Bounds in the Theory of Finite Covers

David M. Evans, School of Mathematics, University of East Anglia,

Norwich NR4 7TJ, England,

email: d.evans@uea.ac.uk

Osama A. Rashwan, Department of Mathematics, Faculty of Science, University of Assiut, Assiut, Egypt, email: orashwan@aun.eun.eg

Abstract

We give an upper bound for the number of conjugacy classes of closed subgroups of the full wreath product F Wr_{W} Sym(Ω) which project onto Sym(Ω). Here, Ω is infinite, W is the set of n-tuples of distinct elements from Ω (for some finite n), F is a finite nilpotent group and the topology on the wreath product is that of pointwise convergence in its imprimitve permutation action. The result addresses a problem which arises in a natural model-theoretic context about classifying certain types of finite covers. 2000 Mathematics Subject Classification: 20B27, 20E22, 20C32, 03C35.

1 Introduction

This paper is a sequel to [7] and [9], but can be read independently of these. It is motivated by questions arising in model theory concerning finite covers, and can be seen as providing new information about disintegrated totally categorical structures. However, the problems and results can be phrased naturally in terms of permutation groups, and as the methods we adopt are purely group-theoretic, we avoid model-theoretic terminology in most of the paper. A brief commentary on the model theory is given in Section 6.2.

Given an infinite set Ω, a natural number n, and a finite group F acting faithfully on a finite set Δ, consider the symmetric group G = Sym(Ω) acting on W = Ω^{(n)} and form the full wreath product Γ_{0} = F Wr_{W} G acting on C_{0} = Δ × W. There is a natural topology on the symmetric group on any (infinite) set making it into a topological group in which pointwise stabilisers of finite sets give a base of open neighbourhoods of the identity. With this topology, Γ_{0} is a closed subgroup of Sym(C_{0}) and the natural homomorphism ρ : Γ_{0} → G is continuous. In particular, its kernel, K_{0}, the base group of the wreath product, is closed (and is the product F^{W}, so is compact and in fact, profinite).

Let C be the set of closed subgroups Γ ≤ Γ_{0} with ρ(Γ) = G. The main purpose of this paper is to give an explicit upper bound on the number of Γ_{0}-conjugacy classes of subgroups in C in terms of n and |F |, at least in the case where F is nilpotent. For general F, it is known that any Γ ∈ C contains some Σ ∈ C with Σ ∩ K_{0} nilpotent ([7], 5.2 and 5.3), so there may be some possibility to extend our results to the general case.

To get our bounds we first use representation theory of the infinite symmetric group (cf. [9]) to quantify the cohomology calculations in [7] for the case where F is abelian. We then give an inductive argument and a little more cohomological machinery to extend this to the case where F is nilpotent. A key extra ingredient here is a result of J. Krajíček ([14]) on the weight of generating sets for submodules of tabloid modules for the infinite degree symmetric group.

The proof is by induction on the composition length of F, so the base case is where F is 𝔽_{p}, the cyclic group of prime order p, regarded as the additive group of the field with p elements. We regard K_{0} = 𝔽_{p}^{W} as a topological 𝔽_{p}G-module, and if Γ ∈ C then K_{Γ} = Γ ∩ K_{0} is a closed submodule. We have two steps to describe the possible (conjugacy classes of) Γ. First, bound the number of closed submodules K of K_{0}. Then for each such K bound the number of Γ_{0}-conjugacy classes of Γ ∈ C with K_{Γ} = K. The latter number is given by the first (continuous) cohomology group H_{c}^{1}(G, K_{0}/K), and we use dimension-shifting together with the submodule bounds to bound this. This is done in Section 4.

To bound the number of closed submodules of K_{0} we consider instead the dual problem: bound the number of 𝔽_{p}G-submodules of the permutation module 𝔽_{p}Ω^{(n)}. To do this, we use the extension of the representation theory of the finite symmetric groups initiated in [9]. To make inductive arguments work, we need to work with permutation modules 𝔽_{p}Ω^{λ} for more general partitions λ. This is done in Section 3.

The passage from the case F = 𝔽_{p} to arbitrary nilpotent F is done in the proof of Theorem 5.1.

Acknowledgements: Most of the material in this paper is from the Ph.D. thesis of the second Author at UEA. He would like to thank the Egyptian government for support during his period at UEA. Both Authors would like to thank Jan Krajíček for detailed discussions about his paper [14].

2 General Results

In this section, we give some general results about modules with finitely many submodules. Throughout, R denotes a ring (with identity 1), G a group, 𝔽 a field and A an 𝔽-algebra. For R-modules M and N, Hom_{R}(M, N) is the group of R-module homomorphisms from M to N and End_{R}(M) is Hom_{R}(M, M).

Lemma 2.1 (i) If M is an R-module which has a finite composition series of length m, then any submodule and any factor module of M has a composition series of length at most m. (ii) If m_{1} and m_{2} are the (finite) lengths of every composition series of N and M/N respectively, then M has a finite composition series of length m_{1} + m_{2}. (iii) Let M_{1}, M_{2}, …, M_{s} be R-modules. If M_{i} has a finite composition series of length m_{i} (for i = 1, 2, …, s), then M_{1} ⊕ M_{2} ⊕ ⋯ ⊕ M_{s} has a composition series of length ∑_{i=1}^{s} m_{i}. (iv) If ϕ_{i} : M ⟶ M_{i}, (i = 1, 2, …, s) are R-homomorphisms, then M/ ⋂_{i=1}^{s} ker ϕ_{i} is isomorphic to a submodule of M_{1} ⊕ M_{2} ⊕ ⋯ ⊕ M_{s}. □

Definition 2.2 Say that an A-module M is good if it has a finite composition series and End_{A}(S) = {α id_{S} : α ∈ 𝔽} for all composition factors S of M, (where id_{S} denotes the identity map on S).

Lemma 2.3 Suppose M is a completely reducible good A-module of composition length c. Then dim𝔽(Hom_{𝔽G}(A, A)) ≤ c^{2}. □

Lemma 2.4 Let 𝔽 be a finite field and M a completely reducible good A-module with composition length c. Then | 𝔽 |^{c^{2}} is an upper bound for the number of submodules of M.

Proof: Let M = S_{1} ⊕ S_{2} ⊕ ⋯ ⊕ S_{l}, written according to isomorphism of irreducible submodules, that is S_{i} ≅ I ⊕ I ⊕ ⋯ ⊕ I for some irreducible submodule I. Then the number of submodules of M is Π_{i=1}^{l}n_{i}, where n_{i} is the number of submodules of S_{i}, by ([5], Lemma 15.2) and the fact that the S_{i} have no composition factors in common. So it is enough to prove that the number of submodules of I^{t} is at most |𝔽|^{t^{2}}. This is done by induction on t, and the base case is of course trivial. Suppose it holds for t and let S = I^{t+1}.

Let H be a submodule of S and let T be the projection of H onto the first t coordinates. Suppose first that the kernel of this projection is {0}. Then there is a unique module homomorphism ψ : T → I with H = {(x̄, ψ(x̄)) : x̄ ∈ T }. Now, T is isomorphic to I^{r} for some r ≤ t so the number of possibilities for such homomorphisms is at most |𝔽|^{t}, for fixed T. By inductive hypothesis, the number of possibilities for T is at most |𝔽|^{t^{2}}, thus the number of such H is at most |𝔽|^{t^{2}+t}.

If the kernel of the projection of H onto the first t factors is not zero, then H contains the last direct summand of S. By inductive hypothesis, the number of possibilities for such H is at most |𝔽|^{t^{2}}. Thus, the number of submodules of S is at most |𝔽|^{t^{2}+t} + |𝔽|^{t^{2}} ≤ |𝔽|^{(t+1)^{2}}. □

Lemma 2.5 Let 𝔽 be a finite field and let M be a good A-module with composition length c. Then | 𝔽 |^{c^{3}} is an upper bound for the number of submodules of M.

