Locally Finite Simple Groups of 1-Type

Stefaan Delcroix University of Wyoming Laramie, WY 82071 delcroix@uwyo.edu

Ulrich Meierfrankenfeld Michigan State University East Lansing, MI 48824 meier@math.msu.edu

Abstract A locally finite, simple group G is called of 1-type if every Kegel cover for G has a factor which is an alternating group. In this paper we study the finite subgroups of locally finite simple groups of 1-type. We also introduce the concept of ”blockdiagonal embeddings” for groups of alternating type. We show that the groups of 1type are exactly the groups which have an alternating Kegel cover with block diagonal embeddings.

1 Introduction

Let G be a group. G is locally finite if every finite subset of G lies in a finite subgroup of G. G is finitary if there exist a field 𝕂 and a faithful 𝕂G-module V so that V/C_{V} (g) is finite dimensional for all g ∈ G.

If H is a group and Ω is an H-set, we denote by H^{Ω} the image of H in Sym(Ω). So H^{Ω} ≅ H/C_{H}(Ω).

Let G be an infinite, locally finite, simple group. Let A be the set of pairs (H, Ω) so that H is a finite subgroup of G, Ω is an H-set, |Ω| ≥ 7 and H^{Ω} = Alt(Ω).

We say that G is of alternating type if G is non-finitary and if for each finite subgroup F of G there exists (H, Ω) ∈ A such that F ≤ H and F acts faithfully on Ω.

Let G be of alternating type and F ≤ G finite. We say that F is non-regular if there exists a finite subgroup F^{∗} ≤ G with F ≤ F^{∗} and so that for all (H, Ω) ∈ A with F^{∗} ≤ H, F has no regular orbit on Ω.

G is called of non-regular alternating type if G is of alternating type and G has a non-regular finite subgroup.

Our first theorem (proven in section ??) describes the normal closure of a non-regular subgroup in (large enough) finite over-groups.

Theorem 1.1 Let G be a locally finite simple group of alternating type and F a finite nonregular subgroup of G. Then there exists a finite subgroup F ≤ F^{∗} ≤ G such that for all finite F^{∗} ≤ L ≤ G

(a) There exist normal subgroups R_{1}, …, R_{n} of ⟨F^{L}⟩ such that

⟨F^{L}⟩ = R_{1}R_{2} … R_{n}

and

R_{i} ≅ (K_{i} ≀_{Ω_{i}} Alt(Ω_{i}))′

for some finite group K_{i} and some finite set Ω_{i}.

(b) For i = 1, …, n let B_{i} be the base group of R_{i} and choose notation so that [R_{i}, F] ≰ B_{i} if and only if i ≤ m. Then

R_{1} … R_{m} = R_{1} × R_{2} × … × R_{m}.
□

Recall that a Kegel cover for G is a set K such that

(a) Each K ∈ K is a pair (H, M), where H is a finite subgroup of G and M is maximal normal subgroup of H.

(b) For each finite subgroup F of G there exists (H, M) ∈ K with F ≤ H and F ∩ M = 1.

The groups H/M, (H, M) ∈ K, are called the factors of K. K is alternating if all the factors of K are alternating groups. If K is an alternating Kegel cover, we view K as a subset of A. Indeed, if (H, M) ∈ K with H/M ≅ Alt(Ω), then H acts on Ω (with M = C_{H}(Ω)) and (H, Ω) ∈ A. This also reveals that a non-finitary locally finite simple group G is of alternating type if and only if G has an alternating Kegel cover.

Our next theorem (proven in section ??) shows that non-regular subgroups can be detected from a given alternating Kegel cover.

Theorem 1.2 Let G be a locally finite, simple group of alternating type and F a finite subgroup of G. Then F is non-regular if and only if there exists an alternating Kegel cover K and a non-negative integer t such that for all (H, Ω) ∈ K with F ≤ H, F has at most t regular orbits on Ω. □

The preceding theorem, together with [?, Proposition 1.33], shows that the groups Brian Hartley called Mf-groups of ’visual diagonal alternating type’ [?, Definition 1.31], are in fact of non-regular alternating type. Hence (see section ?? for the details) some of the non-absolutely simple, locally finite simple groups constructed in [?, Section 6] are of non-regular alternating type :

Theorem 1.3 There exist non-absolutely simple, locally finite, simple groups of non-regular alternating type. □

Let G be of alternating type

Let F ≤ G be finite. Let A_{reg}(F) be the set of all (H, Ω) ∈ A so that F ≤ H and F has a regular orbit on Ω. We say that F is regular if A_{reg}(F) is a Kegel cover for G. Note that the definition of a Kegel cover implies that F is non-regular if and only if F is not regular. G is of regular alternating type if G is locally regular, that is if every finite subgroup of G is regular.

We say that G is of ∞-type if G has the following property :

Let S be any class of finite simple groups such that every finite group can be embedded into a member of S. Then there exists a Kegel cover for G all of whose factors are isomorphic to a member of S.

We say that G is of 1-type if every Kegel cover for G has a factor which is an alternating group.

The next theorem (proven in section ??) shows the relationship between groups of 1-, ∞-, regular- and non-regular type.

Theorem 1.4 Let G be a locally finite, simple group of alternating type.

(a) G is of non-regular alternating type if and only if G is of 1-type.

(b) G is of regular alternating type if and only if G is of ∞-type.

□

Let p be a prime and G a non-finitary, locally finite, simple group. G is of p-type if every Kegel cover for G has a factor which is a classical group in characteristic p. From Theorem ?? and [?, Theorem A] we have

Theorem 1.5 Let G be a locally finite, simple group. Then exactly one of the following holds:

1. G is finitary.

2. G is of 1-type.

3. G is of p-type for a unique prime p.

4. G is of ∞-type. □

In [?] ”pseudo natural orbits” have been introduced. They are used in [?, Theorem 3.4] to devide alternating Kegel covers into two classes which Brian Hartley [?, Defintion 2.8] called RA- and DA- type. Unfortunately these two types are not disjoint. For example suppose {(G_{i}, Ω_{i}) | i = 1, 2 …} is a Kegel cover so that G_{i} = Alt(Ω_{i}), G_{i} ≤ G_{i+1} and G_{i} acts semiregulary on Ω_{i}, then this Kegel cover is both of RA and DA type. This comes from the fact that a regular orbit also is a pseudo natural orbit. In this paper we define ”block natural orbits” which avoid this problem:

