ASYMPTOTIC RESULTS ON MODULAR REPRESENTATIONS OF SYMMETRIC GROUPS AND ALMOST SIMPLE MODULAR GROUP ALGEBRAS

A.A. BARANOV, A.S. KLESHCHEV, AND A.E. ZALESSKII

1. Main results

Let G be a group and F be a field. When is the lattice of the (two-sided) ideals of the group algebra F G as simple as possible? More precisely, every group algebra has the augmentation ideal. It is natural to ask when F G has no other proper nonzero ideals. In this case the group algebra F G is called almost simple. The following problem goes back to Kaplansky (1965).

Problem. For a field F, find a group-theoretic characterization of the groups G for which F G is almost simple.

In fact, this is a question about infinite simple groups because it is easy for G finite, and because a non-trivial normal subgroup gives rise to a non-trivial ideal different from the augmentation ideal. Also note that the problem reduces easily to the question on when the augmentation ideal is simple as a ring.

The first interesting class of groups with almost simple F G was discovered in [3]. This class is rather exotic and contains groups like the universal Hall group and algebraically closed groups. For locally finite groups G a substantial progress was achieved recently by using representation theory of finite groups, see [16, 17, 14], and others. The representation theory approach transforms the problems on ideals to certain problems on asymptotic behavior of representations of finite groups, which are often of independent interest. The method is more effective over fields of characteristic zero as the theory of ordinary representations is much better elaborated than the modular theory.

This paper is devoted to the modular aspect of the theory. One of our main results deals with the case where G is a direct limit of alternating groups and F is any field of characteristic p > 2. We note that the case where char F = 0 was considered in [16].

Let ℕ be the set of natural numbers. Denote by Alt(Ω) and Sym(Ω) (or, simply, A_{n} and Σ_{n}) the alternating and symmetric groups, respectively, on a set Ω with |Ω| = n. Let

Alt(Ω_{1}) ⊂ Alt(Ω_{2}) ⊂ ⋯ ⊂ Alt(Ω_{i}) ⊂ …  (1)

be a chain of (strict) embeddings of finite alternating groups. Then we can consider the union (more precisely, the direct limit)

G = ⋃_{i=1}^{∞} Alt(Ω_{i}),

which is a locally finite group. We call such G a limit alternating group. We emphasize that the embeddings in (1) do not have to be natural embeddings. If the embeddings are natural then G is the finitary alternating group Alt_{∞}, which consists of all permutations of ℕ fixing all but finitely many numbers. In general G has no natural permutation representation, like in the case of non-diagonal limit permutation groups defined below, see [16] for more details. We note that for diagonal G the natural permutation set may be identified with ℕ but the action of each Alt(Ω_{i}) has infinitely many non-trivial orbits unless G is the finitary alternating group.

Mathematics Subject Classification: 16S34, 20C20, 20C30.

The first author has been supported by INTAS (Project 93-183-Ext) and by the Institute of Mathematics of the National Academy of Sciences of Belarus in the framework of the state program “Mathematical structures”. The second author was supported by the NSF.

Definition 1.1. (i) An embedding Alt(Ω_{1}) → Alt(Ω_{2}) is called diagonal if the orbits of Alt(Ω_{1}) on Ω_{2} have lengths 1 or |Ω_{1}|.

(ii) A limit alternating group G is called diagonal if all but finitely many embeddings Alt(Ω_{i}) → Alt(Ω_{i+1}) are diagonal. Otherwise G is called non-diagonal.

The next theorem explains the ring theoretic importance of non-diagonal groups.

Theorem A. Let char F ≠ 2, and G be a limit alternating group. Then F G is almost simple if and only if G is non-diagonal.

The proof of Theorem A (see also a more general Theorem 3.13) relies on four powerful tools. The first one is a general result of Passman and Zalesskii [13] which implies that the Jacobson radical of F G is trivial. The second one is the inductive systems techniques introduced by Zalesskii (see e.g. the exposition [18] and references there). These tools allow us to reduce the problem to some questions on the asymptotic behavior of representations of finite alternating groups. To answer these questions we need one of the main results of [14]. This result shows that if the embeddings in (1) are non-diagonal, then for any n ∈ ℕ there exists N > n such that, for any i > N, the group Alt(Ω_{n}) has a regular orbit on Ω_{i}, i.e. an orbit of length | Alt(Ω_{n})|. Finally, we need to prove some new asymptotic results on the branching rules for symmetric groups in characteristic p. We believe these results might be of independent interest so we present them in this section.

Recall that the representations 1_{Σ_{n}} : g ↦ 1 and sgn_{n} : g ↦ sign(g) are the only 1-dimensional representations of Σ_{n}. All the other representations of Σ_{n} are faithful. Suppose first that char F = 0. Then the natural permutation F Σ_{n}-module M (of dimension n) splits as a direct sum M ≅ 1_{Σ_{n}} ⊕ V where V is the natural irreducible Σ_{n}-module. Denote by V^{σ} the module V ⊗ sgn_{n}. Using the classical branching rule one can easily deduce the following simple but useful fact.

Asymptotic Theorem. Let char F = 0.

(i) For any n ∈ ℕ there exists M > n such that, for every i ≥ M, the restriction of any Σ_{i}-module to Σ_{n} has a 1-dimensional composition factor.

(ii) For any n ∈ ℕ there exists N > n such that, for every i ≥ M, the restriction of any faithful irreducible Σ_{i}-module to Σ_{n} has either the natural irreducible module V or the module V^{σ} as a composition factor.

It is the analogue of this theorem which we need to find in characteristic p in order to prove Theorem A. Note that in positive characteristic the result is wrong as stated.

From now on let char F = p > 0.

If p does not divide n then the natural permutation module M splits, as in characteristic 0, into a direct sum of 1_{Σ_{n}} and an irreducible module V of dimension n − 1. Otherwise M is uniserial with composition factors 1_{Σ_{n}}, V, 1_{Σ_{n}} where V is an irreducible module of dimension n − 2. In any case we still call V the natural irreducible (Σ_{n}-)module.

We recall from [8] that the irreducible F Σ_{n}-modules are labelled by the p-regular partitions of n, i.e. the partitions of n in which each part appears with multiplicity < p. If λ is such a partition, we denote the corresponding irreducible module by D^{λ}. We make use of an important class of irreducible Σ_{n}-modules introduced in [10].

Definition 1.2. Let λ = (λ_{1} ≥ λ_{2} ≥ ⋯ ≥ λ_{s} > 0) be a p-regular partition. The number of rows s is called the height of λ. We say λ is completely splittable if λ_{1} − λ_{s} + s ≤ p. In this case the corresponding irreducible module D^{λ} is also called completely splittable. (Clearly, 1 ≤ s ≤ p − 1).

Observe that the natural irreducible module V = D^{(n−1,1)} and the module V^{σ} are not completely splittable (for n > p). Also note that the 1-dimensional modules 1_{Σ_{n}} = D^{(n)} and sgn_{n} are the only completely splittable modules of heights 1 and p−1, respectively (see Section 2). On the other hand, all completely splittable modules of heights 2, 3, …, p−2 are faithful. These facts and the following result proved in [10] show that the straightforward generalization of the Asymptotic Theorem above to characteristic p is wrong.

Proposition 1.3. [10, 2.8] Let N > n > (p − 1)^{2}, and let λ be a completely splittable partition of height s. Then the restriction D^{λ}↓_{Σ_{n}}^{Σ_{N}} is a direct sum of completely splittable modules of height s.

Proposition 1.3 shows that essentially the best result one can hope for as a modular analogue of the Asymptotic Theorem above is

Theorem B (Modular Asymptotic Theorem). Let n > (p − 1)^{2}.