Proof: The proof is by induction on c. If c = 1, then M is irreducible and the result holds. Let T = J(M) be the intersection of all maximal submodules of M. Clearly T is a proper submodule and M/T is completely reducible. By the previous lemma, we may assume that T ≠ {0}.

Let X ≤ M, X_{1} = X ∩T, X_{2} = X +T and ϕ_{X} ∈ Hom_{A}(X_{2}/T, M/X_{1}) be given by ϕ_{X}(x+T) = x + X_{1}. Then X is the union of the cosets in the image of ϕ_{X}, so is determined by X_{1}, X_{2} and ϕ_{X}. By induction, the number of possibilities for X_{1} is at most |𝔽|^{(c−1)^{3}}. As X_{2}/T is completely reducible (of composition length at most c − 1) the image of ϕ_{X} lies in the socle of M/X_{1}, and this has composition length at most c − 1. Thus the number of possibilities for ϕ_{X} is at most |𝔽|^{(c−1)^{2}}. By the previous lemma the number of possibilities for X_{2} is at most |𝔽|^{(c−1)^{2}}. The inductive step follows readily. □

We will use the following to prove that our permutation modules M^{λ} are good. It is of course just a version of Schur’s Lemma.

Lemma 2.6 Let M be an absolutely irreducible A−module and φ ∈ End_{A}(M) have a non-zero invariant finite dimensional subspace M_{0} (i.e φ(M_{0}) ⊆ M_{0}). Then φ = αid_{M} for some α ∈ 𝔽, where id_{M} : M ⟶ M is the identity map.

Proof: Let 𝔼 be an algebraically closed field containing 𝔽. Then φ extends to an endomorphism φ̃ of M ⊗_{𝔽} 𝔼 and M_{0} ⊗_{𝔽} 𝔼 is a finite dimensional, non-zero, φ̃-invariant 𝔼-subspace. As 𝔼 is algebraically closed, φ̃ restricted to M_{0} ⊗_{𝔽} 𝔼 has an eigenvalue α in 𝔼. Thus, by irreducibility of M̃ = M ⊗_{𝔽} 𝔼, we have ker(φ̃ − αidM̃) = M̃. It follows easily that α ∈ 𝔽 and φ = αid_{M}. □

3 Modules for the symmetric group

3.1 Tabloid modules

In this section Ω is an infinite set and G = Sym(Ω). Following the notation of ([9], Section 1.2) we work with partitions λ = (∞ − n, λ_{2}, …, λ_{r}) where n ∈ ℕ, and (λ_{2}, …, λ_{r}) is a partition of n. We denote by Ω^{λ} the set of λ-tabloids (as in the finite case these are ordered partitions of Ω into sets of size ∞ − n, λ_{2}, …, λ_{r}), and M^{λ} the 𝔽G-permutation module with Ω^{λ} as 𝔽-basis. We will occasionally embelish various notations with a subscript to indicate the field over which we are working.

Note that there are natural bijections between the set Ω^{(n)} of n-tuples of distinct elements from Ω and Ω^{(∞−n,1,...,1)}, and between [Ω]^{n}, the set of subsets of size n from Ω, and Ω^{(∞−n,n)}. Moreover these bijections commute with the G-actions on these sets, and we shall use these bijections to make identifications between these sets, without further comment.

If λ_{2} ≥ … ≥ λ_{r} > 0 we say that λ is a proper partition. For any λ there is a unique proper partition λ̂ obtained by rearranging the λ_{i} and there is a canonical isomorphism M^{λ} → M λ̂.

Next we give an analogue to ([11], Definition 3.2) for infinite partitions.

Definition 3.1 Let λ = (∞ − n, λ_{2}, …, λ_{r}) and μ = (∞ − m, μ_{2}, …, μ_{s}) be proper partitions of an infinite set Ω, where ∑_{i=2}^{r} λ_{i} = n and ∑_{i=2}^{s} μ_{i} = m. If ∑_{i>k}^{s} μ_{i} ≤ ∑_{i>k}^{r} λ_{i} for all k ≥ 1, we say that μ dominates λ, and write μ ≥ λ.

We next recall certain key homomorphisms with domain M^{λ}. If i < r and 1 ≤ δ ≤ λ_{i+1} define μ = (μ_{1}, μ_{2}, …, μ_{r}) where μ_{i} = λ_{i} + δ, μ_{i+1} = λ_{i+1} − δ and all other μ_{j} equal the corresponding λ_{j}. The following is elementary, but crucial to all inductions.

Lemma 3.2 With the above notation μ̂ ≥ λ. □

Continuing with this notation, we define an 𝔽G-homomorphism ψ̂_{i,δ} : M^{λ} → M^{μ} by first defining its effect on the tabloid basis of M^{λ} and extending it linearly. If {t} ∈ Ω^{λ} then ψ̂_{i,δ}({t}) is the sum of all μ-tabloids which can be obtained by tranferring δ elements from the i + 1-th row of t to the i-th row. If we follow this by the canonical isomorphism we obtain ψ_{i,δ} : M^{λ} → M μ̂.

Part of the importance of these homomorphisms is that they characterise the Specht submodule S^{λ} of M^{λ}. Recall that this is defined as the submodule of M^{λ} generated by the λ-polytabloids (see [11, 12] for definitions, and [9] for the straightforward extension to the infinite case). The following is an immediate consequence of ([9], Theorem 2.4):

Lemma 3.3 Let λ = (∞ − n, λ_{2}, …, λ_{r}) be a proper partition of an infinite set Ω. Then

S^{λ} = ⋂_{i=1}^{r−1} ⋂_{δ=1}^{λ_{i+1}} ker ψ_{i,δ}.□

We denote by (., .) the G-invariant bilinear form on M^{λ} which has Ω^{λ} as an orthonormal basis, and let ⊥ denote orthogonality with respect to this form. Define D^{λ} = S^{λ}/S^{λ} ∩ (S^{λ})^{⊥}. Recall that James’ Submodule Theorem ([12]) holds for our infinite partitions:

Theorem 3.4 ([9], Theorem 2.1.) If U is a submodule of M^{λ} then either U ≥ S^{λ} or U ≤ (S^{λ})^{⊥}. In particular, D^{λ} is either {0} or irreducible. □

Next we give the characterisation of (S^{λ})^{⊥} analogous to the characterisation of S^{λ} in Lemma 3.3. Again, this is a straightforward extension of the finite case ([13]). For 1 < l < r and 0 ≤ δ ≤ λ_{l} define ν = (ν_{1}, …, ν_{r}) where ν_{l} = λ_{l} + δ, ν_{l+1} = λ_{l+1} − δ, and ν_{j} = λ_{j} for all other j. As before:

Lemma 3.5 With the above notation ν̂ ≥ λ. □

Continuing with this notation, we define an 𝔽G-homomorphism ϕ̂_{i,δ} : M^{ν} → M^{λ} by first defining its effect on the tabloid basis of M^{ν} and extending it linearly. If {t} ∈ Ω^{ν} then ϕ̂_{i,δ}({t}) is the sum of all λ-tabloids which can be obtained by tranferring δ elements from the i-th row of t to the i + 1-th row. If we precede this by the inverse of the canonical isomorphism we obtain ϕ_{i,δ} : M ν̂ → M^{λ}. The following then follows immediately from ([9], Theorem 2.7):

Lemma 3.6 Let λ = (∞ − n, λ_{2}, …, λ_{r}) be a proper partition of a set Ω with r > 2. Then (S^{λ})^{⊥} = ∑_{i=2}^{r−1} ∑_{δ=1}^{λ_{i+1}} Im ϕ_{i,δ}. If r = 2, then (S^{λ})^{⊥} = {0}. □

The following is the basis for our induction:

Definition 3.7 Let λ = (∞ − n, λ_{2}, …, λ_{r}) be a proper partition of a set Ω. Define L(λ) to be the maximal length of a chain of proper partitions μ^{1} > μ^{2} > … > μ^{L} > λ.