Let (H, Ω) ∈ A. By [?, Lemma 2.8], there exists a unique minimal (sub)normal supplement R to C_{H}(Ω) in H. Let Λ be an H-set. An orbit Σ for H on Λ is called Ω-essential if C_{H}(Σ) ≤ C_{H}(Ω). That is if and only if R acts non-trivially on Σ. Σ is called Ω-natural if Σ is isomorphic to Ω as an H-set. Σ is called Ω-block-natural if there exists an H-invariant partition Δ of Σ so that Δ is Ω-natural and such that N_{H}(D) = C_{H}(D)C_{H}(Ω) for all D ∈ Δ. In this case, Δ is just the set of orbits of C_{H}(Ω) on Σ. Indeed, since H is transitive on Σ, N_{H}(D) is transitive on D. Hence C_{H}(Ω) is transitive on D. We remark that, since N_{H}(D)/C_{H}(Ω) ≅ Alt(|Ω| − 1) is simple, the condition N_{H}(D) = C_{H}(D)C_{H}(Ω) is equivalent to C_{H}(D) ≰ C_{H}(Ω). Λ is called Ω-block-diagonal if all the Ω-essential orbits are Ω-block-natural.

Theorems ?? and ?? reveal that groups of 1-type are loosely speaking the groups of alternating type with ”block-diagonal” embeddings.

Some of the results in this paper first appeared in [?] and some of the arguments have been developed in [?].

2 The Set-up

Proposition 2.1 (Hall’s Finitary Lemma) A locally finite simple group G which has a sectional cover composed of alternating groups and classical groups of unbounded dimension in which the natural degrees of the element g ≠ 1 are bounded, has a faithful representation as a finitary linear group.

Proof: This is [?, Corollary 3.13]. □

The reader might consult [?] for the definition of a sectional cover. For our purposes it is enough to know that every Kegel cover is a sectional cover. If H/M is a classical group or an alternating group, pdeg_{H/M}(g) denotes the natural degree of g in H/M. So if H/M = Alt(Ω), then pdeg_{H/M}(g) = deg_{Ω}(g) is the number of elements in Ω not fixed by g; if H/M is a classical group defined over a 𝕂-space V, then pdeg_{H}(g) is the minimum of all dim_{𝕂} V/W, where W is a 𝕂-subspace of V on which g acts projectively trivially. If g ∉ H, we put pdeg_{H/M}(g) = 0.

Corollary 2.2 Let G be a non-finitary, locally finite, simple group and F a finite subgroup of G. Let K be a Kegel cover for G all of whose factors are alternating or classical groups.

Let s be a positive integer. Then

K(F, s) := {(H, M) ∈ K | F ≤ H and pdeg_{H/M}(f) ≥ s, ∀1 ≠ f ∈ F }

is a Kegel cover for G.

Proof: For 1 ≠ f ∈ F, let K_{f} = {(H, M) ∈ K | pdeg_{H/M}(f) ≤ s}. Suppose that K_{f} is Kegel cover for G. Then by Hall’s Finitary Lemma applied to the sectional cover K_{f}, G is finitary, a contradiction. So K_{f} is not a Kegel cover for G. Since

K = K(F, s) ∪ ⋃_{1≠f∈F} K_{f},

the Coloring Argument [?, Lemma 3.3] implies that K(F, s) is a Kegel cover for G. □

Let D be a subset of A, F a finite subgroup of G and s a positive integer. Define

D(F, s) = {(H, Ω) ∈ D | F ≤ H and deg_{Ω}(f) ≥ s, ∀1 ≠ f ∈ F }

and

D(F) = D(F, 1).

Lemma 2.3 Let G be a locally finite, simple group of alternating type, F ≤ G finite and s a positive integer. Then there exists a finite F ≤ F^{∗} ≤ G so that A(F^{∗}) ⊆ A(F, s).

Proof: Let l be the function from [?, Lemma 2.5]. By ??, we can choose (F^{∗}, Λ) ∈ A(F, l(s)). Let (H, Ω) ∈ A(F^{∗}). Then by [?, Lemma 2.5], deg_{Ω}(f) ≥ s for all elements f of prime order in F. Since every non-trivial cyclic group contains an element of prime order, deg_{Ω}(f) ≥ s for all 1 ≠ f ∈ F. □

For the remainder of the paper, let G be locally finite, simple group of alternating type.

The following result forms the technical basis for the investigations in this paper.

Lemma 2.4 There exists an increasing function f_{reg} : ℤ^{+} → ℤ^{+} with f_{reg}(n) ≥ 9n^{2} and so that the following statement holds :

Let F be a finite subgroup of G and (H, Ω) ∈ A(F, f_{reg}(|F |)). Suppose that H is transitive and Ω-essential on a set Λ. Then one of the following holds :

1. F has a regular orbit on Λ.

2. There exist 1 ≤ t ≤ |F | − 2 and an H-invariant partition Δ of Λ so that the action of H on Δ is isomorphic to the action of H on the subsets of Ω of size t.

3 BLOCK-DIAGONALITY IN GROUPS OF NON-REGULAR ALTERNATING TYPE6

Proof: See [?, Lemma 2.14]. □

For A ∈ A, we define H_{A} and Ω_{A} by A = (H_{A}, Ω_{A}). Let D ⊆ A. We say that D is Kegel cover for G if {(H, C_{H}(Ω)) | (H, Ω) ∈ D} is Kegel cover for G.

Let D ⊆ A be a Kegel cover for G. Let F be a finite subgroup of G. Put

D^{∗}(F) = D(F, f_{reg}(|F |).

We say that F is D-regular if there exists D ∈ D^{∗}(F) so that F has a regular orbit on Ω_{D}. In other words, F is D-regular if and only if D^{∗}(F) ∩ A_{reg}(F) ≠ ∅. We will prove in Theorem ?? that F is D-regular if and only if F is regular.

3 Block-Diagonality in Groups of Non-Regular Alternating Type

We continue to use the notation introduced in the previuos section. In particular, G is a locally finite, simple group of alternating type.

Proposition 3.1 Let F be a finite subgroup of G, (H, Ω) ∈ A^{∗}(F) ∩ A_{reg}(F) and Σ an H-set. Then F has a regular orbit on each Ω-essential orbit for H on Σ. In particular, A(H) ⊆ A_{reg}(F).