(i) Assume p > 2. There exists M = M(n) > n such that, for any i ≥ M, the restriction of any Σ_{i}-module to Σ_{n} contains a completely splittable Σ_{n}-module as a composition factor.

(ii) Assume p > 3. There exists N = N(n) > n such that, for any i ≥ N, the restriction of any faithful irreducible Σ_{i}-module to Σ_{n} contains either the natural irreducible module V, or the module V^{σ}, or a faithful completely splittable Σ_{n}-module as a composition factor.

(ii′) Assume p = 3. There exists N = N(n) > n such that, for any i ≥ N, the restriction of any faithful irreducible Σ_{i}-module to Σ_{n} contains as a composition factor either the natural irreducible module V, or the module V^{σ}, or an irreducible Σ_{n}-module D^{λ} or D^{λ} ⊗ sgn_{n} with λ = (λ_{1}, λ_{2}), 2 ≤ λ_{1} − λ_{2} ≤ 7.

Remark. The proof of the Modular Asymptotic Theorem relies on the modular branching rules for symmetric groups [9] and the Mullineux Conjecture proved in [7] (see also [2]). The Mullineux bijection becomes trivial in characteristic 2. This is the reason why our methods fail in this case. In characteristic 3 we use the results of Sheth [15].

2. Inductive systems and asymptotic branching

In this section we prove the Modular Asymptotic Theorem.

Notation. Throughout the paper F is an arbitrary field of characteristic p > 0.

If G is a group and M is an F G-module, we denote by IrrM the set of the isomorphism classes of the composition factors of M. If G is a subgroup of a group H and Φ = {M_{1}, M_{2}, …, M_{k}} is a set of H-modules, we denote by Φ↓_{G} the set ⋃_{i=1}^{n} Irr(M_{i}↓_{G}).

If λ = (λ_{1} ≥ λ_{2} ⋯ ≥ λ_{s} > 0) is a partition we write h(λ) for s. We do not distinguish between λ and its Young diagram.

If A is a removable node of a partition λ, i.e. one obtains a Young diagram after removing A, we write λ_{A} for λ \ {A}, which is a partition of n − 1.

Let m, f ∈ ℕ. Express m in the form

m = (p − 1)d + r, d ∈ ℤ, 0 < r ≤ p − 1.

We need to consider certain classes of partitions. Set

β(m, f) = ((f + d)^{p−1}, (f + d − 1)^{p−1}, …, (f + 1)^{p−1}, f^{r}),
γ(m, f) = (f + (m − 1)(p − 1), f + (m − 2)(p − 1), …, f + (p − 1), f),

γ′(m, f) = (f + (m − 2)(p − 1), f + (m − 2)(p − 1), f + (m − 3)(p − 1), …, f + (p − 1), f). Obviously all these partitions are of height m. If λ^{i} = (λ_{1}^{i}, …, λ_{t_{i}}^{i}), i = 1, …, k, are partitions and λ_{t_{i}}^{i} ≥ λ_{1}^{i+1} for all i < k, put

(λ^{1}; …; λ^{k}) = (λ_{1}^{1}, …, λ_{t_{1}}^{1}, …, λ_{1}^{k}, …, λ_{t_{k}}^{k})

(glue λ_{2} to the bottom of λ_{1}, λ_{3} to the bottom of λ_{2}, and so on). We write β(m, f; μ) and γ(m, f; μ) instead of (β(m, f); μ) and (γ(m, f); μ), respectively.

Denote by ε_{n} the partition corresponding to the sign representation of Σ_{n}. By [7],

ε_{n} = (l^{a_{1}}, (l − 1)^{a_{2}})

where l, a_{1} and a_{2} are determined from n = (p−1)(l−1)+a_{1}, 0 < a_{1} ≤ p−1, a_{1}+a_{2} = p−1. So

β(m, f) = (ε_{k_{1}}; …; ε_{k_{d}}; (f^{r}));
γ(m, f) = ((n_{1}); (n_{2}); …; (n_{m}));
γ′(m, f) = ((n_{2}); (n_{2}); (n_{3}); …; (n_{m})).

where k_{i} = (f + d + 1 − i)(p − 1) and n_{j} = f + (p − 1)(m − j).

Some properties of the Mullineux bijection. For a p-regular partition λ we denote by M(λ) its Mullineux image, which is the p-regular partition defined by

D^{λ} ⊗ sgn_{n} ≅ D^{M(λ)}.

The map λ ↦ M(λ) is called the Mullineux bijection. We shall use an explicit description of the bijection M, see [7]. For a partition μ we denote by μ_{1} the first part of μ.

Lemma 2.1. Let m ≥ p−1, ε_{m} = (l^{a_{1}}, (l−1)^{a_{2}}) and μ be a partition such that μ_{1} ≤ l−1. If the partition α = (ε_{m}; μ) is p-regular then M(α) = ((m); M(μ)).

Proof. Recall that M(ε_{m}) = (m). We shall prove by induction on m that if μ satisfies the assumptions of the lemma, then M(α) = (M(ε_{m}); M(μ)) and M(μ)_{1} ≤ m − (p − 1) where M(μ)_{1} is the length of the first row in M(μ). If m = p − 1, then μ = ∅ and everything is clear.

Assume that p ≤ m < 2p − 1. Then l = 2 and μ_{1} ≤ 1, so m = p − 1 + a_{1} and μ = (1^{b}). We may assume that b > 0. Then the Mullineux symbol G(α) of α is

G(α) = ({p + b \atop p − 1 + b} {m − p \atop m − p}) or ({p + 1 \atop p})

(the second case arises if m = p, when b must be 1 since otherwise α is not p-regular). Therefore

G(M(α)) = ({p + b \atop 2} {m − p \atop 1}) or ({p + 1 \atop 2}).

We have b ≤ a_{1} = m − (p − 1). So G(M(α)) is the Mullineux symbol of the partition (m, b) in both cases. Thus M(α) = (m, b).

Assume now that m ≥ 2p − 1. Let

α′ = α\{p-edge of α}.

Observe that α′ = (ε_{m−p}; μ′). Note that m − p ≥ p − 1. By inductive hypothesis,

M(α′) = (M(ε_{m−p}); M(μ′)) = ((m − p); M(μ′))

and

M(μ′)_{1} ≤ (m − p) − (p − 1).  (2)

We have

G(α) = ({p + a \atop p − 1 + b} G(α′)) where ({a \atop b} G(μ′)) = G(μ).

Therefore

G(M(α)) = ({p + a \atop 1 + c} G(M(α′))) where ({a \atop c} G(M(μ′))) = G(M(μ)).

Now, in view of (2), in constructing M(α) one glues a nodes to the rows of M(μ′) (obtaining M(μ)) and p nodes to the first row (obtaining (m)). Therefore M(α) = (M(ε_{m}); M(μ)), and M(μ)_{1} ≤ m − (p − 1), as required.

Lemma 2.2. Let α = (ε_{m_{1}}; …; ε_{m_{k}}; μ) be a p-regular partition with m_{i} ≤ m_{i+1} − (p − 1) for all i. Then M(α) = ((m_{1}, …, m_{k}); M(μ)).

Proof. Apply Lemma 2.1 (k times).

Lemma 2.3. Let m = d(p − 1) + r where 0 ≤ r < p − 1. Then M(β(m, f)) = γ(d, e; ν) where e = (f + 1)(p − 1), ν = M((f^{r})).

Proof. This follows from Lemma 2.2 with μ = (f^{r}).