Henceforth, we shall work with proper partitions unless we explicitly state otherwise.

Lemma 3.8 If n ∈ ℕ and λ = (∞ − n, λ′) where λ′ is a partition of n, then n ≤ L(λ) ≤ n^{2}. □

Theorem 3.9 Let λ = (∞ − n, λ_{2}, …, λ_{r}) be a partition of an infinite set Ω. Then:

(a) any composition factor of M^{λ} is of the form D^{μ} for some μ ≥ λ;

(b) M^{λ} has a finite composition series of length at most 2^{L(λ)[L(λ)+4]/2}.

(c) The composition length of M^{λ} is at most 2^{n^{4}}.

Proof: There is a total order (of order type ω^{∗}) on the partitions, which contains the partial ordering ≤. We prove parts (a) and (b) of the result by downwards induction on this ordering. Part (c) then follows from the previous lemma. The base step is trivial, so we assume as inductive hypothesis that the result is true for partitions μ > λ, and deduce it for λ.

By Lemma 3.2 and Lemma 3.3, there are proper partitions μ^{1}, μ^{2}, …, μ^{s}, where s ≤ n and 𝔽Sym(Ω)-homomorphisms ψ_{i,δ} : M^{λ} ⟶ M^{μ^{i}} such that S^{λ} = ⋂_{i=1}^{s} ker ψ_{i} and μ^{i} > λ. From the definition of L(λ), we get L(μ^{i}) ≤ L(λ) − 1. Therefore by inductive hypothesis M^{μ^{i}} has a finite composition series of length m_{1} ≤ 2^{[L(λ)−1][L(λ)+3]/2}, and all the composition factors are of the form D^{ν} for various ν > λ. So M^{λ}/S^{λ}, which is isomorphic to a sub-module of M^{μ^{1}} ⊕ M^{μ^{2}} ⊕ ⋯ ⊕ M^{μ^{s}} by Lemma 2.1(iv), has composition series of length m_{2} ≤ s2^{[L(λ)−1][L(λ)+3]/2} ≤ n2^{[L(λ)−1][L(λ)+3]/2} with the composition factors satisfying (a). Also we have t ≤ n and ν^{1}, …, ν^{t} with ν^{i} > λ and (S^{λ})^{⊥} is isomorphic to a submodule of a homomorphic image of M^{ν^{1}} ⊕ M^{ν^{2}} ⊕ ⋯ ⊕ M^{ν^{t}}, by Lemma 3.6. So (S^{λ})^{⊥} has a finite composition series of length ≤ n2^{[L(λ)−1][L(λ)+3]/2}, with factors satisfying (a). Then S^{λ} ∩ (S^{λ})^{⊥} has a finite composition series of length ≤ n 2^{[L(λ)−1][L(λ)+3]/2} by Lemma 2.1(i). Since D^{λ} = S^{λ}/(S^{λ} ∩ (S^{λ})^{⊥}) is irreducible (or zero) by Theorem 3.4, it follows that S^{λ} has a composition series of length m_{3} ≤ n2^{[L(λ)−1][L(λ)+3]/2} + 1. So M^{λ} has a finite composition series of length m_{2} + m_{3} ≤ 2n2^{[L(λ)−1][L(λ)+3]/2} + 1. Thus, (using Lemma 3.8) m_{2} + m_{3} ≤ 2L(λ)2^{[L(λ)−1][L(λ)+3]/2} + 1 ≤ 2^{L(λ)[L(λ)+4]/2}. Moreover all composition factors are of the form D^{ν} for various ν ≥ λ.□

Corollary 3.10 Let m, n ∈ ℕ with m ≥ n and λ = (∞ − n, λ_{2}, …, λ_{r}) be an infinite partition. Let w = (w_{1}, …, w_{m}) ∈ Ω^{(m)}, and G_{w} the stabiliser of w in Sym(Ω). Then the composition length of 𝔽Ω^{λ} as an 𝔽G_{w}-module is at most (mn)^{n}2^{n^{4}}.

Proof: Let Ω′ = Ω \ {w_{1}, …, w_{m}}. The G_{w}-orbit containing a particular λ-tabloid is determined by the intersections of its rows with {w_{1}, …, w_{m}}. So a crude upper bound on the number of G_{w}-orbits on Ω^{λ} is:

( \begin{matrix}m \\ n \end{matrix} ) r^{n} ≤ m^{n}r^{n} ≤ (mn)^{n}.

The action of G_{w} on such an orbit is equivalent to the action of Sym(Ω′) on (Ω′)^{μ} for some μ ≥ λ and as an 𝔽G_{w}-module, 𝔽Ω^{λ} decomposes as a direct sum of such modules 𝔽Ω^{μ}, with one summand for each orbit. The result follows, by Theorem 3.9. □

Theorem 3.11 Each M^{λ} is a good 𝔽G-module.

Proof: By Theorem 3.9 it is enough to prove that any 𝔽G-endomorphism of D^{λ} is given by multiplication by some scalar in 𝔽. We do this using Lemma 2.6, so the proof divides into two steps.

(i) First we show that D𝔽^{λ} is absolutely irreducible. By James’ Submodule Theorem, D𝔼^{λ} is irreducible (or zero) for any field 𝔼. So it is enough to show that if 𝔼 is an extension field of 𝔽 then D𝔽^{λ} ⊗ 𝔼 is isomorphic to D𝔼^{λ}. Note that by exactness of − ⊗ 𝔼 we have:

Claim 1. For any 𝔽-vector spaces A ⊇ B with a group G acting on them, there is an 𝔼G−isomorphism (A/B) ⊗𝔽 𝔼 ≅_{𝔼G} (A ⊗𝔽 𝔼)/(B ⊗𝔽 𝔼).

Identify M𝔽^{λ} with a subset of M𝔼^{λ} (via x ↦ x⊗1𝔼). We now show that that ⟨S𝔽^{λ} ∩(S𝔽^{λ})^{⊥𝔽}⟩𝔼 = S𝔼^{λ} ∩ (S𝔼^{λ})^{⊥}. Note firstly that ⟨S𝔽^{λ}⟩𝔼 = S𝔼^{λ} and ⟨S𝔽^{λ} ∩ (S𝔽^{λ})^{⊥}⟩𝔼 ⊆ S𝔼^{λ} ∩ (S𝔼^{λ})^{⊥}. For the converse, suppose that V𝔽 is an arbitrary vector space over 𝔽. Let {x_{i} : i ∈ I} be an 𝔽−basis of V𝔽. Then {x_{i} ⊗ 1𝔼 : i ∈ I} is an 𝔼−basis for V𝔼 = V𝔽 ⊗ 𝔼. If V𝔽 has a bilinear form ( , )𝔽, then V𝔼 has a bilinear form ( , )𝔼 inherited from that of V𝔽, given by (x_{i} ⊗ 1𝔼, x_{j} ⊗ 1𝔼)𝔼 = (x_{i}, x_{j})𝔽.

Claim 2. Let U = {u ∈ V𝔽 : (u, v)𝔽 = 0 ∀v ∈ V𝔽}. Then (V𝔼)^{⊥𝔼} = ⟨U⟩𝔼.

