Hence there are only finitely many possibilities for the structure of H^{∗}(G, M_{i}) as an H^{∗}(G, k)-module.

It follows from Theorem 2.3 that there are only finitely many possibilities for the ring structure on H^{∗}(G, M_{p}) ≅ H^{∗}(H, k). □

4. Root groups

In this section we want to investigate the cohomology rings of some of the groups at the stems of the trees associated to the coclass classification of p-groups. For p = 2 these groups are all wreath products of dihedral groups or of extensions of abelian groups by quaternion groups. Several results are already known. We begin by reminding the reader of an old calculation [CE].

Lemma 4.1. If G is a dihedral group of order 2^{n} for some n ≥ 2, then H^{∗}(G, k) ≅ k[z, y, x]/(zy), where z and y have degree 1 and x has degree 2.

Let D_{1} be the split extension

0 ⟶ T_{1} ⟶ D_{1} ⟶ C_{2} ⟶ 1

where T_{1} ≅ ℤ_{2} is a copy of the 2-adic integers, and C_{2} acts on T_{1} by inverting elements (multiplication by −1). Let D_{1,j} = D_{1}/2^{j}T_{1}. Then D_{1,j} is a dihedral group of order 2^{j+1}.

Let D_{2} = D_{1} ≀ C_{2} be the wreath product of a D_{1} with a cyclic group of order 2. More generally, let D_{n} = D_{n−1} ≀ C_{2}. Then D_{n} is a uniserial 2-adic space group with translation group T_{n} ≅ T_{1}^{2^{n−1}} and point group P_{n} ≅ C_{2} ≀ C_{2} ≀ ⋯ ≀ C_{2}. Let D_{n,j} = D_{n}/2^{j}T_{n}. The following should be obvious.

Lemma 4.2. For n ≥ 2 and j ≥ 1, we have that D_{n,j} ≅ D_{n−1,j} ≀ C_{2}.

Proposition 4.3. For all n ≥ 1 and i, j ≥ 2, we have that H^{∗}(D_{n,i}, k) ≅ H^{∗}(D_{n,j}, k) where the isomorphism is an isomorphism of rings.

Proof. The proposition is a straightforward application of Nakaoka’s Theorem ([CT] Theorem 6.2.4) which says that the cohomology ring of a wreath product such as D_{n,j} ≅ D_{n−1,j} ≀C_{2} is isomorphic as a ring to the the E_{2} page of the spectral sequence of the group extension

1 ⟶ D_{n−1,j} × D_{n−1,j} ⟶ D_{n,j} ⟶ C_{2} ⟶ 1

Hence the result is a consequence of Lemma 4.1 and repeated application of Lemma 4.2 and Nakaoka’s Theorem. □

There is one other set of space groups that we must consider. These are slightly more complicated. Let Q_{1} be the split extension

1 ⟶ T_{1} ⟶ Q_{1} ⟶ Q_{16} ⟶ 1

where Q_{16} is a quaternion group of order 16 and T_{1} ≅ ℤ_{2}^{4} is a sum of four copies of the 2-adic integers. The action of Q_{16} on T_{1} is given in the paper [LMP2]. The details of that action are not really important to us here. For j ≥ 1, let Q_{1,j} = Q_{1}/2^{j}T_{1}. As before, for n > 1 we let Q_{n} = Q_{n−1} ≀ C_{2} and Q_{n,j} = Q_{n}/2^{n}T_{n} where T_{n} ≅ ℤ_{2}^{2^{n+1}} is the translation group of Q_{n}. It is straightforward to prove the following.

Lemma 4.4. For n ≥ 2 and j ≥ 1, we have that Q_{n,j} ≅ Q_{n−1,j} ≀ C_{2}.

The first result we want to prove is the following. Among other things, the proposition implies that there is an infinite collection of the groups Q_{1,j} with isomorphic mod-2 cohomology rings.

Proposition 4.5. There are only finitely many possible cohomology rings of the form H^{∗}(Q_{1,j}, k) for all j ≥ 2.

The proof requires two steps. The first we state as a lemma.

Lemma 4.6. Let S be the split extension

0 ⟶ T_{1} ⟶ S ⟶ C_{8} ⟶ 0

where C_{8} is the maximal subgroup of order 8 in Q_{16}. Thus, S is a subgroup of index 2 in Q_{1}. Let S_{j} = S/2^{j}T_{1}. There are only finitely many cohomology rings of the form H^{∗}(S_{j}, k) for all j.