Lemma 2.4. Let μ = (μ_{1} ≥ ⋯ ≥ μ_{k}) be a p-regular partition, and m be an integer satisfying 0 ≤ m ≤ μ_{k} − (p − 1). Set α = (μ; (m)). Then M(α) = (M(μ); ε_{m}).

Proof. The proof is similar to that of Lemma 2.1. Proceed by induction on m, the case m = 0 being trivial. Set

α′ = α\{p-edge of α}.

Observe that α′ = (μ′; (m − p)) (or μ′ if m ≤ p). Moreover,

μ′_{k} − (p − 1) ≥ μ_{k} − p − (p − 1) ≥ m − p

if m > p. Therefore by inductive hypothesis, M(α′) = (M(μ′); ε_{m−p}) (or M(μ′) if m ≤ p). We consider the generic case m > p (the case m ≤ p being similar). We have

G(α) = ({a + p \atop b + 1} G(α′)) where ({a \atop b} G(μ′)) = G(μ).

Note that the assumption μ_{k} − (p − 1) ≥ m > 0 implies μ_{k} ≥ p, whence a is divisible by p. Therefore

G(M(α)) = ({a + p \atop a − b + p − 1} G(M(α′)))

Now in constructing M(α), one glues p nodes to the p − 1 rows of ε_{m−p} (obtaining ε_{m}) and a nodes to the rows of M(μ′) (obtaining M(μ)). Therefore M(α) = (M(μ); ε_{m}), as required.

Lemma 2.5. Let μ = (μ_{1} ≥ ⋯ ≥ μ_{l}) be a p-regular partition, α = (μ; (m_{1}, …, m_{k})) where μ_{l} − (p − 1) ≥ m_{1}, and m_{i} − (p − 1) ≥ m_{i+1} for 1 ≤ i < k. Then M(α) = (M(μ); ε_{m_{1}}; …; ε_{m_{k}}).

Proof. Apply Lemma 2.4 (k times).

Lemma 2.6. Let f, d ∈ ℕ, e = f(p − 1), and μ = (μ_{1}, …, μ_{k}) be a p-regular partition with μ_{k} ≥ e + d(p − 1). Then

M((μ; γ(d, e))) = (M(μ); β(d(p − 1), f)).

In particular,

M(γ′(d, e)) = (ν; β((d − 2)(p − 1), f))

where ν = M((l^{2})) with l = e + (d − 2)(p − 1).

Proof. This follows from Lemma 2.5.

Lemma 2.7. Let k ∈ ℕ, and λ = (λ_{1}, …, λ_{p−1}) be a partition with λ_{p−1} ≥ kp and λ_{1} − λ_{p−1} > 1. If M(λ) = (μ_{1}, μ_{2}, …), then μ_{p+1} ≥ k.

Proof. Define λ^{(1)} = λ, λ^{(i)} = λ^{(i−1)}\{p-edge of λ^{(i−1)}} for 1 < i ≤ k. Since λ_{1} −λ_{p−1} > 1, the number of nodes of the p-edge of λ^{(1)} is n_{1}p with n_{1} ≥ 2. This implies λ_{1}^{(2)} − λ_{p−1}^{(2)} > 1.

Also λ_{p−1}^{(2)} ≥ λ_{p−1}^{(1)} − p = (k − 1)p. Therefore we have

G(λ) = ({n_{1}p \atop p − 1} {n_{2}p \atop p − 1} … {n_{k}p \atop p − 1} …)

where n_{i} ≥ 2 (1 ≤ i ≤ k). Hence

G(M(λ)) = ({n_{1}p \atop (n_{1} − 1)p + 1} {n_{2}p \atop (n_{2} − 1)p + 1} … {n_{k}p \atop (n_{k} − 1)p + 1} …).

Since (n_{i} − 1)p + 1 ≥ p + 1 for 1 ≤ i ≤ k, we have μ_{p+1} ≥ k.

Lemmas on branching. The results of this section rely on the modular branching rules proved in [9, 11], especially Theorem 0.4(ii) from [9] and Theorem 1.4 from [11]. A version of the modular branching rule needed here is as follows (for the definition of normal removable nodes see [9, Definition 0.3]).

Theorem 2.8. Let A be a removable node of a p-regular partition λ such that λ_{A} is also p-regular. Then D^{λ_{A}} is a composition factor of D^{λ}↓_{Σ_{n−1}} if and only if A is normal.

The following two lemmas follow immediately from the modular branching rule.

Lemma 2.9. D^{γ(d,f)} ∈ Irr(D^{γ(d,f+1)}↓_{Σ_{n}}) where n = |γ(d, f)|.

Lemma 2.10. D^{γ′(d,f)} ∈ Irr(D^{γ(d,f)}↓_{Σ_{n}}) where n = |γ′(d, f)|.

Lemma 2.11 ([1, 4.10]). Let μ = (μ_{1}, μ_{2}, …) be a p-regular partition (μ may be ∅), f > μ_{1} + 1. Set β = β(m, f; μ), α = β(m, f − 1; μ), n = |α|. Then

D^{α} ∈ Irr(D^{β}↓_{Σ_{n}}).

Proposition 2.12. Let λ = (λ_{1}, …, λ_{k}) be a p-regular partition, m ∈ {1, 2, …, k}, and

λ_{m} > 2 + ∑_{i=m+1}^{k} λ_{i}.

Set f = λ_{m} − 1 − ∑_{i=m+1}^{k} λ_{i}, β = β(m, f), n = |β|, β′ = β(m, f + 1; (1)), n′ = |β′|. Then

D^{β} ∈ Irr(D^{λ}↓_{Σ_{n}}).

Moreover, if λ_{m+1} ≠ 0, then

D^{β′} ∈ Irr(D^{λ}↓_{Σ_{n′}}).

Proof. The first part is [1, 4.14]. The second part is contained in the proof of [1, 4.14].

Lemma 2.13. Let μ = (μ_{1}, μ_{2}, …) be a non-empty p-regular partition, f > μ_{1} + 1, and let A be the top removable node of (the Young diagram of) μ such that μ_{A} is p-regular. Set γ = γ(m, f; μ), α = γ(m, f − 1; μ_{A}), n = |α|. Then

D^{α} ∈ Irr(D^{γ}↓_{Σ_{n}}).

Proof. The proof is identical to [1, 4.11].

Lemma 2.14. Let μ = (μ_{1}, μ_{2}, …) be a non-empty p-regular partition, f > |μ| + 1, e = f − |μ|, n = |γ(m, e)|, n′ = |γ(m, e + 1; (1))|. Then

D^{γ(m,e+1;(1))} ∈ Irr(D^{γ(m,f;μ)}↓_{Σ_{n′}}),
D^{γ(m,e)} ∈ Irr(D^{γ(m,f;μ)}↓_{Σ_{n}}).

Proof. Apply Lemma 2.13 sufficiently many times.

Lemma 2.15. Let μ = (μ_{1}, …, μ_{m}) and λ = (λ_{1}, …, λ_{m}, …, λ_{h}) be p-regular partitions. Set c = λ_{m+1} + ⋯ + λ_{h}. Assume that D^{μ} ∈ Irr(D^{λ}↓_{Σ_{|μ|}}). Then μ_{m} ≤ λ_{m} + c.

Proof. If D^{ν} ∈ Irr(D^{ρ}↓_{Σ_{|ρ|−1}}), then ν dominates ρ_{A} for some removable node A of ρ. Therefore

ν_{m} + ν_{m+1} + ⋯ ≤ (ρ_{A})_{m} + (ρ_{A})_{m+1} + ⋯ ≤ ρ_{m} + ρ_{m+1} + · · · .