Proof: Let Λ be an Ω-essential orbit for H on Σ. We need to show that F has a regular orbit on Λ. So we may assume that (2) in ?? holds. Since F has a regular orbit on Ω, there exists ω ∈ Ω with C_{F} (ω) = 1. Since |Ω| ≥ f_{reg}(|F |) ≥ 9|F |^{2} ≥ 2|F |, there exists a subset U of Ω of size t with U ∩ ω^{F} = {ω}. Then N_{F} (U) ≤ C_{F} (ω) = 1 and F has a regular orbit on Δ. Hence F has a regular orbit on Λ. □

Proposition 3.2 Let F be a finite subgroup of G, (H, Ω) ∈ A^{∗}(F) and A ∈ A with H ≤ H_{A}. Suppose that Λ is an Ω-essential orbit for H on Ω_{A} and ω ∈ Ω such that

(i) F has no regular orbit on Λ.

(ii) There does not exist λ ∈ Λ with C_{F} (λ) ⪇ C_{F} (ω).

Then

(a) There exists an H-invariant partition Δ of Λ so that Δ ≅ Ω as an H-set.

(b) If ω̌ is the element in Δ corresponding to ω, then C_{F} (ω) = N_{F} (ω̌) = C_{F} (ω̌) = C_{F} (λ) for all λ ∈ ω̌.

Proof: Note that all the assumptions of ?? are fullfilled. By (i), ??(1) does not hold. So we can choose t and Δ as in ??(2).

Let 1 ≠ f ∈ C_{F} (ω). Suppose that t ≠ 1. Since deg_{Ω}(f) ≥ 9|F |^{2} ≥ 2|F |, there exists ρ ∈ Ω with ρ ∉ ω^{F} and ρ ≠ ρ^{f}. Since t ≤ |F | − 2, there exists a subset U of Ω of size t with ρ ∈ U, ρ^{f} ∉ U and U ∩ ω^{F} = {ω}. Then N_{F} (U) ≤ C_{F} (ω) and f ∉ N_{G}(U). Thus N_{F} (U) ⪇ C_{F} (ω).

Let δ ∈ Δ so that δ corresponds to U. Note that δ is a subset of Λ and pick λ ∈ δ. Then

C_{F} (λ) ≤ N_{F} (δ) = N_{F} (U) ⪇ C_{F} (ω),

a contradiction to the assumptions.

Thus t = 1 and (a) holds. For (b), pick λ ∈ ω̌ and note that

C_{F} (λ) ≤ N_{F} (ω̌) = C_{F} (ω).

This implies that C_{F} (λ) = C_{F} (ω) = N_{F} (ω̌) = C_{F} (ω̌). So (b) holds. □

Theorem 3.3 Let G be a locally finite, simple group of alternating type and F a finite subgroup of G. Then the following are equivalent :

1. F is not A-regular.

2. F is not D-regular for some alternating Kegel cover D for G.

3. There exists an alternating Kegel cover D for G and a non-negative integer t such that for all A ∈ D(F), F has at most t regular orbits on Ω_{A}.

4. F is non-regular.

Proof: Clearly (1) implies (2).

Suppose (2) holds. By ??, D^{∗}(F) is a Kegel cover for G. Hence (3) holds with t = 0 and D^{∗}(F) in place of D.

Suppose (3) holds but (4) does not. Then F is regular and so A_{reg}(F) is a Kegel cover for G. By ??, there exists (H, Ω) ∈ A_{reg}(F) ∩ A^{∗}(F). Let A ∈ D(H). By assumption, F has at most t regular orbits on Ω_{A}. Hence by ??, H has at most t Ω-essential orbits on Ω_{A}. Let R be the minimal normal supplement to C_{H}(Ω) in H. As each Ω-essential H-orbit has size at most |H| and since R acts trivially on the non-Ω-essential orbits, deg_{Ω_{A}}(x) ≤ t|H| for all 1 ≠ x ∈ R. Hence by Hall’s Finitary Lemma ??, G is finitary, a contradiction. So (3) implies (4).

Suppose finally that (4) holds but (1) does not. Then there exists (H, Ω) ∈ A_{reg}(F) ∩ A^{∗}(F). By ??, A(H) ⊆ A_{reg}(F). As A(H) is a Kegel cover for G, so is A_{reg}(F). So F is regular, a contradiction. □

Let F be a finite, non-regular subgroup of G. Let M_{F} be the set of all E ≤ F so that E = C_{F} (ω) for some (H, Ω) ∈ A^{∗}(F) and ω ∈ Ω. Note that E ≠ 1 for all E ∈ M_{F}. Let M_{F}^{∗} be the set of minimal elements of M_{F}. We say that ω is F -extreme if C_{F} (ω) ∈ M_{F}^{∗}. Let B_{F} be the set of (H, Ω) ∈ A^{∗}(F) so that there exists an F -extreme ω ∈ Ω. Let B be the union of the B_{F} ’s as F runs through the non-regular finite subgroups of G.

Theorem 3.4 Let G be a locally finite, simple group of alternating type and F a finite non-regular subgroup of G. Then the following holds :

(a) Let (H, Ω) ∈ B_{F} and A ∈ A^{∗}(F) with H ≤ H_{A} and C_{H}(Ω_{A}) ≤ C_{H}(Ω). Then A ∈ B_{F} and H is Ω-block-diagonal on Ω_{A}.

(b) Let (H, Ω) ∈ B_{F}. Then A(H) ∩ A^{∗}(F) ⊆ B_{F}.

(c) Let A, B ∈ B_{F} with H_{A} ≤ H_{B}. Then H_{A} is Ω_{A}-block-diagonal on Ω_{B}.

(d) Both B_{F} and B are Kegel covers for G.

Proof: (a) Since C_{H}(Ω_{A}) ≤ C_{H}(Ω), there exists an Ω-essential orbit Λ for H on Ω_{A}. Let Λ be any Ω-essential orbit for H on Ω_{A}. Let ω ∈ Ω be F -extreme. Let Δ be the H-invariant partition of Λ, given by ??(a). Let ω̌ be the element of Δ, corresponding to ω and λ ∈ ω̌. By ??(b),

C_{F} (ω) = C_{F} (ω̌) = C_{F} (λ).

In particular, C_{F} (λ) = C_{F} (ω) ∈ M_{F}^{∗}, λ is F -extreme and A ∈ B_{F}.

Since C_{F} (ω̌) = C_{F} (ω) ≠ 1 and F is faithful on Ω, C_{F} (ω̌) ≰ C_{H}(Ω). Hence C_{H}(ω̌) ≰ C_{H}(Ω) and Λ is Ω-block-natural. So Ω_{A} is Ω-block-diagonal.

(b) Follows from (a).

(c) If C_{H_{A}}(Ω_{B}) ≰ C_{H_{A}}(Ω_{A}), H_{A} has no Ω_{A}-essential orbits on Ω_{B}. So (c) holds in this case. If C_{H_{A}}(Ω_{B}) ≤ C_{H_{A}}(Ω_{A}), we can apply (a) and again (c) holds.