Proof. The important thing is that S ⊆ D_{3} and moreover, |D_{3} : S| = 2^{4} = 16. The point is that the element of order 8 that generates the cyclic group at the point group of S is conjugate to an element of order 8 in the point group of D_{3}, whose translation group is also ℤ_{2}^{4}. Now by Theorem 3.5 on subgroups, H^{∗}(S_{j}, k) is determined by H^{∗}(D_{3,j}, k) up to a finite number of possibilities. But now H^{∗}(D_{3,j}, k) ≅ H^{∗}(D_{3,i}, k) for all i and j. Hence H^{∗}(S_{j}, k) is determined by H^{∗}(D_{3,1}, k) up to a finite number or possibilities. □

Proof of Proposition 4.5. We want to consider the spectral sequence of the group extension

1 ⟶ S_{j} ⟶ Q_{1,j} ⟶ C_{2} ⟶ 1

which has the form E_{2}^{r,s} = H^{r}(C_{2}, H^{s}(S_{j}, k)) ⇒ H^{r+s}(Q_{1,j}, k). By Theorem 3.4, the cohomology of Q_{1,j} is determined up to a finite number of possibilities by the cohomology ring of S_{j} provided we can show that the spectral sequence stops after some fixed number of pages, where that number does not depend on j. Consequently, we are done once we show that the spectral sequence stops.

For the spectral sequences, note that E_{2}^{1,0} = E_{∞}^{1,0} is generated by an element η that is inflated from a generator of H^{1}(C_{1}, k). Indeed, we know that for any r > 0, E_{2}^{r,s} = ηE_{2}^{r−1,s}. The inflation from C_{2} to Q_{1,j} factors through the inflation H^{∗}(C_{2}, k) ⟶ H^{∗}(Q_{16}, k) where Q_{16} ≅ Q_{1}/T_{1}. It follows that η^{3} = 0 in E_{4}^{∗,∗} because this is what happens in the cohomology of Q_{16}. The implication is that E_{4}^{r,s} = 0 for r > 3 and hence that E_{4}^{r,s} = E_{∞}^{r,s} for all r and s. This completes the proof of the proposition. □

Proposition 4.7. For any n, there is only a finite number of possible cohomology rings of the form H^{∗}(Q_{n,j}, k) for all j ≥ 2.

Proof. This follows the same ideas as the proof of Proposition 4.3. That is, for each n the cohomology of Q_{n,j} ≅ Q_{n−1,j} ≀ C_{2} is determined entirely from the cohomology ring H^{∗}(Q_{n−1,j}, k), using the spectral sequence. Inductively, we see that there is only a finite number of possibilities for H^{∗}(Q_{n,j}, k) by Proposition 4.5. □

5. The main theorem

Suppose that G is a p-group. The purpose of this section is to show that the cohomology ring of G is determined up to a finite number of choices by the coclass of G. The main theorem is the following.

Theorem 5.1. Let k be a field of characteristic 2. For any natural number r there are only finitely many graded commutative k-algebras R with the property that R ≅ H^{∗}(G, k), where G is a 2-group of coclass r.

Hence the theorem says that among all of the groups G of coclass r there is only a finite number of cohomology rings H^{∗}(G, k) up to isomorphism. The proof of the theorem relies very heavily on the classification of p-groups by coclass [LG]. The classification can be expressed as follows. Note that we are identifying a group with its isomorphism class.

Theorem 5.2. Let r be a positive integer and let G_{r} be the collection of all 2-groups of coclass r. Then G_{r} = G′_{r} ∪ G′′_{r} where G′_{r} and G′′_{r} satisfy the following properties.

(1) G′′_{r} has only a finite number of elements.

(2) If G is in G′_{r}, then there exist a normal subgroup N ⊆ G, a uniserial 2-adic

space group S of coclass r and a normal subgroup A of S such that a. A is contained in the translation subgroup of S, b. |N| ≤ 2^{2^{r−1}(2^{r−1}+r+3)}, and c. G/N ≅ S/A.

Proof. The theorem is a rewrite of Theorem 7.6 of [LG], except that we have incorporated a definition of “constructible” (see the paragraph preceding Lemma 5.1 of [LG]. □

Recall that a 2-adic space group is a group S that has a normal subgroup T ≅ ℤ_{2}^{t} for some t, such that B = S/T is a finite 2-group. The action of B on T should be irreducible. In particular, for each n there is one and only one P -invariant subgroup of T of index 2^{n} in T where P is the point group of S. That subgroup is an element of the lower central series for S. So we have an exact sequence

0 ⟶ T ⟶ S ⟶ B ⟶ 1.