The lemma follows by induction.

Inductive systems. Let G be a countable locally finite group. Then G can be considered as a union

G = ∪_{n=1}^{∞}G_{n}

where G_{1} ⊂ G_{2} ⊂ … are finite subgroups of G.

Definition 2.16. [18, 1.1] Let Φ_{n} be a non-empty set of the isomorphism classes of irreducible F G_{n}-modules (n = 1, 2, …). We say that the collection Φ = {Φ_{n}}_{n∈ℕ} is an inductive system (for G) if for any n ∈ ℕ we have

Φ_{n} = Irr(Φ_{n+1}↓_{G_{n}}).

The following result explains our interest in inductive systems. Note that G always has at least two inductive systems: the inductive system Γ = {Γ_{n}}_{n∈ℕ} with Γ_{n} = Irr G_{n} for all n and the unitary inductive system with Φ_{n} = {1_{G_{n}}} for all n. Any inductive system distinct from Γ is called proper. The following result was proved by the third author, see [18].

Theorem 2.17. Let G = ∪_{n=1}^{∞}G_{n} be a locally finite group as above. Assume additionally that G is simple. Then the group algebra F G is almost simple if and only there are no proper inductive systems for G, except for the unitary one.

Proof. Assume there are no proper inductive systems, except for the unitary one. Then by [17, Theorem 1], (see also [18, 1.25]) every proper subideal of the augmentation ideal is contained in the Jacobson radical of F G. But this radical is trivial by a theorem of Passman and Zalesskii [13, 12]. Now since the augmentation ideal has codimension 1, we conclude that F G is almost simple. The converse follows immediately from [18, 1.25].

We denote by Σ_{∞} the group of all finitary permutations of ℕ. Clearly, Σ_{∞} is a locally finite group and

Σ_{∞} = ∪_{n>N}Σ_{n}

where Σ_{n} is the group of all permutations of the set {1, 2, …, n}, and N is any natural number. So we may consider the inductive systems for Σ_{∞}.

Definition 2.18. Let Φ = {Φ_{n}}_{n>N} be an inductive system for Σ_{∞}. Define the inductive system Φ^{σ} = {Φ_{n}^{σ}}_{n>N} by

Φ_{n}^{σ} = {D^{λ} ⊗ sgn_{n} | D^{λ} ∈ Φ_{n}}.

One can easily check that Φ^{σ} so defined is indeed an inductive system.

We shall use some special inductive systems introduced in [10, 1]. For n > (p − 1)^{2} let Φ(s)_{n} be the set of all completely splittable Σ_{n}-modules of height s (see Definition 1.2), and let Φ(s) = {Φ(s)_{n}}_{n>(p−1)^{2}}. It has been proved in [1] that Φ(1), Φ(2), …, Φ(p − 1) are the minimal inductive systems for Σ_{∞} = ∪_{n>(p−1)^{2}}Σ_{n}.

Lemma 2.19. [1, 5.6] Let s ∈ {1, 2, …, p − 1}. Then Φ(s)^{σ} = Φ(p − s).

Note that Φ(1)_{n} = {1_{Σ_{n}}}. It follows from Lemma 2.19 that Φ(p − 1)_{n} = {sgn_{n}}.

It is convenient to call the inductive systems Φ(1), Φ(p − 1), and Φ(1) ∪ Φ(p − 1) (for Σ_{∞}) trivial. One of our goals in the next subsection is to describe the minimal non-trivial inductive systems for Σ_{∞}.

Put

Π_{n} = {D^{(n)}, D^{(n−1,1)}}, n > (p − 1)^{2}.

The following result is straightforward.

Lemma 2.20. Π = {Π_{n}}_{n>(p−1)^{2}} is an inductive system for Σ_{∞}.

The inductive systems for Σ_{∞} consisting of modules D^{λ} with λ having at most two parts have been classified by Sheth [15]. We need a very special corollary of his result:

Lemma 2.21. [15, Theorem 7.7] Let p > 2. Set

Θ_{n} = {D^{(l_{1},l_{2})} | l_{1} + l_{2} = n, 0 ≤ l_{1} − l_{2} ≤ p^{2} − 2}.

Then Θ = {Θ_{n}}_{n∈ℕ} is an inductive system for Σ_{∞}, Θ ⊃ Φ(2), and there are no inductive systems between Θ and Φ(2).

Minimal non-trivial inductive systems. Set

Φ(β, m)_{n} = ∪_{f>n} Irr(β(m, f)↓_{Σ_{n})},
Φ(β, m; (1))_{n} = ∪_{f>n} Irr(β(m, f; (1))↓_{Σ_{n}});
Φ(β, m) = {Φ(β, m)_{n}}_{n>(p−1)^{2}},
Φ(β, m; (1)) = {Φ(β, m; (1))_{n}}_{n>(p−1)^{2}}.

It follows from Lemma 2.11 that Φ(β, m) and Φ(β, m; (1)) are inductive systems for the group Σ_{∞} = ∪_{n>(p−1)^{2}}Σ_{n} (for any m ∈ ℕ). Recall the systems Φ(1), Φ(2), …, Φ(p − 1), and Π, introduced before.

Lemma 2.22. Let 1 ≤ m ≤ p − 1. Then Φ(β, m) = Φ(m).

Proof. Clearly, D^{β(m,f)} ∈ Φ(m) for any f. So the definition of Φ(β, m) ensures that Φ(β, m) ⊆ Φ(m). Now the result follows from the minimality of Φ(m).

Lemma 2.23. Φ(β, p − 1; (1)) = Π^{σ}.

Proof. Let n = |β(p − 1, f; (1))|. By Lemma 2.1 we have M(β(p − 1, f; (1))) = (n − 1, 1). The lemma follows.

Definition 2.24. Let Φ = {Φ_{n}}_{n>N} be an inductive system for Σ_{∞} = ∪_{n>N}Σ_{n}. We define its height h(Φ) as

h(Φ) = sup{h(λ) | D^{λ} ∈ Φ_{n} for some n > N}.

Proposition 2.25 ([1, 5.2]). Let Φ = {Φ_{n}}_{n>N} be an inductive system for the group Σ_{∞} = ∪_{n>N}Σ_{n}. Assume that h(Φ) = +∞. Then Φ_{n} = Irr Σ_{n} for all n > N.

The proposition above shows that all proper inductive systems for Σ_{∞} have finite heights. So assume from now on that Φ is an inductive system with h(Φ) < +∞. It follows that for any M ∈ ℕ there exists n ∈ ℕ and D^{λ} ∈ Φ_{n} such that λ_{1} > M (in other words, the first part of the partitions involved gets arbitrarily large). This shows that the set

V (Φ) = {k ∈ ℕ | for any M ∈ ℕ there exists n ∈ ℕ and D^{λ} ∈ Φ_{n}
with λ = (λ_{1}, λ_{2} …) such that λ_{k} > M}

is not empty (it contains k = 1). Since also V (Φ) ⊆ {1, 2, …, h(Φ)} we can define

m(Φ) = max V (Φ).

It follows from the definition of m = m(Φ) and h = h(Φ) that there exists c = c(Φ) ∈ ℕ such that

∑_{j=m+1}^{h} λ_{j} ≤ c  (3)

for any D^{λ} ∈ Φ_{n} (for any n). Let N_{0} ≥ c + 2. By definition of m, there exists n ∈ ℕ and D^{λ} ∈ Φ_{n} such that λ_{m} > N_{0}. Set

f = λ_{m} − ∑_{j=m+1}^{h} λ_{j} − 1,
β = β(m, f), n = |β|,
β′ = β(m, f + 1; (1)), n′ = |β′|.