(d) Let (H, Ω) ∈ B_{F}. By ??, A(H) ∩ A^{∗}(F) is a Kegel cover for G. By (b), A(H) ∩ A^{∗}(F) ⊆ B_{F} ⊆ B. So (d) holds. □

4 Groups Acting Block-Diagonally on a Set

Let (H, Ω) ∈ A and Σ ⊆ Ω. If |Ω \ Σ| ≥ 5, let R_{Σ} be the minimal normal supplement to C_{H}(Ω) in C_{H}(Σ); otherwise put R_{Σ} = 1. Put R = R_{∅}.

Suppose that H is Ω-block-diagonal on a set Λ. Let Λ^{∗} be the union of the Ω-essential orbits for H on Λ. As every orbit for H on Λ^{∗} is Ω-block-natural, there exists an Hinvariant partition Δ of Λ^{∗} so that Ω and Δ are isomorphic as H-sets. Note that this isomorphism is unique. Let Σ̃ denote the image of Σ in Δ under this H-isomorphism. Each D ∈ Σ̃ is a subset of Λ^{∗}. Let Σ̂ = ⋃ Σ̃ be the union of these subsets. So Σ̂ ⊆ Λ^{∗} and N_{H}(Σ̂) = N_{H}(Σ̃) = N_{H}(Σ). Define HΣ̂ = C_{H}(Σ̂) and H_{Σ} = C_{H}(Σ). Then HΣ̂ ≤ H_{Σ}. Note that C_{H}(Ω) = H_{Ω} ≤ H_{Σ} but H_{Ω} ≰ HΣ̂, unless H_{Ω} acts trivially on Λ^{∗} or Σ = ∅.

Lemma 4.1 Let (H, Ω) ∈ A. Suppose that H is Ω-block-diagonal on a set Λ. Let Σ ⊆ Ω.

(a) If |Ω \ Σ| ≥ 5, then

H_{Σ} = HΣ̂H_{Ω} = (HΣ̂ ∩ R)H_{Ω} and R_{Σ} ≤ HΣ̂ ∩ R.

(b) Both ⋂_{ω∈Ω} R_{ω} and R ∩ HΩ̂ act trivially on Λ.

(c) Let Σ_{1}, Σ_{2} ⊆ Ω with Ω = Σ_{1} ∪ Σ_{2}. If H is faithful on Λ, then

[R_{Σ_{1}}, R_{Σ_{2}}] = [HΣ̂_{1} ∩ R, HΣ̂_{2} ∩ R] = [HΣ̂_{1}, HΣ̂_{2}] ∩ R = 1.

Proof: (a) Let ω ∈ Ω. Let Ξ be an orbit for H on Λ such that R acts non-trivial on Ξ. Then Ξ is Ω-essential for H and as Λ is Ω-block-diagonal, Ξ is Ω-block-natural. Hence

H_{ω} = C_{H}(Ξ ∩ ω̂)H_{Ω}.

Thus R_{ω} ≤ C_{H}(Ξ ∩ ω̂). As Ξ was an arbitrary Ω-essential orbit for H on Λ, R_{ω} ≤ Hω̂. Thus

H_{ω} = R_{ω}H_{Ω} = Hω̂H_{Ω}.

Let ω ∈ Σ. We compute

H_{Σ} = H_{ω} ∩ H_{Σ} = (Hω̂H_{Ω}) ∩ H_{Σ} = (Hω̂ ∩ H_{Σ})H_{Ω}.

By minimality of R_{Σ}, R_{Σ} ≤ Hω̂∩H_{Σ} ≤ Hω̂. As this is true for all ω ∈ Σ and HΣ̂ = ⋂_{ω∈Σ} Hω̂, R_{Σ} ≤ HΣ̂. As H_{Σ} = R_{Σ}H_{Ω}, H_{Σ} = HΣ̂H_{Ω}. Finally, H = RH_{Ω} implies H_{Σ} = (R ∩ H_{Σ})H_{Ω} and so R_{Σ} ≤ R ∩ H_{Σ}. Hence R_{Σ} ≤ HΣ̂ ∩ R and H_{Σ} = R_{Σ}H_{Ω} = (HΣ̂ ∩ R)H_{Ω}. This proves (a).

(b) Note that Ω̂ = Λ^{∗} and so HΩ̂ acts trivially on Λ^{∗}. Since R acts trivially on each of the non-Ω-essential orbits for H on Λ, R acts trivially on Λ \ Λ^{∗}. So HΩ̂ ∩ R acts trivially on Λ. By (a), ⋂_{ω∈Ω} R_{ω} ≤ ⋂_{ω∈Ω}(Hω̂ ∩ R) = HΩ̂ ∩ R. So (b) holds.

(c) Note that HΣ̂_{1} ≤ H_{Σ_{1}} ≤ H_{Ω\Σ_{2}} ≤ N_{H}(Σ_{2}) ≤ N_{H}(HΣ̂_{2}). Similarly, HΣ̂_{2} ≤ N_{H}(HΣ̂_{1}). So

[HΣ̂_{1}, HΣ̂_{2}] ≤ HΣ̂_{1} ∩ HΣ̂_{2} ≤ H{ \atop Σ_{1}∪Σ_{2}} = HΩ̂.

Using (a) we get that

[R_{Σ_{1}}, R_{Σ_{2}}] ≤ [HΣ̂_{1} ∩ R, HΣ̂_{2} ∩ R] ≤ [HΣ̂_{1}, HΣ̂_{2}] ∩ R ≤ HΩ̂ ∩ R.

Since H is faithful on Λ, (b) implies HΩ̂ ∩ R = 1. Thus (c) holds. □

Theorem 4.2 Let G be a locally finite, simple group of alternating type and D an alternating Kegel cover for G. Suppose that there exists (H, Ω) ∈ A such that for all A ∈ D(H), H is Ω-block-diagonal on Ω_{A}. Then H is non-regular and G is of non-regular alternating type.

Proof: By ??, D^{∗}(H) is a Kegel cover for G. By ??(2), it suffices to show that for all A ∈ D^{∗}(H), H does not have a regular orbit on Ω_{A} . Pick A ∈ D^{∗}(H) and λ ∈ Ω_{A}. Put Λ = λ^{H}. Suppose that Λ is regular. If Λ is not Ω-essential, then 1 ≠ R ≤ C_{H}(λ), a contradiction. Hence Λ is Ω-essential and so λ ∈ ω̂ for some ω ∈ Ω. By Lemma ??(a), 1 ≠ R_{ω} ≤ Hω̂ ≤ C_{H}(λ), a contradiction. So Λ is not regular. □

Theorem 4.3 Let (H, Ω) ∈ A and suppose that H is faithful and Ω-block-diagonal on some set. Let R be the minimal normal supplement to H_{Ω} in H. Let ω ∈ Ω and put K = C_{R}(ω)/R_{ω}. Then

R ≅ (K ≀_{Ω} Alt(Ω))′.