The subgroup T is called the translation subgroup and B is called the point group. In the examples of the last section, for D_{n}, the translation subgroup is ℤ_{2}^{2^{n−1}} while the point group is C_{2} ≀ ⋯ ≀ C_{2} (n copies), and for Q_{n} the translation group is ℤ_{2}^{2^{n+1}} and the point group is Q_{16} ≀ C_{2} ≀ ⋯ ≀ C_{2}. The result we want is the following.

Theorem 5.3. Suppose that S is a 2-adic uniserial space group with translation subgroup T ≅ ℤ_{2}^{2^{n−1}} and point group P. Then either S ⊆ D_{n} or S ⊆ Q_{n−2}. That is, S is isomorphic to a subgroup of either D_{n} or Q_{n−2}. Moreover, the embedding is such that T is a normal subgroup of the translation group of D_{n} or Q_{n−2} as appropriate.

Proof. The proof can be found in the literature, but does not appear to be succinctly written in any one location. For that reason, we present a brief sketch. As on page 76 of [LMP1], we can blow up T to V ≅ T ⊗_{ℤ_{2}} ℚ_{2}, yielding a split extension X of V by the point group P. Now P is conjugate to a subgroup Q of the point group of either D_{n} or Q_{n−2} (see [LP] or [LMP2]). So assume that P ⊆ Q and let T̃ be a Q-invariant superlattice

T ⊆ T̃ ⊆ V ≅ ℚ_{2}^{2^{n−1}}.

We can assume that T̃ is chosen so that the extension R, given as

0 ⟶ T̃ ⟶ R ⟶ Q ⟶ 1,

is isomorphic to either D_{n} or Q_{n−2}. Thus, S ⊆ R, where both are regarded as subgroups of X.

As to the invariance of T, we note that by the uniseriality assumption, T is the only P -invariant subgroup of T̃ having index |T̃ : T |. Likewise, there is one and only one Q-invariant sublattice of T̃ having that index, and the sublattice must be T. □

In considering coclass we need the following. It is Conjecture E of [LG] and was first proved in [LMP2], though it also follows from the alternate approach to the subject presented in [S].

Theorem 5.4. For any r > 0, there are only finitely many isomorphism classes of 2-adic uniserial space groups of coclass r.

Lemma 5.5. Let r be a positive integer. There exist integers γ(r) and δ(r) with the following properties. Suppose that G is a finite 2-group in G′_{r} and that the translation subgroup of the 2-adic uniserial space group associated to G in Theorem 5.2(2) has rank 2^{n−1}. Then G has a normal subgroup N such that

(1) |N| ≤ γ(r), and

(2) G/N is isomorphic to a subgroup of a group Q where either Q = D_{n,j} or Q = Q_{n−2,j} for some j, and where |Q : G/N| ≤ δ(r).

Proof. By Theorem 5.2, there exists a 2-adic uniserial space group S of coclass r, a subgroup A of the translation subgroup of S and a normal subgroup N′ of G such that G/N′ ≅ S/A. In turn, S is a subgroup of a space group R such that R ≅ D_{n} or R ≅ Q_{n−2} and A is a normal subgroup of the translation subgroup T of R. Therefore, G/N′ is isomorphic to a subgroup of R/A and its index in R/A is the same as the index of S in R. Note that because there are only finitely many 2-adic uniserial space groups S of coclass r, we may assume that the index |R/A : G/N′| is bounded by a number B that depends only on the coclass r, and not on the particulars of G or N′.

Suppose that T is the translation subgroup of R. Then for some j,

2^{j+1}T ⊆ A ⊆ 2^{j}T and |2^{j}T : A| < 2^{n}.

Let N be the kernel of the composition G ⟶ R/A ⟶ R/(2^{j}T). Note here that R/(2^{j}T) is isomorphic to either D_{n,j} or to Q_{n−2,j}. Moreover, the order of N/N′ is at most 2^{n} while the index |R/(2^{j}T) : G/N| is at most B. Finally by Theorem 5.2, the order N′ is bounded by 2^{2^{r−1}(2^{r−1}+r+3)}. So the theorem is proved by letting δ(r) be B and letting γr be 2^{u}2^{2^{r−1}(2^{r−1}+r+3)} where 2^{u} is the largest rank of the translation subgroup of any uniserial 2-adic space group of coclass r. □

With this result in mind, we can prove the main theorem.