By Proposition 2.12,

D^{β} ∈ Irr(D^{λ}↓_{Σ_{n}}) ⊆ Φ_{n}.

Moreover, if λ_{m+1} ≠ 0, then

D^{β′} ∈ Irr(D^{λ}↓_{Σ_{n′}}).

So we have

Lemma 2.26. Let Φ be an inductive system for Σ_{∞}, and let m = m(Φ). Then Φ ⊇ Φ(β, m). Moreover, if for each M ∈ ℕ there exists a partition λ = (λ_{1}, λ_{2}, …) such that D^{λ} ∈ Φ_{|λ|}, λ_{m} > M, and λ_{m+1} ≠ 0, then Φ ⊇ Φ(β, m; (1)).

Lemma 2.27. Let p > 2, m = m(Φ). Assume that m = d(p − 1) + r for d ≥ 1 and 0 < r < p − 1. Then Φ ⊃ Φ(β, d(p − 1)).

Proof. By Lemma 2.26, Φ ⊇ Φ(β, m). It follows from Lemma 2.3 that the inductive system Φ^{σ} contains D^{γ(d,(f+1)(p−1);μ(f))} for all f ∈ ℕ, where μ(f) = M((f^{r})). Note that |μ(f)| = fr, whence (f + 1)(p − 1) > |μ(f)| + 1. So, by Lemma 2.14, Φ^{σ} contains D^{γ(d,(f+1)(p−1)−fr)} for all f ∈ ℕ. Since (f + 1)(p − 1) − fr → +∞ as f → +∞, Lemma 2.9 implies that Φ^{σ} contains all D^{γ(d,e)}, e ∈ ℕ. In particular, Φ^{σ} contains all D^{γ(d,f(p−1))}, f ∈ ℕ. Therefore, by Lemma 2.6, the inductive system Φ = (Φ^{σ})^{σ} contains all D^{β(d(p−1),f)}, f ∈ ℕ, so Φ ⊃ Φ(β, d(p − 1)).

Lemma 2.28. Let p > 2, m = m(Φ) = d(p − 1) for d ≥ 2. Then Φ ⊃ Φ(β, m − p).

Proof. As in the proof of Lemma 2.27 we see that Φ^{σ} contains D^{γ(d,f(p−1))} for all f ∈ ℕ. By Lemma 2.10, Φ^{σ} contains all D^{γ′(d,f(p−1))}, f ∈ ℕ. Set

μ(f) = M(γ′(d, f(p − 1)).

By Lemma 2.6,

μ(f) = (ν(f); β((d − 2)(p − 1), f))

where

ν(f) = M((l^{2})) ∈ Φ(2)^{σ}, l = f(p − 1) + (d − 2)(p − 1).

By Lemma 2.19, h(ν(f)) = p − 2, so

h(μ(f)) = p − 2 + (d − 2)(p − 1) = m − p.

Observe that the length of the last row of μ(f) is f. By Proposition 2.12 (applied with m = k), D^{β(m−p,f−1)} ∈ Irr(D^{μ(f)}↓_{Σ_{|β(m−p,f−1)|}}). Therefore Φ ⊃ Φ(β, m − p).

Theorem 2.29. Let F be a field of characteristic p > 2.

(i) Let p > 3. Then any non-trivial inductive system contains one of the following systems:

Π, Π^{σ}, Φ(s) (1 < s < p − 1).  (4)

(ii) Let p = 3. Then any non-trivial inductive system contains one of the following systems:

Π, Π^{σ}, Θ, Θ^{σ}.  (5)

Proof. Let Φ be a proper non-trivial inductive system. In view of Lemma 2.19, it suffices to prove that either Φ or Φ^{σ} contains one of the inductive systems (4) (or (5) if p = 3). So we may assume without loss of generality that m(Φ) ≥ m(Φ^{σ}). Set m = m(Φ). Consider the following cases.

Case 1: m = 1. Since Φ ≠ Φ(1), we have h(Φ) > 1. Therefore for each i there exists a partition λ = (λ_{1}, λ_{2}, …) such that D^{λ} ∈ Φ_{i} and λ_{2} ≠ 0. Hence by Lemma 2.26, Φ ⊇ Φ(β, 1; (1)) = Π.

Case 2: 1 < m < p − 1. By Lemmas 2.26 and 2.22, we have Φ ⊇ Φ(β, m) = Φ(m).

Case 3: m = p − 1. We may assume that Φ does not contain Φ(β, p − 1; (1)) = Π^{σ} (see Lemma 2.23 for the last equality). So, in view of Lemma 2.26, there exists N_{1} ∈ ℕ such that for any partition λ = (λ_{1}, λ_{2}, …) with D^{λ} ∈ Φ_{|λ|} and λ_{p} ≠ 0 we have λ_{p−1} < N_{1}.

Furthermore, take D^{λ} ∈ Φ_{|λ|} with λ_{p−1} very large. By Lemma 2.7, D^{λ} ⊗ sgn = D^{μ} with μ_{p+1} very large, unless λ_{1} − λ_{p−1} ≤ 1. But Φ^{σ} may not contain D^{μ} with arbitrarily large μ_{p+1} since then m(Φ^{σ}) ≥ p + 1 > m(Φ) which contradicts our assumptions. Thus, there exists N_{2} > N_{1} such that for each λ with D^{λ} ∈ Φ_{|λ|} and λ_{p−1} ≥ N_{2} we have λ_{1} −λ_{p−1} ≤ 1, and so λ = ε_{|λ|}, as in addition λ_{p} = 0.

By Lemma 2.26, Φ ⊇ Φ(β, p − 1). But Φ(β, p − 1)_{n} = {sgn_{n}}. Set

Ψ_{n} = Φ_{n}\{sgn_{n}}.

We are going to show that Ψ = {Ψ_{n}}_{n>N} is an inductive system for Σ_{∞} = ∪_{n>N}Σ_{n}, provided N is large enough. It suffices to check that sgn_{j} ∉ Irr(Ψ_{i}↓_{Σ_{j}}) for any pair i > j > N. Assume not, i.e. there exists λ = (λ_{1}, λ_{2}, …) with D^{λ} ∈ Ψ_{i} such that sgn_{j} ∈ Irr(D^{λ}↓_{Σ_{j}}). Let c = c(Φ), see (3). Since N is large enough, we may assume that the length of the (p − 1)-st row of ε_{j} is at least N_{2} + c. So, by Lemma 2.15, we have λ_{p−1} ≥ N_{2}. Hence by the choice of N_{2}, D^{λ} = sgn_{i} ∉ Ψ_{i}. This yields a contradiction. So Ψ is an inductive system. Observe that Ψ ∪ Φ(p − 1) = Φ. Therefore Ψ is non-trivial. Since λ_{p−1} < N_{2} for each partition λ with D^{λ} ∈ Ψ_{|λ|}, we have m(Ψ) < p − 1. So Ψ (and Φ) contains one of the systems (4) by what has already been proved.

Case 4: p−1 < m < 2(p−1). Set r = m−(p−1). By Lemma 2.26, Φ ⊇ Φ(β, m). It follows from Lemma 2.3 that Φ^{σ} contains all D^{γ(1,(f+1)(p−1);μ(f))}, f ∈ ℕ, where μ(f) = M((f^{r})). We have |μ(f)| = fr. Hence

(f + 1)(p − 1) > |μ(f)| + 1.

So Lemma 2.14 implies that Φ^{σ} contains all D^{γ(1,n(f);(1))}, f ∈ ℕ, where n(f) = (f +1)(p− 1) − fr + 1. Note that

D^{γ(1,n(f);(1))} = D^{(n(f),1)} ∈ Π_{n(f)+1}.