Proof: To simplify notation, we assume that Ω = {1, …, n} and ω = 1. For i ∈ Ω, pick r_{i} ∈ R with 1^{r_{i}} = i. Let π : R → Alt(Ω) be the homomorphism arising from the action of R on Ω. Note that r_{i}gr_{i^{g}}^{−1} ∈ C_{R}(1) for all g ∈ R and all i ∈ Ω. Hence we obtain a map

ϕ : R → K ≀_{Ω} Alt(Ω) : g → ((r_{i}gr_{i^{g}}^{−1}R_{1})_{i∈Ω}, π(g)).

We will first show that ϕ is a homomorphism. Indeed let g, h ∈ R. Then

ϕ(g)ϕ(h) = ((r_{i}gr_{i^{g}}^{−1}R_{1})_{i∈Ω}, π(g))((r_{i}hr_{i^{h}}^{−1}R_{1})_{i∈Ω}, π(h))
= ((r_{i}gr_{i^{g}}^{−1}r_{i^{g}}hr_{i^{gh}}^{−1}R_{1})_{i∈Ω}, π(g)π(h))
= ((r_{i}ghr_{i^{gh}}^{−1}R_{1})_{i∈Ω}, π(gh))
= ϕ(gh).

If ϕ(g) = 1, then π(g) = 1 and so

ϕ(g) = ((r_{i}gr_{i}^{−1}R_{1})_{i∈Ω}, 1).

Thus r_{i}gr_{i}^{−1} ∈ R_{1} and g ∈ R_{1}^{r_{i}} = R_{i} for all i ∈ Ω. By ??(b), g = 1. So ϕ is one-to-one.

Put D = ϕ(R) and S = K ≀_{Ω} Alt(Ω). It remains to show that D = S′. For i ∈ Ω let D_{i} = ϕ(R_{i}). Also let B be the base group of S. Then B = ⨁_{i=1}^{n} B_{i} with B_{i} ≅ K for all i ∈ Ω. Note that B ∩ D = ϕ(C_{R}(Ω)). Since C_{H}(1) = R_{1}C_{H}(Ω) and R_{1} ≤ R we have C_{R}(1) = R_{1}C_{R}(Ω). As K = C_{R}(1)/R_{1}, we conclude that B ∩ D projects onto B_{i} for all i ∈ Ω. For i ∈ Ω, put Q_{i} = B_{i} ∩ D. As B ∩ D normalizes Q_{i}, Q_{i} ⊴ B_{i} for all i ∈ Ω. Let Q = ⨁_{i∈Ω} Q_{i}. Then Q ⊴ BD = S.

Put E = ⟨B ∩ D_{i} | i ∈ Ω⟩ and D^{∗} = ⟨D_{i} | i ∈ Ω⟩. Note that both E and D^{∗} are normal in D and D = D^{∗}(B ∩ D). By the definition of R, R has no proper normal supplement to C_{R}(Ω). Thus D = D^{∗}.

Let i ∈ Ω. Then [C_{R}(Ω), R_{i}] ≤ C_{R}(Ω) ∩ R_{i} and so [B ∩ D, D_{i}] ≤ B ∩ D_{i}. As D = D^{∗}, we get [B ∩ D, D] ≤ E and D/E is a perfect central extension of D/B ∩ D ≅ Alt(Ω). Note that D_{1}/D_{1} ∩ B ≅ Alt(Ω \ {1}). Since D_{1} ≰ E and D_{1} ∩ E = D_{1} ∩ B we conclude D_{1}E/E ≅ D_{1}/D_{1} ∩ E = D_{1}/D_{1} ∩ B ≅ Alt(Ω \ {1}). From the Schur multiplier of Alt(Ω) [?] and since |Ω| ≥ 7 we see that no non-trivial,perfect, central extension of Alt(Ω) has a subgroup isomorphic to Alt(Ω \ {1}). Thus B ∩ D = E. Let i ≠ j ∈ Ω. As [B ∩ D, D_{i}] ≤ B ∩ D_{i}, D_{i} normalizes (B ∩ D_{j})(B ∩ D_{i}). Since D_{i} acts transitively on Ω \ {i}, we get B ∩ D_{k} ≤ (B ∩ D_{j})(B ∩ D_{i}) for all k ∈ Ω. Thus

E = (B ∩ D_{i})(B ∩ D_{j}), ∀i ≠ j ∈ Ω.

Put

X = [B ∩ D, B ∩ D_{2}, …, B ∩ D_{n}].

Since B ∩ D_{j} projects trivially onto B_{j} for all 2 ≤ j ≤ n, X ≤ B_{1} and so X ≤ Q_{1}. Since B ∩ D = E = (B ∩ D_{j})(B ∩ D_{1}) for all 2 ≤ j ≤ n,

X(B ∩ D_{1}) = \underbrace{[B ∩ D, …, B ∩ D]}_{n terms}(B ∩ D_{1}).

Projecting this equation on B_{1} and using that the projection of B ∩ D_{1} is trivial, we get

\underbrace{[B_{1}, …, B_{1}]}_{n terms} = X ≤ Q_{1}.

So B_{1}/Q_{1} is nilpotent. Hence also B/Q is nilpotent.

Suppose for the moment that B is abelian. Let B_{0} = {b ∈ B | ∑ b_{i} = 0}. So [B, S] = [B, D] = B_{0} and S′ = B_{0} Alt(Ω). As D is perfect, D ≤ S′ and so E = B ∩ D ≤ B_{0}.

Put L = ⟨e_{i}−e_{j} | e ∈ E, 1 ≤ i < j ≤ n⟩ ≤ K and Y = {b ∈ B | b_{i} ∈ L}. Note that Y ∩B_{0} is generated by elements of the form (0, …, 0, e_{i} − e_{j}, 0, …, 0, e_{j} − e_{i}, 0, …, 0) where the non-zero entries are in arbitrary positions and (e_{1}, …, e_{n}) ∈ E. Let e = (e_{1}, …, e_{n}) ∈ E. Pick s_{1}, s_{2} ∈ R with π(s_{1}) = (1, 2, 3) and π(s_{2}) = (1, 4)(5, 6). Then

[e, ϕ(s_{1}), ϕ(s_{2})] = [(e_{3} − e_{1}, e_{1} − e_{2}, e_{2} − e_{3}, 0, …, 0), ϕ(s_{2})] = (e_{1} − e_{3}, 0, 0, e_{3} − e_{1}, 0, …, 0).