Proof of Theorem 5.1. Because the collection G′′_{r} is finite, it is only necessary to show that there are only finitely many rings of the form H^{∗}(G, k) for G a 2-group in G′_{r}. For G ∈ G′_{r}, there is a normal subgroup N ⊆ G such that G/N is isomorphic to a subgroup of either D_{n,j} or Q_{n−2,j} for some n and some j. Note that n is the rank of the translation group of some uniserial 2-adic space group of coclass r. Hence n is bounded. Therefore by Propositions 4.3 and 4.7 there are only finitely many possible cohomology rings for D_{n,j} or Q_{n−2,j}. Then by Theorem 3.5, there are only finitely many possibilities for the cohomology ring H^{∗}(G/N, k). Finally, by Proposition 3.3 and the bound on the order of N in Lemma 5.5, we are finished. □

6. Problems and remarks

It seems difficult to believe that the Main Theorem 5.1 is not also true in the case of p-groups for p an odd prime. That is, it would seem that for any prime p and integer r, there should be only a finite number of cohomology ring H^{∗}(G, k) (up to isomorphism) for all p-groups G of coclass r and any field k of characteristic p. Unfortunately, the proof given in the last section does not work for odd p. There are two reasons.

The first difficulty is that for odd primes p, the classification is slightly different. In the case that p is odd we must allow for the possibility that G/N is only isomorphic to a twisted version of S/A in Part (2) of Theorem 5.2. That is, suppose T is the translation subgroup of S. Let B be a subgroup of T that contains A. It is possible to form nonabelian central extensions

0 ⟶ B/A ⟶ H ⟶ T/B ⟶ 1

where the cocycle defining the extension is only altered from that defining T/A by addition of nontrivial commutators. Then, in the classification of p-groups for odd p, the collection G′_{r} include groups G with G/N ≅ W where W is an extension of the form

0 ⟶ B/A ⟶ W ⟶ S/B ⟶ 1

that is determined by cocycles as above. See Section 5 of [LG] for more explicit details. The point is that proof of Theorem 5.1 for p-groups with p odd would have to accommodate this difference in the classification. However, this part can be easily handled by methods that are similar to those in the proof of Theorem 3.5. The other difficulties may be more fundamental.

The other problem with proving an analog of Theorem 5.1 in odd characteristics is that there is no analog to Propositions 4.3 and 4.7. What is true for odd p is that every uniserial p-adic space group of coclass r is a subgroup of finite index in one of the group R_{n} which can be described as follows (see [LMP2]). Let T_{1} be the free ℤ_{p}-lattice of rank p − 1 on which C_{p} acts by the companion matrix to the polynomial x^{p−1} + ⋯ + x + 1. As a ℤ_{p}C_{p}-module it can be regarded as the kernel of the augmentation map ℤ_{p}C_{p} ⟶ ℤ_{p}. Let R_{1} be the split extension

0 ⟶ T_{1} ⟶ R_{1} ⟶ C_{p} ⟶ 1

with the given action of C_{p}. Then inductively, let R_{n+1} = R_{n} ≀ C_{p}. One approach to a proof of Theorem 5.1 in odd characteristics would be to verify an analog of Proposition 4.5 for the groups R_{1,j} = R_{1}/(p_{j}T_{1}). So we are left with the following question.

Question 6.1. Let p and a field k of characteristic p be given. Is it possible that all of the cohomology rings H^{∗}(R_{1,j}, k) are isomorphic?

There is a tiny amount of experimental evidence to support a conjecture in this direction. Informally, David Green, at the authors suggestion, computed the cohomology rings for p = 3 and j = 1, 2, using his own implementations of cohomology programs [Gr], and the rings appear to be isomorphic.

Because every p-group of coclass r must have a metabelian subgroup of bounded index, another approach might be to attempt to answer the following question.

Question 6.2. Suppose that we have an extension of the form

1 ⟶ H ⟶ G ⟶ C_{p} ⟶ 1

where H is a p-group. Is the cohomology ring of G (over a field of k characteristic p) determined up to a finite number of possibilities by the cohomology ring of H?

It turns out that this is the same question as asking if the spectral sequence of the extension must stop after a certain number of steps, where that number depends only on p and the cohomology ring H^{∗}(H, k).

As noted in the introduction, there is some experimental evidence to suggest that the cohomology rings of successive quotients of the 2-adic uniserial space groups have isomorphic cohomology rings more often than can be justified using the methods of the proof of Theorem 5.1. The 2-adic uniserial space groups of coclass 1, 2 and 3 were enumerated by Newman and O’Brien in [NO]. Eamonn O’Brien has also provided the author with the data base suitable for input into MAGMA for the 2-adic uniserial space groups of coclass 1, 2 and 3. The quotients R/γ_{i}(R), where R is a 2-adic unserial space group and γ_{i}(R) is the i^{th} term in its lower central series, can be obtained using the ANU p-quotient algorithm.