Since n(f) → +∞ as f → +∞, Φ^{σ} ⊃ Π, so Φ ⊃ Π^{σ}, as desired.

Case 5: p > 3 and m = 2(p−1). Then by Lemmas 2.28 and 2.22, Φ ⊇ Φ(β, p−2) = Φ(p−2), as desired.

Case 6: p = 3 and m = 2(p − 1) = 4. By Lemma 2.26, Φ ⊇ Φ(β, 4). Recall that Φ(β, 4) is generated by all β(4, f) = (ε_{2(f+1)}; ε_{2f}), f ∈ ℕ. Therefore by Lemma 2.2, Φ(β, 4)^{σ} is generated by all partitions η(f) = (2f + 2, 2f), f ∈ ℕ. We shall use Lemma 2.21. Note that

D^{η(f)} ∈ (Θ_{4f+2}\Φ(2)_{4f+2})

for all f. Since there are no inductive systems between Θ and Φ(2), we have Φ(β, 4)^{σ} = Θ. It follows that Φ ⊇ Θ^{σ}, as required.

Case 7: m > 2(p − 1). Represent m in the form m = d(p − 1) + r where 0 ≤ r < p − 1. Assume that r = 0. Then d ≥ 3. By Lemma 2.28, Φ(β, m − p) ⊂ Φ. Since m(Φ(β, m − p)) = m − p > p − 1, Φ(β, m − p) is non-trivial, and we can apply induction. If r ≠ 0, then the theorem follows from Lemma 2.27.

Set

T_{n} = \{ \begin{matrix}∪_{s=2}^{p−2}Φ(s)_{n} ∪ \{D^{(n−1,1)}, D^{(n−1,1)} ⊗ sgn_{n}\}, & if p > 3 \\ Θ_{n} ∪ Θ_{n}^{σ} ∪ \{D^{(n−1,1)}, D^{(n−1,1)} ⊗ sgn_{n}\}, & if p = 3. \end{matrix}

Corollary 2.30. Let p > 2 and n > (p − 1)^{2}. Then there exists M = M(n) > n such that, for any i ≥ M, the restriction of any faithful irreducible Σ_{i}-module to Σ_{n} contains a composition factor D ∈ T_{n}.

Proof. For each i ≥ n denote by Ψ_{i} the set of all D^{λ} ∈ Irr Σ_{i} such that

Irr(D^{λ}↓_{Σ_{n}}) ∩ T_{n} = ∅

Observe that for all i ≥ n we have Irr(Ψ_{i+1}↓_{Σ_{i}}) ⊆ Ψ_{i}. For i ≥ n we set

Φ_{i} = ∩_{j≥i} Irr(Ψ_{j}↓_{Σ_{i}})

We show that Φ = {Φ_{i}}_{i≥n} is an inductive system. Let i ≥ n. Note that

Ψ_{i} ⊇ Irr(Ψ_{i+1}↓_{Σ_{i}}) ⊇ Irr(Ψ_{i+2}↓_{Σ_{i}}) ⊇ …

Since the set Ψ_{i} is finite, there exists k = k(i) such that for any j ≥ k we have

Φ_{i} = Irr(Ψ_{j}↓_{Σ_{i}}).  (6)

Choose j ≥ max(k(i), k(i + 1)). Then

Irr(Φ_{i+1}↓_{Σ_{i}}) = Irr(Irr(Ψ_{j}↓_{Σ_{i+1}})↓_{Σ_{i}}) = Irr(Ψ_{j}↓_{Σ_{i}}) = Φ_{i},

so Φ is an inductive system.

If the corollary is false, there exists j ≥ k(n) such that Ψ_{j} contains a faithful module D. In view of (6), Φ_{n} contains a faithful module. Therefore Φ is non-trivial. By Theorem 2.29, Φ contains one of the systems (4). Therefore Φ_{n} ∩ T_{n} ≠ ∅. On the other hand, since Φ_{n} ⊆ Ψ_{n},

Φ_{n} ∩ T_{n} ⊆ Ψ_{n} ∩ T_{n} = ∅.

The contradiction obtained proves the corollary.

Now parts (ii) and (iii) of the Modular Asymptotic Theorem stated in Section 1 follow from Corollary 2.30. Part (i) follows from the following proposition.

Proposition 2.31. Let p > 2 and n > (p−1)^{2}. There exists N = N(n) > n such that, for any i ≥ N, the restriction of any simple Σ_{i}-module to Σ_{n} contains a completely splittable D^{λ} as a composition factor (see Section 1).

Proof. The proof is similar to that of Corollary 2.30, but uses [1, 5.7] instead of Theorem 2.29.

3. Diagonal and Non-diagonal groups

Throughout this section we denote by M_{n} and V_{n} the natural permutation module (of dimension n) and the natural irreducible module, respectively, for both F A_{n} and F Σ_{n}. If G and H are two groups, L is an F G-module and M is an F H-module we write L ⊠ M for the outer tensor product of L and M (which is a module over G × H). If N is another F G-module we write L⊗N for the inner tensor product of L and N (which is a G-module).

Diagonal embeddings. In this subsection we prove the easier direction of Theorem A.

Any diagonal embedding A_{n} → A_{m} is obtained by composing a diagonal embedding A_{n} → \underbrace{A_{n} × ⋯ × A_{n}}_{k times}, the embedding \underbrace{A_{n} × ⋯ × A_{n}}_{k times} → A_{kn} as a ‘Young subgroup’, and

the natural embedding A_{kn} → A_{m} (m ≥ kn). So

Lemma 3.1. Let φ : A_{n} → A_{m} be a diagonal embedding. Then the only possible composition factors of the restriction V_{m}↓_{A_{n}} are V_{n} and 1_{A_{n}}. Moreover, V_{n} always occurs.

Proof. Note that the restriction of V_{m} to A_{kn} has composition factors V_{kn} and 1_{A_{kn}}, and V_{kn} always occurs (1_{A_{kn}} also occurs providing kn < m). It now suffices to observe that all composition factors of the restriction of V_{kn} to A_{n} × ⋯ × A_{n} are of the form

1_{A_{n}} ⊠ ⋯ ⊠ 1_{A_{n}} or 1_{A_{n}} ⊠ ⋯ ⊠ 1_{A_{n}} ⊠ V_{n} ⊠ 1_{A_{n}} ⊠ ⋯ ⊠ 1_{A_{n}},

and modules of the second form occur.

The next lemma is only needed in the proof of Proposition 3.10.

Lemma 3.2. M_{k} ⊠ F ⊠ F is a subfactor of the restriction V_{3k}↓_{Σ_{k}×Σ_{k}×Σ_{k}}^{Σ_{3k}}.

Proof. Let {e_{1}, e_{2}, …, e_{3k}} be the natural basis of M_{3k}. Set L to be the 1-dimensional submodule F (e_{1}+e_{2}+⋯+e_{3k}) if the characteristic p divides 3k, and set L = (0) otherwise. Denote ē_{j} = e_{j} + L ∈ M_{3k}/L. Then V_{3k} is a submodule of M_{3k}/L (of codimension 1) generated by the elements ē_{i} − ē_{3k}, 1 ≤ i ≤ 3k − 1. Set

N_{1} = span{ē_{i} − ē_{3k} | 2k + 1 ≤ i ≤ 3k − 1}, N_{2} = span{N_{1}, ē_{i} − ē_{3k} | 1 ≤ i ≤ k}.