Proof of claim 2: Without loss of generality let {x_{i} : i ∈ I_{0} ⊆ I} be an 𝔽−basis for U, and assume that w = ∑_{i∈I_{0}} α_{i}(x_{i} ⊗ 1𝔼) + ∑_{j∈I_{1}} β_{j}(x_{j} ⊗ 1𝔼) ∈ (V𝔼)^{⊥𝔼}, where I_{1} = I \ I_{0} and α_{i}, β_{j} ∈ 𝔼. To prove our claim we may assume each α_{i} = 0 and show that β_{j} = 0 for all j ∈ I_{1}. Suppose not and β_{i_{1}}, β_{i_{2}}, ⋯, β_{i_{r}} ≠ 0. As (w, x_{j} ⊗ 1𝔼)𝔼 = 0 for all j ∈ I, we have a collection of linear equations

β_{i_{1}}(x_{i_{1}}, x_{j})𝔽 + β_{i_{2}}(x_{i_{2}}, x_{j})𝔽 + ⋯ + β_{i_{r}}(x_{i_{r}}, x_{j})𝔽 = 0, j ∈ I

in β_{i_{1}}, ⋯, β_{i_{r}} with coefficient in 𝔽 which has a non-trivial solution in 𝔼^{r}. So there is a non-zero

solution βi_{1}^{′}, ⋯, βi_{r}^{′} in 𝔽^{r}. Thus w = ∑_{j∈J} βj^{′}(x_{j} ⊗ 1𝔼) ∈ (V𝔼)^{⊥}, where J = {i_{1}, i_{2}, ⋯, i_{r}} and

so ∑_{j∈J} βj^{′}x_{j} ∈ (V𝔽)^{⊥} as βj^{′} ∈ 𝔽 which is a contradiction to the choice of the basis of V𝔽. Hence β_{j} = 0 for all j ∈ I_{1} and therefore (V𝔼)^{⊥} = ⟨U⟩𝔼. This is the end the proof of claim 2.

From claim 2 by replacing V𝔽 by S𝔽^{λ} and U = S𝔽^{λ} ∩(S𝔽^{λ})^{⊥}, we get S𝔼^{λ} ∩(S𝔼^{λ})^{⊥} = ⟨S𝔽^{λ} ∩(S𝔽^{λ})^{⊥}⟩𝔼. Now by using claim 1, we get D𝔽^{λ}⊗𝔼 ≅_{𝔼G} S𝔼^{λ}/⟨S𝔽^{λ}∩(S𝔽^{λ})^{⊥}⟩𝔼 = S𝔼^{λ}/S𝔼^{λ}∩(S𝔼^{λ})^{⊥} = D𝔼^{λ}, as required.

(ii) Next, suppose D^{λ} ≠ {0} and let ϕ ∈ End_{𝔽G}(D𝔽^{λ}).

Claim 3. There is a ϕ−invariant non-zero, finite dimensional subspace of D^{λ}.

Proof of claim 3: Let X be a finite subset of Ω. If this is large enough, there is some non-zero x ∈ D^{λ} which is fixed by G_{(X)}, the pointwise stabiliser of X in G. By Corollary 3.10, the fixed-point set of G_{(X)} on D^{λ} is finite dimensional. Moreover, it is clearly ϕ-invariant.

The result now follows by Lemma 2.6.□

Suppose now that 𝔽 is a finite field, n ∈ ℕ and λ = (∞−n, λ′). As a corollary to Theorem 3.11, Theorem 3.9 and Lemma 2.5 we have the following:

Corollary 3.12 (i) The number of submodules of M^{λ} is bounded above by | 𝔽 |^{α}, where α = 2^{3n^{4}}. (ii) Let λ^{1}, λ^{2}, ⋯, λ^{r} be partitions of Ω with λ^{i} = (∞ − n_{i}, ν^{i}). Let m = ∑_{i} n_{i}. The number of submodules of M^{λ^{1}} ⊕ M^{λ^{2}} ⊕ ⋯ ⊕ M^{λ^{r}} is at most | 𝔽 |^{β}, where β = 2^{3m^{4}}. □

The final result we need from the representation theory of the symmetric group is due to J. Krajíček ([14]), following earlier work of M. Ajtai ([2]). To state it, we need some more terminology.

Definition 3.13 Suppose μ is a partition of the infinite set Ω. Let x = ∑_{T ∈Ω^{μ}} a_{T} T ∈ M^{μ}. We shall say that a tabloid T is in the support of x if a_{T} ≠ 0. Suppose any element in the support of x has the elements of all its rows apart from the first contained in the finite set A. Abusing terminology, we say that the support of x is contained in A, and define the minimum size of such an A to be the weight of x.

The following is a part of ([14], Theorem 3.4) but expressed in our notation:

Theorem 3.14 For any submodule U of M^{λ} where λ = (∞ − n, 1^{n}), there is a generating set H of U such that any vector in H has weight at most

n^{2^{n+1}}.

Proof: In the proof of ([14], Theorem 3.4), Krajíček gives the bound

2^{(n+1)^{2}}n^{2^{n}+2}

and for n ≥ 4 this is dominated by the above expression (which we have chosen for its slightly simpler form). For n < 4 Krajíček’s proof can be inspected to show that the bound we give is rather extravagant. □

Corollary 3.15 Suppose λ = (∞ − n, λ′), where λ′ is a partition of n, and U ≤ M^{λ}. Then there exists a homomorphism ϕ : (𝔽_{p}Ω^{(m)})^{r} → M^{λ} with image U, where m = n^{2^{n+1}} and r = 2^{n^{4}}.

Proof: Let μ = (∞ − n, 1^{n}). There is a natural surjection M^{μ} → M^{λ} (which does not increase the weight of a vector). Thus we may assume that λ = μ. By Theorem 3.14 there are generators x_{1}, …, x_{s} of U each of weight at most m. By Theorem 3.9 we may take s ≤ r here. The result follows: ϕ takes some generator of the i-th summand whose support contains the support of x_{i} to x_{i}. □

3.2 Duality

We now return to the problem originally stated in the Introduction, where F is abelian. In this case, we are concerned with closed G-invariant subgroups of F^{W}, where W = Ω^{(n)} = Ω^{(∞−n,1^{n})}. (The group action is given by ^{g}f(w) = f(g^{−1}w) for f ∈ F^{W}, w ∈ W and g ∈ G.) As this is a compact (continuous) G-module, Pontriagin duality gives an inclusion-reversing correspondence between the closed G-submodules of F^{W} and the submodules of its dual (F^{W})^{∗}. Moreover the duality sends exact sequences of compact G-modules to exact sequences of (discrete) G-modules, so the topological G-composition factors of F^{W} are the duals of the G-composition factors of its dual.

Any finite abelian group is self-dual, so in fact the dual of F^{W} is the permutation module F W (with coefficients in F), which in our case is one of the tabloid modules. So if F = 𝔽_{p} we have dual versions of the various results for the M^{λ}: the same bounds on the topological G-composition length and number of closed submodules hold, and all topological composition factors are of the form (D^{μ})^{∗} for various μ ≥ λ.

The reader is referred to ([6], Section 6.3) for more details. See also the remarks before Lemma 6.4.

4 Cohomology Groups

We refer the reader to ([6], Sections 6 and 7) for more background to the material in this section.

If Ω is any set then there is a natural topology on Sym(Ω) which makes it into a topological group. The open sets are unions of cosets of pointwise stabilizers of finite subsets of Ω. We then make any permutation group G on Ω into a topological group by giving it the relative topology.