We conclude that Y ∩ B_{0} ≤ E. Pick e = (e_{1}, …, e_{n}) ∈ B ∩ D_{i} ≤ E. Since B ∩ D_{i} projects trivially on B_{i}, e_{i} = 0. Hence e_{j} = e_{j}−e_{i} ∈ L for all j ∈ Ω and e ∈ Y. So B∩D_{i} ≤ Y and the definition of E implies E ≤ Y. Thus Y ∩ B_{0} ≤ E ≤ Y ∩ B_{0} and E = Y ∩ B_{0}. As E = B ∩ D projects onto B_{1}, we get that K = L, B = Y and E = B_{0}. So D ≤ S′ ≤ B_{0}D = D and S′ = D.

Thus the theorem holds if B is abelian. More importantly, note that all the above arguments are valid in S/B′ and so S′ = DB′. Hence B_{0} = EB′ where B_{0} = S′ ∩ B. For j = 2, 3, pick d_{j} ∈ D with 1^{d_{j}} = j. Then for j = 2, 3, [B_{1}, d_{j}] ≤ B_{1}B_{j} and [B_{1}, d_{j}] projects onto B_{1}. Thus

[[B_{1}, d_{2}], [B_{1}, d_{3}]] = [B_{1}, B_{1}].

Hence B′_{0} = B′ and B_{0} = EB′_{0}. As B/Q is nilpotent and Q ≤ E ≤ B_{0}, B′_{0}Q/Q ≤ Φ(B_{0}/Q). So B_{0} = EQ = E and S′ = DB′ = DB′_{0} = DE′ = D. □

The preceding theorem is false for |Ω| = 6 and thus our assumption that |Ω| ≥ 7 for all (H, Ω) ∈ A. Indeed, let 3. Alt(6) be the 3-cover of Alt(6). Then 3. Alt(6) has Alt(5) as a subgroup. Also, 3. Alt(6) acts faithfully and block-natural on the cosets of Alt(5). But 3. Alt(6) is not the derived group of a wreath-product.

5 GROUPS OF 1-TYPE AND ∞-TYPE 12

5 Groups of 1-Type and ∞-Type

The main goal of this section is to prove theorems ?? and ??. For this, we first prove a couple of technical lemmas.

Lemma 5.1 Let H be a group, 𝕂 a field and 0 ≤ X ≤ Y a chain of 𝕂H-modules. Let P be its stabilizer in GL_{𝕂}(Y). If H acts projectively non-trivially on Y/X, then

X = [Y, [P, H]].

Proof: Let Z = [Y, [P, H]] and suppose that Z ≠ X. Then Z ≤ X. Replacing 0 ≤ X ≤ Y by 0 ≤ X/Z ≤ Y/Z, we may assume that Z = 0. So [Y, [P, H]] = 0 and [P, H] = 1. It is well-known and easily verified that C_{GL_{𝕂}(Y)}(P) ≤ Z(GL_{𝕂}(Y))P. But this implies that H acts as scalars on Y/X, a contradiction. □

Lemma 5.2 Let T be a finite group, n ∈ ℤ^{+} and T_{1}, T_{2}, … T_{n} non-trivial subgroups of T. Suppose that T ≤ PGL_{𝕂}(V) and that pdeg_{V} (t) ≥ (n + 1)|T |^{2} for all 1 ≠ t ∈ T. Then there exists a T -invariant unipotent subgroup Q ≤ PGL_{𝕂}(V) with T ∩ Q = 1 and

[[Q, T_{1}], [Q, T_{2}]], …, [Q, T_{n}]] ≠ 1.

Proof: For 1 ≠ t ∈ T, let U_{t} be a one-dimensional subspace of V such that (U_{t})^{t} ≠ U_{t}. Put V_{1} = ⟨U_{t}^{T} | 1 ≠ t ∈ T ⟩. Then dim_{𝕂} V_{1} ≤ |T |^{2}, T acts projectively faithfully on V_{1} and pdeg_{V/V_{1}}(t) ≥ n|T |^{2} for all 1 ≠ t ∈ T.

An easy induction argument now shows that there exists an ascending chain

0 = V_{0} ≤ V_{1} ≤ V_{2} ≤ V_{n} ≤ V_{n+1} ≤ V

of T -submodules so that T acts (projectively) faithfully on V_{i+1}/V_{i} and dim_{𝕂} V_{i+1}/V_{i} ≤ |T |^{2} for all 0 ≤ i ≤ n. Let P ≤ GL_{𝕂}(V) be the stabilizer of this chain and Q the image in PGL_{𝕂}(V) of P. For i = 1, …, n, let S_{i} be the pre-image in GL_{𝕂}(V) of T_{i} and P_{i} = [P, S_{i}]. Put A_{0} = C_{P} (V/V_{1}), A_{1} = [A_{0}, S_{1}] ≤ P_{1} and inductively A_{i} = [A_{i−1}, P_{i}] for i = 2, …, n.

We will prove by induction that

(∗) [V_{i}, A_{i}] = 0 and [V, A_{i}] = [V_{i+1}, A_{i}] = V_{1}

for all 1 ≤ i ≤ n. Note first that A_{i} ≤ A_{0} for all i and so [V_{i+1}, A_{i}] ≤ [V, A_{i}] ≤ [V, A_{0}] ≤ V_{1}. By ?? applied to 0 ≤ V_{1} ≤ V_{2} and H = S_{1}, [V_{2}, [A_{0}, S_{1}]] = V_{1}. Thus (*) holds for i = 1.

Suppose that (∗) holds for i − 1. Then

[V_{i}, P_{i}, A_{i−1}] ≤ [V_{i−1}, A_{i−1}] = 0

and

[V_{i}, A_{i−1}, P_{i}] = [V_{1}, P_{i}] = 0.

Thus by the Three Subgroup Lemma,

[V_{i}, [A_{i−1}, P_{i}]] = 0.

So the first statement in (*) holds.

By ?? applied to 0 ≤ V_{i}/V_{i−1} ≤ V_{i+1}/V_{i−1} and H = S_{i},

[V_{i+1}, [P, S_{i}]] + V_{i−1} = V_{i}.

Taking the commutator with A_{i−1} on both sides (and using the induction assumption), we conclude

[V_{i+1}, P_{i}, A_{i−1}] = V_{1}.