Then N_{1} ⊂ N_{2} are submodules of V_{3k}↓_{Σ_{k}×Σ_{k}×Σ_{k}}^{Σ_{3k}}, and N_{2}/N_{1} ≅ M_{k} ⊠ F ⊠ F.

Proposition 3.3. Assume that G is a diagonal limit alternating group. Then F G is not almost simple.

Proof. By Theorem 2.17, it is enough to construct a proper inductive system for G, different from the unitary one. We may assume that all embeddings in (1) are diagonal (just take the union starting from i large enough, which does not change the group G). Set Φ_{n} = {1_{Alt(Ω_{n})}, V_{|Ω_{n}|}} for all n. Then Φ = {Φ_{n}}_{n∈ℕ} is a (proper non-unitary) inductive system, thanks to Lemma 3.1.

Non-diagonal embeddings. The main result of this subsection is Theorem 3.12, which claims that a non-diagonal limit alternating group G has no proper inductive systems different from the unitary one. Since G is simple, Theorem A follows from Proposition 3.3, Theorem 3.12 and Theorem 2.17.

Lemma 3.4. Let X be a finite group. Then each irreducible Σ_{n} × X-module is of the form D ⊠ L with D ∈ Irr Σ_{n} and L ∈ Irr X.

Proof. Since any field is a splitting field for Σ_{n}, the lemma follows from [5, 7.10 and 10.38].

We cite an important result of [14]

Lemma 3.5. [14, Theorem 1.7] Let Alt(Ω_{1}) ⊂ Alt(Ω_{2}) ⊂ ⋯ ⊂ Alt(Ω_{i}) ⊂ … be nondiagonal embeddings. Then for every i ∈ ℕ there exists N = N(i) > i such that, for any n > N, the group Alt(Ω_{i}) has a regular orbit on Ω_{n}.

We shall use the following result of Bryant and Kovacs [4] (the proof can also be found in [6, III.2.16]).

Lemma 3.6. [4] Let X be a finite group and let V_{g} be an F X-module on which g does not act as a scalar, for all g ∈ X \ {1}. Then the regular module F X is a direct summand of ⊗_{g∈X\{1}}V_{g}.

Lemma 3.7. Let k, n ∈ ℕ, n > (p−1)^{2}, and λ be a completely splittable partition of kn of height s. Denote by H the Young subgroup \underbrace{Σ_{n} × ⋯ × Σ_{n}}_{k times} < Σ_{kn}. Then any composition

factor of the restriction D^{λ}↓_{H} is of the form

D^{λ(1)} ⊠ ⋯ ⊠ D^{λ(k)}

where λ(1), … λ(k) are completely splittable partitions of n of height s.

Proof. It is proved in [10] that the composition factors of the restriction D^{λ}↓_{Σ_{n}} are of the form D^{μ} where μ is a completely splittable partition of n of height s. Hence the composition factors of the restriction D^{λ}↓_{Σ_{n}×···×Σ_{n}} are as claimed.

The case p = 3 in the Asymptotic Theorem is somewhat exceptional. We need the following lemma for this case only.

Lemma 3.8. Let p = 3; k, n ∈ ℕ; n > 9. We say a partition λ is nice if λ = (λ_{1}, λ_{2}) and 1 < λ_{1} − λ_{2} ≤ 7. Let λ be a nice partition of kn. Denote by H the Young subgroup \underbrace{Σ_{n} × ⋯ × Σ_{n}}_{k times} < Σ_{kn}. Then the restriction D^{λ}↓_{H} has a composition factor of the form

D^{λ(1)} ⊠ ⋯ ⊠ D^{λ(k)} where λ(1), …, λ(k) are nice.

Proof. Without loss of generality we may assume that k = 2. If λ = (λ_{1}, λ_{2}) is a partition of r and s = λ_{1} − λ_{2}, we shall write (in this proof only) D_{s}^{(r)} for D^{λ}. Also let N ∼ m_{1}D_{1} + ⋯ + m_{l}D_{l} mean that D_{1}, …, D_{l} are the composition factors of the module N with multiplicities m_{1}, …, m_{l}, respectively. We use the following special case of the main result from [15]: if r > 9 then

D_{0}^{(r)}↓_{Σ_{r−1}} ∼ D_{1}^{(r−1)};
D_{1}^{(r)}↓_{Σ_{r−1}} ∼ D_{0}^{(r−1)};
D_{2}^{(r)}↓_{Σ_{r−1}} ∼ D_{1}^{(r−1)} + 2D_{3}^{(r−1)} + D_{7}^{(r−1)};
D_{3}^{(r)}↓_{Σ_{r−1}} ∼ D_{2}^{(r−1)} + D_{4}^{(r−1)};
D_{4}^{(r)}↓_{Σ_{r−1}} ∼ D_{3}^{(r−1)};
D_{5}^{(r)}↓_{Σ_{r−1}} ∼ D_{4}^{(r−1)} + 2D_{6}^{(r−1)};
D_{6}^{(r)}↓_{Σ_{r−1}} ∼ D_{5}^{(r−1)} + D_{7}^{(r−1)};
D_{7}^{(r)}↓_{Σ_{r−1}} ∼ D_{6}^{(r−1)}.

It follows that the composition factors of D^{λ}↓_{Σ_{n}×Σ_{n}} are all of the form D_{s_{1}}^{(n)} ⊠ D_{s_{2}}^{(n)} where s_{1}, s_{2} ≤ 7. Moreover, for any 1 < s ≤ 7 and 9 ≤ m < 2n, we get by induction on 2n − m that the restriction D_{s}^{(2m)}↓_{Σ_{m}} contains more composition factors D_{t}^{(m)} with 1 < t ≤ 7 than composition factors D_{t}^{(m)} with 0 ≤ t ≤ 1. Since D_{1}^{(m)} and D_{0}^{(m)} are the sign modules sgn_{m} (for m odd and even, respectively), they are 1-dimensional. Now we can conclude that some D_{s_{1}}^{(n)} ⊠ D_{s_{2}}^{(n)} with 1 < s_{1}, s_{2} ≤ 7 is a composition factor of D^{λ}↓_{Σ_{n}×Σ_{n}}.

Let φ_{1}, …, φ_{d−1} : X → Σ_{k} be embeddings of a group X of order d into symmetric groups and φ : X → Σ_{r} be any homomorphism. Consider the embedding

X → \underbrace{Σ_{k} × ⋯ × Σ_{k}}_{d−1 times} ×Σ_{r}, g ↦ (φ_{1}(g), …, φ_{d−1}(g), φ(g)).  (7)

Then the following lemma holds.

Lemma 3.9. Let D_{1}, …, D_{d−1} be faithful irreducible Σ_{k}-modules (k > 4), and D be a Σ_{r}- module. Then the restriction (D_{1} ⊠ ⋯ ⊠ D_{d−1} ⊠ D)↓_{X} contains the regular F X-module as a direct summand.

Proof. We have

D_{1} ⊠ ⋯ ⊠ D_{d−1} ⊠ D↓_{X} ≅ (D_{1}↓_{φ_{1}(X)}) ⊗ ⋯ ⊗ (D_{d−1}↓_{φ_{d−1}(X)}) ⊗ (D↓_{φ(X)}).

Since D_{i} is faithful, Σ_{k} is a subgroup of End D_{i}. As Σ_{k} has trivial center, each 1 ≠ g ∈ X does not act as a scalar on D_{i}. Therefore by Lemma 3.6, (D_{1}↓_{φ_{1}(X)})⊗⋯⊗(D_{d−1}↓_{φ_{d−1}(X)}) contains the regular X-module as a direct summand.