Definition 4.1 (cf. [6], Definition 6.4.1.) If G is a group and M is a G-module, then a derivation from G to M is a map d : G ⟶ M which satisfies d(gh) = d(g) + gd(h) for all g, h ∈ G. An inner derivation is a derivation of the form d_{a}( for a ∈ M) where d_{a}(g) = ga − a for all g ∈ G. The set of all such derivations forms an abelian group Der(G, M) (with the pointwise addition of functions), and the inner derivations form a subgroup Inn(G, M). The quotient group is denoted by H^{1}(G, M), and is referred to as the first cohomology group of G on M. If G is a topological group and M a topological G-module (that is, M is a topological abelian group and the action G × M → M is continuous) then the continuous derivations form a subgroup of the group of all derivations, and this clearly contains all the inner derivations. We denote the quotient group of continuous derivations modulo the inner derivations by H_{c}^{1}(G, M).

Definition 4.2 If G is a group and M is a G-module then we define the zero-th cohomology group H^{0}(G, M) to be the submodule of G−fixed elements of M. This is also denoted by M^{G}.

The long exact sequence holds for H^{0} and H_{c}^{1} ([6], Lemma 7.1.2), as well as a version of Shapiro’s Lemma ([6], Lemma 7.1.4). The following is also well-known for discrete cohomology groups. We omit the proof that it holds in our context.

Lemma 4.3 If M is a G−module and H is a normal subgroup of G, then there is an action of G/H on H^{1}(H, M) and an exact sequence

0 ⟶ H^{1}(G/H, M^{H}) ⟶ H^{1}(G, M) ⟶ (H^{1}(H, M))^{G/H}.

If, moreover, G is a topological group and M is a topological G-module, then there is an exact sequence as above in which the H^{1} terms are replaced by H_{c}^{1}. □

The long exact sequence gives:

Lemma 4.4 Let 𝔽 be a field, G be a topological group and M a compact topological 𝔽G-module.

(i) For any closed submodule N of M

| (H_{c}^{1}(G, M)) |≤| (H_{c}^{1}(G, M/N)) || (H_{c}^{1}(G, N)) | .

(ii) Let N be a closed submodule of M and suppose that H_{c}^{1}(G, M/N) and H_{c}^{1}(G, N) are trivial. Then H_{c}^{1}(G, M) is trivial. (iii) Suppose M has a finite topological G-composition series with S the multiset of composition factors. Then

dim𝔽(H_{c}^{1}(G, M)) ≤ ∑_{S∈S} dim𝔽(H_{c}^{1}(G, S)).□

Lemma 4.5 Suppose S is a Hausdorff topological group and N a compact normal subgroup. Let H = {H ≤ S : H is closed , H ∩ N = 1, NH = S}, the set of closed complements to N in S.

(i) If H ∈ H, the map μ : N × H → S given by μ(n, h) = nh is a homeomorphism. Suppose also that N is abelian. (ii) If H ∈ H there is a bijection θ : Der_{c}(H, N) → H such that any δ_{1}, δ_{2} ∈ Der_{c}(H, N) are equal modulo an inner derivation if and only if θ(δ_{1}) and θ(δ_{2}) are conjugate in S. (iii) If H ∈ H the number of S-conjugacy classes of elements of H is |H_{c}^{1}(H, N)|.

Proof: (i) Clearly μ is a continuous bijection. To show that it is a homeomorphism, it is enough to prove that it is a closed map. Suppose A ⊆ N and B ⊆ H are closed. Then A is a compact subset of S and B is a closed subset of S, so AB is a closed subset of S, by ([10], (4.4)). Thus μ(A × B) is closed, and the result follows. (ii) This is well-known, except that we require the correspondence between continuous derivations and closed complements to N. First, given δ ∈ Der_{c}(H, N) let θ(δ) = H_{1} = {δ(h)h : h ∈ H}. This is a subgroup of S. As δ is continuous, its graph gives a closed subset of N × H. Now H_{1} is the image of this subset under μ, so is closed, by (i). Conversely, if H_{1} ∈ H, then by (i) we have a homeomorphism μ_{1} : N ×H_{1} → S. The corresponding derivation δ is given by δ(h) = −π_{1}(μ_{1}^{−1}(h)), where π_{1} is projection to N. This is continuous. (iii) This follows immediately from (ii). □

We now compute some bounds associated with these cohomology groups for sections of the continous G-modules F^{Ω^{λ}}, when F = 𝔽_{p}.

Theorem 4.6 Let λ = (∞ − n, λ_{2}, …, λ_{r}) be a partition of an infinite set Ω and W = Ω^{λ} be the set of all λ-tabloids. Let G = Sym(Ω). If K is a closed submodule of K_{0} = 𝔽_{p}^{W}, then the dimension of H_{c}^{1}(G, K_{0}/K) over 𝔽 is bounded above by 2^{18n^{4}}.

Proof: The composition factors of K_{0}/K are of the form (D^{μ})^{∗} for various μ ≥ λ, and we have a bound on their number. So (by Lemma 4.4) it is enough to obtain a bound on the 𝔽-dimension of H_{c}^{1}(G, (D^{μ})^{∗}).

By the long exact sequence we have:

dim(H_{c}^{1}(G, (D^{μ})^{∗})) ≤ dim(H^{0}(G, (S^{μ})^{∗}/(D^{μ})^{∗})) + dim(H_{c}^{1}(G, (S^{μ})^{∗})).

The first of these is bounded by the composition length of 𝔽_{p}^{Ω^{μ}}, and this is at most 2^{n^{4}}. For the second, suppose μ = (∞ − m, μ′), and let ν = (∞ − 2m, 1^{2m}). As the Specht module is generated by any polytabloid, there is a surjective homomorphism φ : M^{ν} → S^{μ}. Dualizing, we have an embedding φ^{∗} : (S^{μ})^{∗} → (M^{ν})^{∗}. So by the long exact sequence again,

dim(H_{c}^{1}(G, (S^{μ})^{∗}) ≤ dim(H^{0}(G, (M^{ν})^{∗}/imφ^{∗})) + dim(H_{c}^{1}(G, (M^{ν})^{∗})).

By Shapiro’s lemma

H_{c}^{1}(G, (M^{ν})^{∗}) = H_{c}^{1}(G, 𝔽^{Ω^{(2m)}}) ≅ H_{c}^{1}(G_{w}, 𝔽_{p}) = Hom(G_{w}, 𝔽_{p}) = {0}

for any w ∈ Ω^{(2m)}, as G_{w} has no proper subgroup of finite index. Also dim(H^{0}(G, (M^{ν})^{∗}/imφ^{∗})) is bounded by the composition length of M^{ν}, so by Theorem 3.9:

dim(H_{c}^{1}(G, (S^{μ})^{∗})) ≤ 2^{16m^{4}} ≤ 2^{16n^{4}}.

Therefore

dim(H_{c}^{1}(G, (D^{μ})^{∗})) ≤ 2^{n^{4}} + 2^{16n^{4}}

and so

dim(H_{c}^{1}(G, K_{0}/K)) ≤ 2^{n^{4}}(2^{n^{4}} + 2^{16n^{4}}) ≤ 2^{18n^{4}}. □

5 The Main Result

We return to the problem stated in the Introduction. Thus, Ω is an infinite set, n ∈ ℕ, and λ = (∞ − n, λ′) an infinite partition, where λ′ is a partition of n. We consider G = Sym(Ω) acting on Ω^{λ}, and some group F of permutations on a finite set Δ. We form the unrestricted wreath product Γ_{0} = F Wr_{W} G in its imprimitive action on Δ × W. The projection ρ : Γ_{0} → G is continuous with kernel K_{0} = F^{W}, and we consider C = {Γ ≤ Γ_{0} : Γ closed , ρ(Γ) = G}. Clearly Γ_{0} acts by conjugation on C. The main result is:

Theorem 5.1 If F is nilpotent, then the number of Γ_{0}-conjugacy classes of subgroups in C is at most |F | to the power 2^{k^{4}b_{n}}, where k is the composition length of F and b_{n} = 6n^{2^{(n+2)^{4}}} .