The cohomology rings H^{∗}(G, k) for G = G_{R,i} ≅ R/γ_{i}(R) and k = 𝔽_{p} were computed for several of the families of 2-groups and for some values of i using the authors cohomology ring programs. In a surprisingly large number of cases, it happens that H^{∗}(G_{R,i}, k) did not depend on the value of i as long as i was sufficiently large. In particular, to have any meaning, the order of G_{R,i} should be at least 2^{q+c} where 2^{q} is the order of the point group of R and c is the rank of the translation subgroup of R. The computer was unable to handle the calculations of the cohomology rings for groups of order larger than 256, and could compute the cohomology of the larger groups only if the degrees of the cohomology generators were not too large. Hence the body of evidence is not huge. But here is a sampling of what we can say.

For the family number 23 in the scheme of [NO] the groups of order 32, 64, 128, and 256 all have isomorphic mod-2 cohomology rings. In this case, the translation subgroup T has rank 2 and R/T has order 8. Because the rank is 2, it might be expected that the two quotients of orders 64 and 256 would have isomorphic cohomology rings, and likewise for the quotients of order 32 and 128. However, the cohomology rings are all isomorphic to each other. In this case, the cohomology rings have the following form. Let P = k[z, y, x, w, v, u, t, s] where the variables are in degrees 1, 1, 2, 2, 2, 3, 3, 4. Then H^{∗}(G, k) ≅ P/I where I is the ideal generated by the elements

zy, y^{2}, z^{3} + yx, zx + yx, yw, z^{2}v + zt,
x^{2}, yu, yt, z^{2}u + z^{2}t + xt, xu,
z^{2}w^{2} + zvt + t^{2}, xw^{2} + zvu + zvt + ut + t^{2},
zwu + zvu + zvt + z^{2}s + u^{2} + ut + t^{2}.

We should note that the computer did not obtain exactly the same relations for cohomology rings in the different cases. Instead, the isomorphisms were established by making changes of variables.

For the family number 21, it also appears that the cohomology rings for the groups of order 32, 64, 128, and 256 are isomorphic. In this case, the rings have 11 generators and 34 relations. The relations were sufficiently complicated that it would be very tedious to verify an actual isomorphism. However, the rings are very similar. In this case Rank(T) = 2 and |R/T | = 8. A similar thing happened for the groups of order 64. 128 and 256 for family number 24 (Rank(T) = 2 and |R/T | = 16). In other cases it was not difficult to be more definite. The groups of order 32, 64, 128 and 256 had isomorphic cohomology rings for the families with numbers 22 (Rank(T) = 2 and |R/T | = 8), 25 (Rank(T) = 1 and |R/T | = 8), 27 (Rank(T) = 1 and |R/T | = 8), 47 (Rank(T) = 1 and |R/T | = 8), and 48 (Rank(T) = 1 and |R/T | = 8). For family number 63 (Rank(T) = 2 and |R/T | = 8) the groups of order 64 and 128 have isomorphic cohomology rings.

In all of the cases above we looked only at the quotients G = R/γ_{i}(R) for R a 2adic uniserial space group. There is also the question of what happens to isomorphic cohomology rings under extension that have the same extension class in H^{2}(G, k). Do we obtain isomorphic cohomology rings?

We end with a remark about the Steenrod algebra. The cohomology ring H^{∗}(G, k) is an unstable module over the Steenrod algebra and we can ask the question of whether there are only finitely many algebras over the Steenrod algebra that can be the cohomology rings of 2-groups of fixed coclass r. The answer is yes and the proof follows directly from the Main Theorem 5.1. That is, the Cartan formula for the action of the Steenrod algebra implies that the action is determined entirely by the action on the generators of H^{∗}(G, k). Since the Steenrod algebra is defined over the prime field 𝔽_{2}, there are only finitely many choices that can be made for the image of any of the Steenrod operators on a generator for the cohomology ring. Of course, the Steenrod algebra is infinitely generated, but the operators in high enough degree must annihilate all of the generators for the cohomology ring. Hence there is only a finite number of choices to be made.

References

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[LP] C. Leedham-Green and W. PleskinSome remarks on Sylow groups of general linear groups, Math. Z. 191(1986), 529-535.

[NO] M. Newman and E. O’Brien, Classifying 2-groups by coclass, Trans. Amer. Math. Soc. 351(1999), 131/–161.

[S] A. Shalev, The structure of finite p-groups: constructive proof of the coclass conjectures.

[W] T. Weigel, p-central groups and Poincaré duality, Trans. Amer. Math. Soc. 352(2000), 4143-4154.

Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA

E-mail address: jfc@@sloth.math.uga.edu