Proposition 3.10. Let p > 2 and let X be a subgroup of Sym(Ω), d = |X| and d > (p − 1)^{2}. Assume that X has at least d − 1 regular orbits on the set Ω, and |Ω| ≫ d(d − 1). Then for any faithful Sym(Ω)-module D we have Irr(D↓_{X}) = Irr(X).

Proof. Set n = |Ω|. We may assume that Ω = {1, 2, …, n}, and

{1, 2, …, d}, {d + 1, …, 2d}, …, {(d − 2)d + 1, …, (d − 1)d}

are regular orbits of X on Ω.

Assume first that p > 3. By Theorem B, the restriction D↓_{Σ_{d(d−1)}} contains (as a composition factor) either the natural module V = V_{d(d−1)} or V^{σ} or a completely splittable module C of height s with 1 < s < p−1. So, if we put r = n−d(d−1) then the restriction of D to the Young subgroup Σ_{d(d−1)} × Σ_{r} must contain a module of the form V ⊠ B or V^{σ} ⊠ B or C ⊠ B where B is some Σ_{r}-module.

In the first two cases it follows from Lemma 3.2 that the restriction of D to the Young subgroup H = \underbrace{Σ_{d} × ⋯ × Σ_{d}}_{(d−1) times} ×Σ_{r} contains a subfactor M_{d} ⊠ F ⊠ ⋯ ⊠ F ⊠ B or M_{d}^{σ} ⊠ F ⊠

⋯ ⊠ F ⊠ B. Note that the image of X in Sym(Ω) is contained in H. Since {1, 2, …, d} is a regular orbit of X, the restriction of D to X contains a tensor product of the regular module with some other module, hence contains the regular module as a direct summand. In particular, we get the claim of the lemma.

In the second case, Proposition 1.3 shows that the restriction of D to H contains a composition factor of the form C_{1} ⊠ ⋯ ⊠ C_{d−1} ⊠ B where all modules C_{i} are faithful completely splittable. So the result follows from Lemma 3.9.

If p = 3 then, by Theorem B, we have two possibilities: either the restriction D↓_{Σ_{d(d−1)}} contains the natural module V = V_{d(d−1)} or V^{σ} or one of the modules module D^{λ}, D^{λ}⊗sgn_{n} where λ = (λ_{1}, λ_{2}) with 1 < λ_{1} − λ_{2} ≤ 7. The first case is considered as for p > 3. In the second case Lemma 3.8 ensures that the restriction of D to H contains a composition factor of the form C_{1} ⊠ ⋯ ⊠ C_{d−1} ⊠ B with faithful modules C_{1}, …, C_{d−1}. So we again can argue as in the case p > 3.

Corollary 3.11. Let p > 2, and let X be a subgroup of Alt(Ω), d = |X| and d > (p − 1)^{2}. Assume that X has at least 2(d − 1) regular orbits on the set Ω, and |Ω| ≫ 2d(d − 1). Then, for any faithful Alt(Ω)-module E we have Irr(E↓_{X}) = Irr(X).

Proof. Obviously, there are X-invariant subsets Ω_{1}, Ω_{2}, and Ω_{3} of Ω such that |Ω_{1}| = |Ω_{2}| = n ≫ d(d − 1), Ω_{i} (i = 1, 2) involves at least d − 1 regular X-orbits, and the permutation representations X → Sym(Ω_{1}) and X → Sym(Ω_{2}) are equivalent. Therefore

X < H = {(σ, σ, τ) | σ ∈ Σ_{n}, τ ∈ Alt(Ω_{3})} < Alt(Ω).

Denote Σ = {(σ, σ, 1) | σ ∈ Σ_{n}}. Then Σ ≅ Σ_{n} and H = Σ × Alt(Ω_{3}). Clearly, the restriction E↓_{Σ} has a faithful composition factor. Therefore the restriction of E to H contains a subfactor D ⊠ B with faithful D ∈ Irr Σ. Now by Proposition 3.10, Irr(D↓_{X}) = Irr X. Therefore Irr(E↓_{X}) = Irr X, as required.

Theorem 3.12. Let p > 2 and G be a non-diagonal limit alternating group. Then G has no proper inductive systems, different from the unitary one.

Proof. Let G_{i} = Alt(Ω_{i}), i=1,2, …, and Φ = {Φ_{i}}_{i∈ℕ} be an inductive system for G = ⋃_{i∈ℕ} G_{i}. Assume that Φ_{i} contains a faithful G_{i}-module (i.e. a module different from 1_{G_{i}}). Then Φ_{j} contains a faithful G_{j}-module for any j ≥ i. We have to prove that in this case Φ_{i} = Irr(G_{i}) for every i ∈ ℕ. We may assume that i is large enough since for any k < i we have Irr((Irr G_{i})↓_{G_{k}}) = Irr(G_{k}). By Lemma 3.5, the group G_{i} has arbitrarily many regular orbits on Ω_{j}, provided j is big enough. Now Corollary 3.11 shows that the restriction of any faithful irreducible G_{j}-module to G_{i} contains all irreducible G_{i}-modules as composition factors, provided j is big enough.

It is well known that any countably infinite, locally finite, simple group G is a union of finite subgroups {G_{i} | i ∈ ℕ} such that for each i we have G_{i} ⊂ G_{i+1} and G_{i} ∩ N_{i+1} = {1} for some maximal normal subgroup N_{i+1} of G_{i+1}. The set of pairs K = {(G_{i}, N_{i}) | i ∈ ℕ} is called a Kegel sequence for G. Assume that G_{i}/N_{i} ≅ Alt(Ω_{i}) for all i. We say that G is diagonal if there exists n ∈ ℕ such that for all i > n the embedding G_{i} ⊂ G_{i+1} is diagonal, i.e. for every point α ∈ Ω_{i+1} the index |G_{i} : (G_{i})_{α}N_{i}| is 1 or |Ω_{i}|, and nondiagonal otherwise. Here (G_{i})_{α} denotes the stabilizer of α in the natural action of G_{i} on Ω_{i+1}. Assume that char F = p > 0 and all N_{i} are p-groups. Then N_{i} acts trivially on any irreducible G_{i}-module, in particular Irr G_{i} can be identified with Irr Alt(Ω_{i}). Clearly, if G is diagonal then it has a proper inductive system different from the unitary one (cf.

Proposition 3.3), so F G is not almost simple. The converse is also true. We have the following general result.

Theorem 3.13. Let {(G_{i}, N_{i}) | i ∈ ℕ} be a Kegel sequence for G with G_{i}/N_{i} ≅ Alt(Ω_{i}) for all i. Assume that G is non-diagonal and one of the following holds:

(a) char F = 0 and each N_{i} is abelian;

(b) char F = p > 0, each N_{i} is abelian and is not a p-group;

(c) char F = p > 2 and each N_{i} is a p-group.

Then F G is almost simple.

Proof. The cases (a) and (b) have been proved in [14, 1.6]. Consider the case (c). By [14, 1.7 and 3.1], for each i and k there exists n > i such that G_{i} has at least k regular orbits on Ω_{n}. Now, using Corollary 3.11 as in the proof of Theorem 3.12, we conclude that G has no proper inductive systems different from the unitary one, so F G is almost simple.

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Institute of Mathematics, Academy of Sciences of Belarus, Surganova 11, Minsk, 220072, Belarus

E-mail address: baranov@im.bas-net.by

Department of Mathematics, University of Oregon, Eugene, OR 97403, USA

E-mail address: klesh@math.uoregon.edu

School of Mathematics, University of East Anglia, Norwich, NR4 7TJ, UK

E-mail address: A.Zalesskii@uea.ac.uk