Lemma 5.2 (i) The topology on Γ_{0} is independent of the permutation action of F. (ii) If E is a normal subgroup of F there is a natural, continuous, open epimorphism α : Γ_{0} → (F/E) Wr_{W} G with kernel E^{W}. Thus Γ_{0}/E^{W} is isomorphic to (F/E) Wr_{W} G, as a topological group. (iii) If Γ ∈ C then ρ|Γ : Γ → G is continous, open and maps closed subgroups of Γ to closed subgroups of G.

Proof: (i), (ii) are easy. For (iii), see for example, ([6], Lemma 1.4.2). □

Proof of Theorem: The proof is by induction on k. In the base case, F = 𝔽_{p} is cyclic of prime order p, and K_{0} = 𝔽_{p}^{W} is a continuous G-module. Let Γ ∈ C. Then K_{Γ} = Γ∩K_{0} is a closed G-submodule

of K_{0}. So by the dual version of Corollary 3.12, there are at most p^{2^{3n^{4}}} possibilities for K_{Γ}. Now consider C_{Γ} = {Γ_{1} ∈ C : K_{Γ_{1}} = K_{Γ}}. There is a one-to-one correspondence between this and the set of closed complements to K_{0}/K_{Γ} in Γ_{0}/K_{Γ} given by Γ_{1} ↦ Γ_{1}/K_{Γ}, and this correspondence commutes with Γ_{0}-conjugation. Thus, by Lemma 4.5, the number of Γ_{0}-conjugacy classes in C_{Γ} is |H_{c}^{1}(G, K_{0}/K_{Γ})| (because (Γ/K_{Γ})/(K/K_{Γ}) ≅ Γ/K ≅ G, as topological groups). By Theorem 4.6, this is at most p^{2^{18n^{4}}} . The base case then follows immediately.

For the inductive step, let E ≤ Z(F) be cyclic of prime order p, and let Z_{0} = E^{W} ≤ Γ_{0}. By Lemma 5.2, there is a continuous, open epimorphism Γ_{0} → \overline{Γ}_{0} = (F/E) Wr_{W} G with kernel Z_{0}, and Γ_{0}/Z_{0} ≅ \overline{Γ}_{0}. So by inductive hypothesis the number of Γ_{0}-conjugacy classes of Σ ∈ C with Σ ≥ Z_{0} is at most |(F/E)|^{2^{(k−1)^{4}bn}}.

Note that as Z_{0} is central in K_{0}, conjugation in Γ_{0} makes Z_{0} into a continuous G-module. Fix some Γ_{1} ∈ C and let K_{1} = Γ_{1} ∩ Z_{0}. Note that the latter is a normal subgroup of Γ_{0} (as it is centralised by K_{0} and normalised by Γ_{1}) and a closed G-submodule of Z_{0}. There are therefore at most p^{2^{3n^{4}}} possibilities for K_{1}, by Corollary 3.12. Let Σ = Γ_{1}Z_{0}. We have already bounded the possibilities for this. We next count Σ-conjugacy classes of Γ_{2} ≤ Σ, Γ_{2} ∈ C with Γ_{2} ∩ Z_{0} = K_{1}.

Let Π = Σ/K_{1} and ˜ : Σ → Π the natural map (which is of course continuous and open). Then Γ̃_{1} is a closed subgroup of Π, Z̃_{0} is a compact normal subgroup of Π and Γ̃_{1} ∩ Z̃_{0} = 1, Γ̃_{1}Z̃_{0} = Π. So by Lemma 4.5, H_{c}^{1}(Γ̃_{1}, Z̃_{0}) counts the number of Π-classes of closed complements to Z̃_{0} in Π, and this is the same as the number of Σ-classes of Γ_{2} ∈ C with Γ_{2} ≤ Σ and Γ_{2} ∩ Z_{0} = K_{1}.

Let T_{1} = Γ_{1} ∩ K_{0}. This is a compact normal subgroup of Γ_{1}. Let T = T̃_{1}. Then Γ̃_{1}/T ≅ Γ_{1}/T_{1} ≅ G. By Lemma 4.3 we have an exact sequence

H_{c}^{1}(Γ̃_{1}/T, Z̃_{0}^{T}) → H_{c}^{1}(Γ̃_{1}, Z̃_{0}) → H_{c}^{1}(T, Z̃_{0})^{Γ̃_{1}/T} .

Now, as Z_{0} is central in K_{0}, T acts trivially on Z̃_{0}, so the last term here is the set of continuous group homomorphisms

γ : T → Z̃_{0}

such that γ(^{g}t) = gγ(t) for all t ∈ T and g ∈ Γ̃_{1}. We denote this by Hom_{G}^{c}(T, Z̃_{0}). Thus

|H_{c}^{1}(Γ̃_{1}, Z̃_{0})| ≤ |H_{c}^{1}(G, Z_{0}/K_{1})|| Hom_{G}^{c}(T, Z̃_{0})|.

As Z_{0} ≅ 𝔽_{p}^{W}, the first factor is bounded above by p^{2^{18n^{4}}} , by Theorem 4.6. The following bounds the remaining factor.

Claim A: | Hom_{G}^{c}(T, Z̃_{0})| is at most p to the power s2^{3((k(mn)^{n}2^{n^{4}} +1)^{4}}, where m = n^{2^{n+1}} and s = 2^{n^{4}}.

First, we embed (as continuous G-modules) Z̃_{0} into (𝔽_{p}^{Ω^{(m)}})^{s} for some m, s ∈ ℕ. Thus we require an exact sequence

0 → K_{1} → Z_{0} →^{φ^{∗}} (𝔽_{p}^{Ω^{(m)}})^{s}.

Dually, we require

(𝔽_{p}Ω^{(m)})^{s} →^{φ} Z_{0}^{∗} → K_{1}^{∗} → 0.

Now, Z_{0}^{∗} = 𝔽_{p}Ω^{λ} and m, s have to be chosen so that φ has image a prescribed submodule of this. By Corollary 3.15 we may take m = n^{2^{n+1}} and s = 2^{n^{4}}.

Note that

Hom_{G}^{c}(T, Z̃_{0}) ⊆ Hom_{G}^{c}(T, (𝔽_{p}^{Ω^{(m)}})^{s}) ≅ (Hom_{G}^{c}(T, 𝔽_{p}^{Ω^{(m)}}))^{s}

so our next step is to bound Hom_{G}^{c}(T, 𝔽_{p}^{Ω^{(m)}}).

Take a fixed w ∈ Ω^{(m)} and let U_{w} = {f ∈ 𝔽_{p}^{Ω^{(m)}} : f(w) = 0}. For ϕ ∈ Hom_{G}^{c}(T, 𝔽_{p}^{Ω^{(m)}}) put T_{w,ϕ} = ϕ^{−1}(U_{w}) and T_{1w} = ⋂_{ϕ} T_{w,ϕ}. Then T/T_{w,ϕ} is an abelian group of order at most p since

T/T_{w,ϕ} ↪ 𝔽_{p}^{Ω^{(m)}}/U_{w} ≅ 𝔽_{p}. Also T/T_{1w} is an elementary abelian p−group since T/T_{1w} embeds into ∏_{ϕ} T/T_{w,ϕ}. We seek to bound the size of this.