Also

[V_{i+1}, A_{i−1}, P_{i}] ≤ [V, A_{0}, P_{i}] ≤ [V_{1}, P_{i}] = 0.

Thus by the Three Subgroup Lemma,

[V_{i+1}, [A_{i−1}, P_{i}]] = V_{1}.

Hence (*) holds for all 1 ≤ i ≤ n. In particular, [V, A_{n}] = V_{1} and A_{n} ≰ Z(GL_{𝕂}(V)). Note that

A_{n} ≤ [[P, S_{1}], [P, S_{2}], …, [P, S_{n}]]

and the lemma follows by considering the image of the last equation in PGL_{𝕂}(V). □

Theorem 5.3 Let G be a locally finite, simple group of alternating type and F a finite subgroup of G. Then the following are equivalent :

1. F is regular.

2. Let L be a finite group and ϕ : F → L an embedding. Then there exists a finite subgroup E of G, containing F, and an epimorphism ξ : E → L with ϕ = ξ |_{F}.

Proof: Suppose (1) holds. Let ϕ : F → L be as in (2). By ??(3), there exists (H, Ω) ∈ A(F) so that F has at least |L|/|F | regular orbits on Ω and |Ω| ≥ |L| + 2. In particular, there exists an F -invariant subset Λ of Ω of size |L| so that all the orbits of F on Λ are regular. Let ρ : L → Sym(Λ) be an embedding such that ρ(L) acts regularly on Λ. Let ψ : N_{H}(Λ) → Sym(Λ) be the homomorphism arising from the action of N_{H}(Λ) on Λ. As both ψ(F) and ρϕ(F) act semi-regularly on Λ, there exists an inner automorphism τ of Sym(Λ) with ψ |_{F} = τρϕ. Thus

ψ(F) = τρϕ(F) ≤ τρ(L) ≤ Sym(Λ).

As |Ω| ≥ |L| + 2, |Ω \ Λ| ≥ 2. Since H^{Ω} = Alt(Ω), ψ(N_{H}(Λ)) = Sym(Λ). Let E = ψ^{−1}(τ(ρ(L))). Since ψ(F) ≤ τρ(L), F ≤ E. Since ρ : L → Sym(Λ) is one-to-one, there exists a partial inverse ρ^{∗} : ρ(L) → L. Put ξ = ρ^{∗}τ^{−1}ψ|_{E}: E → L. As ψ, τ^{−1} and ρ^{∗} are onto, ξ(E) = L and

ξ |_{F} = ρ^{∗}τ^{−1}ψ |_{F} = ρ^{∗}τ^{−1}τρϕ = ϕ.

So (1) implies (2).

Suppose that (2) holds. Let s ≥ f_{reg}(|F |) with s even, Ω a set with |Ω| = s|F | and L = Alt(Ω). Let ϕ : F → L be an embedding so that ϕ(F) is semi-regular on Ω. Let E and ξ be given by (2). Then (E, Ω) ∈ A^{∗}(F) and all the orbits for F on Ω are regular. Hence F is A-regular and so by ?? F is regular. □

We remark that the preceding theorem remains true if in part 2. ”ϕ : F → L an embedding” is replaced by ”ϕ : F → L a homomorphism. ” Indeed suppose that α : F → L is a homomorphism. Define ϕ : F → F × L, f → (f, α(f)) and π : F × L → L, (f, l) → l. Then ϕ is one-to-one and α = πϕ. So if ξ : E → F ×L is onto with ξ |_{F} = ϕ, then πξ : E → L is onto with (πξ) |_{F} = α.

Proof of Theorem ??:

Suppose first that G is of non-regular type. Let F be a finite, non-regular subgroup of G. Assume for a contradiction that G is not of 1-type. Then there exists a Kegel cover K for G, none of whose factors are alternating groups. By [?, Proposition 3.2(b)], we may assume that all the factors of K are of the form P SL_{𝕂}(V) for some finite field 𝕂 and a 𝕂-vectorspace V. Let (T, Σ) ∈ B_{F} with |Σ| ≥ 10. By ??, we can choose (L, M) ∈ K(T) so that pdeg_{L/M}(t) ≥ 9|T |^{2} for all 1 ≠ t ∈ T. For i = 1, 2 pick Σ_{i} ⊆ Σ so that Σ = Σ_{1} ∪Σ_{2} and |Σ \ Σ_{i}| ≥ 5. For i = 1, 2, let T_{i}^{∗} be the minimal normal supplement to C_{T} (Σ) in C_{T} (Σ_{i}). By ??, there exists a T -invariant unipotent subgroup Q/M of L/M so that T ∩ Q = 1 and

[[Q/M, T_{1}^{∗}], [Q/M, T_{2}^{∗}]] ≠ 1.

Note that Q/M is a p-group for some prime p. Put H = T Q. Then Σ is an H-set with Q acting trivially on Σ and (H, Σ) ∈ B_{F}. Pick A ∈ B_{F} (H). By ??(c), both T and H are faithful and Σ-block-diagonal on Ω_{A}. We use the notation introduced just before Lemma ??. Let R be the minimal normal supplement to H_{Σ} in H. Note that H_{Σ} = QT_{Σ}. By ??(a),

H_{Σ_{i}} = (HΣ̂_{i} ∩ R)H_{Σ} = (HΣ̂_{i} ∩ R)QT_{Σ}.

Hence T_{Σ_{i}} = (((HΣ̂_{i} ∩ R)Q) ∩ T)T_{Σ} and the minimality of T_{i}^{∗} implies T_{i}^{∗} ≤ (HΣ̂_{i} ∩ R)Q. Since T_{i}^{∗} is perfect and (HΣ̂_{i} ∩ R)Q/(HΣ̂_{i} ∩ R)M is a p-group, T_{i}^{∗} ≤ (HΣ̂_{i} ∩ R)M. Thus

[[Q, T_{1}^{∗}], [Q, T_{2}^{∗}]] ≤ ([HΣ̂_{1}, HΣ̂_{2}] ∩ R)M.

By ??(c), [[Q, T_{1}^{∗}], [Q, T_{2}^{∗}]] ≤ M, a contradiction.

So non-regular type implies 1-type.

Suppose next that G is locally regular. We will show that G is of ∞-type. So let S be a class of finite simple groups such that every finite group is embedded into a member of S.

Let F be a finite subgroup of G. Pick L ∈ S such that F is embedded in L. By ??, there exist a finite subgroup E of G and M ⊴ E such that F ≤ E, F ∩ M = 1 and E/M ≅ L. Put K_{F} = (E, M). Then {K_{F} | F is a finite subgroup of G} is a Kegel cover for G all of whose factors are isomorphic to members of S.