Let Γ_{i}^{w} = {g ∈ Γ_{i} : ρ(g)w = w} (for i = 0, 1). So T_{1} ≤ Γ_{1}^{w} ≤ Γ_{0}^{w}. Consider Γ_{0}^{w} as a topological group with Γ_{1}^{w} acting as a group of operators (by conjugation). Claim B: Γ_{0}^{w} has a topological Γ_{1}^{w}-composition series of length at most c_{1} + 1 where c_{1} = k(mn)^{n}2^{n^{4}}. Claim C: |T/T_{1w}| ≤ p^{c_{1}}. Proof of Claim B: Take a chief series 1 = F_{0} < F_{1} < ⋯ < F_{k} = F for F such that F_{i}/F_{i−1} ≤ Z(F/F_{i−1}) (and p_{i} = |F_{i}/F_{i−1}| is prime). Let Z^{i} = F_{i}^{Ω^{λ}} ≤ K_{0}. Then 1 = Z_{0} < Z_{1} < ⋯ < Z^{k} = K_{0} < Γ_{0}^{w} is a Γ_{1}^{w}-invariant series for Γ_{0}^{w}. Moreover, Γ_{0}^{w}/K_{0} ≅ G_{w} and Γ_{1}^{w} acts on Z^{i}/Z^{i−1} in the same way as G_{w} acts on 𝔽_{p_{i}}^{Ω^{λ}}. The claim follows by the dual version of Corollary 3.10. Proof of Claim C: By Claim B, T_{1} has a topological Γ_{1}^{w}-composition series of length at most c_{1}, so the same is true of T. Each T_{w,ϕ} is a closed, Γ_{1}^{w}-invariant subgroup of index at most p in T. The claim therefore follows from Claim B by a Jordan-Hölder Theorem.

We now return to the proof of Claim A. Let T^{1} = ⋂_{w} T_{1w}. Then T^{1} ≤ Γ̃_{1} and T̂ = T/T^{1} is abelian. Thus conjugation in Γ̃_{1} makes T̂ into a continuous G-module. Each T̂_{w} = T_{1w}/T^{1} is an open G_{w}-invariant subgroup of T̂ and G permutes these transitively. Moreover (as G_{w} has no proper subgroup of finite index) G_{w} acts trivially on the finite quotient T̂/T̂_{w} ≅ T/T_{1w}, and all of these groups are isomorphic to some finite abelian group A. It follows that (as a continuous Gmodule) T̂ is isomorphic to a closed submodule of A^{Ω^{(m)}} and so, by Claim C, to a closed submodule of (𝔽_{p}^{Ω^{(m)}})^{c_{2}} for some c_{2} ≤ c_{1}.

By construction, T^{1} is in the kernel of all ϕ ∈ Hom_{G}^{c}(T, 𝔽_{p}^{Ω^{(m)}}), so Hom_{G}^{c}(T, 𝔽_{p}^{Ω^{(m)}}) can

be considered as Hom_{G}^{c}(T̂, 𝔽_{p}^{Ω^{(m)}}). By the previous paragraph, the graph of any continuous

G−homomorphism from T/T_{1} to 𝔽_{p}^{Ω^{(m)}} corresponds to a closed submodule of (𝔽_{p}^{Ω^{(m)}})^{c_{2}} × 𝔽_{p}^{Ω^{(m)}} =

(𝔽_{p}^{Ω^{(m)}})^{c_{2}+1}. By the dual version of Corollary 3.12 the number of these is bounded by p to the power 2^{3((c_{2}+1)m)^{4}}. Thus | Hom_{G}^{c}(T, Z̃_{0})| is bounded above by p to the power

s2^{3((k(mn)^{n}2^{n^{4}} +1)^{4}},

and this gives Claim A.

We now put all of this together. By Claim A and what immediately preceded it, the number of Σ-classes of Γ_{2} ∈ C with Γ_{2} ≤ Σ and Γ_{2} ∩ Z_{0} = K_{1} is bounded by p to the power

2^{18n^{4}} + s2^{3((k(mn)^{n}2^{n^{4}} +1)^{4}}.

Multiplying this by the number of possibilities for K_{1} and the number of possibilities for (Γ_{0}-classes of) Σ we obtain the bound:

|(F/E)|^{2^{(k−1)^{4}bn}} p^{A}

where

A = 2^{3n^{4}} + 2^{18n^{4}} + s2^{3((k(mn)^{n}2^{n^{4}} +1)^{4})}.

Substituting in for m and s we obtain

2^{3n^{4}} + 2^{18n^{4}} + 2^{n^{4}}2^{3((kn^{n(2^{n+1}+1)}2^{n^{4}} +1)^{4})},

and, very crudely, we can bound this by 2 to the power

6k^{4}n^{2^{(n+2)^{4}}} = k^{4}b_{n}.

The inductive step follows. □

Remarks 5.3 1. All of the machinery we have developed to handle the case F = 𝔽_{p} will work for arbitrary finite abelian F. Thus the argument in the base step of the above proof can be used to show that if F is abelian, then we have the much better bound of

|F |^{2^{18n^{4}+1}}.

The difference arises from the fact that in this case we do not need to use Krajíček’s result.

2. By similar reasoning, the bound in Theorem 5.1 ought to hold with k being the nilpotency class of F, rather than the composition length. We will not pursue this as it does not significantly affect the overall form of the bound.

6 Commentary

6.1 Finite covers

As is well-known, a subgroup of Sym(W) is closed if and only if it is the group of automorphisms of some first-order structure with domain W. Thus we make the following definition (cf. [6]).

A permutation structure is a pair ⟨W; G⟩ where G is a closed subgroup of Sym(W). We refer to G as the automorphism group and write G = Aut(W). When we do this we refer to ‘the permutation structure W,’ the appropriate automorphism group being understood. If A, B are subsets of W (or of some set on which Aut(W) is acting naturally) then Aut(B/A) denotes the group of permutations of B which extend to automorphisms of W fixing every element of A.

Suppose C, W are permutation structures. A finite-to-one surjection π : C → W is a finite cover of W if the set of fibres of π is preserved by Aut(C) and the induced map ρ : Aut(C) → Sym(W) given by ρ(g)(w) = π(g(π^{−1}(w))) has image Aut(W). Fix such a finite cover. Its kernel is the kernel of ρ, namely Aut(C/W). If w ∈ W let C(w) = π^{−1}(w). The binding group at w is B(w) = Aut(C(w)/W) and this is a normal subgroup of the fibre group F (w) = Aut(C(w)/w). If Aut(W) is transitive on W, then these are isomorphic as w varies. There is a continuous epimorphism χ_{w} : Aut(W/w) → F (w)/B(w) ([6], Lemma 2.1.1). Thus if Aut(W/w) has no proper open subgroup of finite index, then F (w) = B(w).

Suppose now that Aut(W) is transitive. Let Γ = Aut(C) and K_{0} = ∏_{w∈W} F (w) ≤ Sym(C). Then Γ normalises K_{0}. Let Γ_{0} = K_{0}Γ, pick w_{0} ∈ W and let X = C(w_{0}), F = F (w_{0}). We can identify C with X × W so that Γ_{0} is the wreath product F Wr_{W} Aut(W) (this is a particular case of the theorem of Krasner and Kaloujnine). Thus we may regard Aut(C) as a closed subgroup of this wreath product.

The cover problem is, given W and data (F (w), B(w), χ_{w}), to determine the possible finite covers with this data. Here ‘determine’ means up to isomorphism of permutation structures (see [6]). By the above, it follows that if we can determine (up to conjugacy) the closed subgroups of the wreath product which project onto Aut(W), then we have a solution to this problem. So in the case where W is ⟨Ω^{λ}; Sym(Ω)⟩ and the fibre groups are nilpotent, our main theorem gives an explicit finite bound on the number of isomorphism types of finite covers of W.

It would be more satisfactory to have such a result expressed in terms of the binding group. Note that we already have this if λ = (∞ − n, 1^{n}) as the stabiliser of an n-tuple of elements from Ω is isomorphic to the symmetric group, and this has no proper subgroup of finite index. Thus in this case, the fibre group equals the binding group.

Theorem 6.1 Let λ′ be a partition of n, let λ = (∞ − n, λ′), and W = ⟨Ω^{λ}; Sym(Ω)⟩. Suppose B is a nilpotent finite group of composition length k. Then the number of isomorphism types of finite cover of W with binding group B is at most |B| to the power 2^{k^{4}b_{n}}, where b_{n} = 6n^{2^{(n+2)^{4}}} .

Proof: If λ′ = 1^{n}, then this follows from what we have already. If not, we convert finite covers of W into finite covers of W_{1} = Ω^{(n)} without changing the binding group.

Suppose π : C → W is a finite cover. There is a trivial finite cover σ_{1} : W_{1} → W (σ_{1} just puts the elements of a tuple w ∈ Ω^{(n)} into the finite rows of a λ-tabloid in a systematic way). We form the fibred product of these two finite covers of W. Specifically, let C_{1} = {(c, w_{1}) ∈ C × W_{1} : π(c) = σ_{1}(w_{1})}; let π_{1} be projection of this to the second coordinate; let Aut(C_{1}) = {(g, h) ∈ Aut(C) × Aut(W_{1}) : ρ(g) = h} (we are using Aut(W) = Sym(Ω) = Aut(W_{1})). Then π_{1} : C_{1} → W_{1} is a finite cover and the projection C_{1} → C is a trivial finite cover. The binding group in π_{1} is B, which is necessarily the same as the fibre group. Moreover, isomorphic finite covers of W_{1} give rise (via σ_{1}) to isomorphic finite covers of W. The result then follows from the first case. □

6.2 The model-theoretic context

Our original reason for studying the problem was to answer a question which arose in model theory. By [3] any disintegrated totally categorical structure is biinterpretable with a finite cover of a Grassmannian of a pure set (i.e. a permutation structure W = ⟨Ω^{(∞−n,n)}; Sym(Ω)⟩). It was known that there are only finitely many isomorphism types of such finite covers once the kernel is specified. Indeed Cherlin and Hrushovski ([4], but see also [1]) prove in a wider context that if one specifies an arity (defined in the next section) and a fibre group there are only finitely many finite covers of W. Their arguments (and the arguments in [7]) do not, however, appear to give bounds: this is what we provide here.

As well as the bounds, our result gives information which does not seem to follow from the model-theoretic analysis: the arity is not required in this case, as the number of isomorphism types of cover is finite once the binding group is specified. So in fact, there must be a bound on the arity of these finite covers. It would be very interesting to have such a bound, but all we are able to provide is a bound when the kernel (or binding group) is abelian.

6.3 Arities

Definition 6.2 For Γ ≤ Sym(C) and a natural number n, let

cl_{n}(Γ) = {g ∈ Sym(C) : (∀x ∈ C^{n})(∃h ∈ Γ)(gx = hx)}.

This is the n-closure of Γ. Note that cl_{n+1}(Γ) ⊆ cl_{n}(Γ) and the closure of Γ in Sym(C) is ⋂_{n} cl_{n}(Γ).

Suppose Γ is closed. If cl_{n}(Γ) = Γ we say that Γ is n-ary and call the least such n the arity of Γ (if there is no such n, the arity is ∞). Denote this by arity(Γ).

Lemma 6.3 Suppose W is such that Aut(W/X) has no proper open subgroup of finite index for all finite X ⊆ W. Let π : C → W be a finite cover with kernel K and write Γ = Aut(C), Σ = Aut(W). Then

arity(Γ) ≤ n = max(2, arity(Σ), arity(K)).

Proof: Of course, we may assume n < ∞. We show that Γ is n-closed. Let Γ^{(n)} = cl_{n}(Γ). As Γ^{(n)} has the same orbits on pairs as Γ, it preserves the set of fibres of π, so there is an induced map τ : Γ^{(n)} → Sym(W). It is easy to show (using the definition of n) that the image of τ is Σ. Thus it remains to prove that the kernel L of τ is equal to K.

For finite X ⊆ W let Γ(X) = Aut(⋃_{x∈X} π^{−1}(x)/X) and K(X) = Aut(⋃_{x∈X} π^{−1}(x)/W). Then K(X) is a closed normal subgroup of finite index in Γ(X) and exactly as in the proof of ([6], Lemma 2.1.1), there is a continuous epimorphism from Aut(W/X) onto Γ(X)/K(X). So by hypothesis, Γ(X) = K(X).

Let k ∈ L. Then for all c_{1}, …, c_{n} ∈ C there is h ∈ Γ with hc_{i} = kc_{i} for i = 1, …, n. As π(kc_{i}) = π(c_{i}), it follows from the above that we may take h ∈ K. So k ∈ cl_{n}(K) = K, as required. □

Henceforth, let W = ⟨Ω^{(n)}; Sym(Ω)⟩. We are interested in bounding the arity of of a finite cover π : C → W. For simplicity, we assume this is a regular finite cover, i.e. the fibre groups acts regularly on the fibres. The structure W is 2-ary: consider for i, j = 1, …, n the relations R_{ij}(x, y) ⇔ x_{i} = y_{j} where x = (x_{1}, …, x_{n}), y = (y_{1}, …, y_{n}) ∈ W. So it remains to bound the arities of the possible kernels. We are able to do this in the case where the fibre group is the cyclic of prime order, using Krajíček’s result.

Let F be a finite abelian group and consider the two F Sym(Ω)-modules F W and F^{W}. Recall that these are dual and their submodules related by Pontriagin duality. If K is a closed submodule of F^{W} let Ann(K) = {∑ a_{w}w ∈ F W : ∑ a_{w}f(w) = 0 for all f ∈ K}. This is a submodule of F W. Conversely, if L is a submodule of F W define L^{∘} to be {f ∈ F^{W} : f(x) = 0 for all x ∈ L} (where f(x) is evaluated linearly, and we are identifying F and its dual). If x = ∑ a_{w}w ∈ F W define θ(x) = |{w : a_{w} ≠ 0}|.

Lemma 6.4 Suppose K is a closed submodule of F^{W} and x_{1}, …, x_{r} a generating set for Ann(K). Then

arity(K) ≤ m = max(θ(x_{i}) : i = 1, …, r).

Proof: Let f ∈ cl_{m}(K) and write x_{i} = ∑ a_{w}w. There exists g ∈ K such that whenever a_{w} ≠ 0 we have f(w) = g(w). So f(x_{i}) = g(x_{i}) = 0. Thus f ∈ (Ann(K))^{∘}. But this is equal to K, by duality. Thus cl_{m}(K) = K, as required. □

It is clear that Krajíček’s result gives a bound on the θ for generating sets of submodules of F_{p}W, and so the above gives a bound on the arities of finite covers of W with fibre groups which are cyclic of prime order. Presumably this can easily be extended to the full abelian case. It would be very interesting to pass to the non-abelian case.

6.4 Problems

Most of the following have already been mentioned in the paper.

1. What happens if the binding group is not nilpotent? Are there still only finitely many possibilities for the kernels, and if so, are there only finitely many isomorphism types of covers?

2. The bounds we give seem rather large: can they be significantly reduced? Is there an approach which does not require Krajíček’s result?

3. Obtain bounds on the arity of a finite cover of ⟨Ω^{(n)}; Sym(Ω)⟩ (with nilpotent binding group).

4. What happens for finite covers where the base is a Grassmannian of a vector space over a finite field? Here one would need to work with covers of bounded arity.

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