So we proved that locally regular implies ∞-type.

Suppose next that G is of 1-type. Then clearly G is not of ∞-type and so also not locally regular. Thus G is non-regular.

Suppose finally that G is of ∞-type. Then G is clearly not of 1-type. As non-regular implies 1-type, we conclude that G is locally regular. □

Proposition 5.4 Let G be a locally finite, simple group of alternating type and F a nonregular subgroup. Then there exists a finite F ≤ F̃ ≤ G such that for all finite F̃ ≤ H ≤ G and all maximal normal subgroups M of H with M ∩ F̃ = 1, there exists a finite set Ω such that

(a) H/M ≅ Alt(Ω).

(b) F has no regular orbit on Ω.

(c) (H, Ω) ∈ B_{F}

Proof: Let U be the set of pairs (H, M) where H is a finite subgroup of G, M is a maximal normal subgroup of H and H/M is not isomorphic to an alternating group. By ??(a), G is of 1-type and so U is not a Kegel cover for G. Hence there exists a finite subgroup F_{1} with U(F_{1}) = ∅. Let s = f_{reg}(|F |). By ??, there exists a finite F ≤ F_{2} ≤ G with A(F_{2}) ⊆ A(F, s) = A^{∗}(F). Since B_{F} is a Kegel cover for G, there exists (F̃, Σ) ∈ B_{F} with ⟨F_{1}, F_{2}⟩ ≤ F̃.

Let H and M be as in the proposition. Since U(F_{1}) = ∅, H/M ≅ Alt(Ω) for some set Ω. Thus (H, M) ∈ A(F̃). Since F_{2} ≤ F̃, (H, Ω) ∈ A^{∗}(F). In particular, since F is non-regular, F has no regular orbit on Ω. Finally, (c) follows from (F̃, Σ) ∈ B_{F} and ??(b). □

Proof Of Theorem ?? :

Let F be a non-regular finite subgroup of G. Let F̃ be given by ??. For 1 ≠ d ∈ G, we have that G = ⟨d^{G}⟩ = ⟨d^{⟨d^{G}⟩}⟩ and so we can choose a finite F̃ ≤ F^{∗} ≤ G with

F̃ ≤ ⟨d^{⟨d^{F^{∗}}⟩}⟩

for all 1 ≠ d ∈ F̃.

Let F^{∗} ≤ L ≤ G be finite and put H = ⟨F^{L}⟩. Then F̃ ≤ H. Let {M_{1}, …, M_{n}} be the set of all maximal normal subgroups of H. Pick i ∈ {1, …, n}. Since ⟨F^{L}⟩ ≰ M_{i}, F ≰ M_{i}^{l_{i}} for some l_{i} ∈ L. Suppose that 1 ≠ d ∈ F̃ ∩ M_{i}^{l_{i}}. Then

F ≤ F̃ ≤ ⟨d^{⟨d^{F^{∗}}⟩}⟩ ≤ ⟨d^{H}⟩ ≤ M_{i}^{l_{i}},

a contradiction. So F̃ ∩ M_{i}^{l_{i}} = 1. Thus by ??(a)(c), H/M_{i}^{l_{i}} ≅ Alt(Ω_{i}) for some set Ω_{i} and (H, Ω_{i}) ∈ B_{F}. Note that M_{i}^{l_{i}} = C_{H}(Ω_{i}). Let R_{i} be the minimal normal supplement to M_{i} in H. By ??(d), there exists A ∈ B_{F} (H). By ??(c), H is faithful and Ω_{i}-block-diagonal on Ω_{A}. By ??, R_{i}^{l_{i}} ≅ R_{i} ≅ (K_{i} ≀_{Ω_{i}} Alt(Ω_{i}))′ for some finite group K_{i}. As R_{1} … R_{n} lies in none of the M_{i}’s, H = R_{1}R_{2} … R_{n}. Thus (a) holds.

Let R = R_{1} and T = R_{2} … R_{m}. To show (b), it suffices to show that R ∩ T = 1. Note that B_{i} = R_{i} ∩ M_{i} and so [R_{i}, F] ≰ B_{i} just means F ≰ M_{i}. Thus we can choose l_{i} = 1 for all 1 ≤ i ≤ m. Suppose that R ∩ T ≠ 1. Pick an orbit Λ for H on Ω_{A} so that R ∩ T acts non-trivially on Λ. Then R_{1} and at least one R_{j} with 2 ≤ j ≤ m act non-trivially on Λ, say j = 2. As Ω_{A} is Ω_{i}-block-diagonal, there exist H-invariant partitions Δ_{i} of Λ so that Δ_{i} is isomorphic to Ω_{i} as an H-set for i = 1, 2. Let ω ∈ Ω_{1} be F -extreme and U_{1} the corresponding element in Δ_{1}. For all λ ∈ U_{1}, C_{F} (λ) ≤ N_{F} (U_{1}) = C_{F} (ω). Since ω is F -extreme, C_{F} (λ) = C_{F} (ω) for all λ ∈ U_{1} and so C_{F} (ω) acts trivially on U_{1}. Let U_{2} ∈ Δ_{2}.

Suppose that U_{2} ∩ U_{1} = ∅. As M_{1} ≠ M_{2}, H = M_{1}M_{2} and so M_{1} acts transitively on Δ_{2}. So we conclude U ∩ U_{1} = ∅ for all U ∈ Δ_{2}, a contradiction. Hence U_{2} ∩ U_{1} ≠ ∅ and so C_{F} (ω) fixes an element in U_{2}. Thus C_{F} (ω) normalizes U_{2}. As U_{2} was arbitrary, C_{F} (ω) acts trivially on Δ_{2} and so on Ω_{2}. Thus 1 ≠ C_{F} (ω) ≤ M_{2}, a contradiction to F ∩ M_{2} = 1. □

6 Non-Absolutely Simple Groups of 1-Type

In this section we present examples of non-absolutely simple, locally finite, simple groups of non-regular alternating type. The existence of such groups also follows from [?, 1.33] and Theorem ??. Our class of examples is slightly larger than the one in [?] and also shows that one does not have any control over the quotient L/⟨F^{L}⟩ in Theorem ??. Since knowledge of most of details of [?, Section 6] is required, they are repeated here.

Lemma 6.1 Let H be a perfect, finite group and Ω a faithful, finite H-set. Then there exist a perfect, finite group H^{∗} containing H, a function X which associates to each subgroup A of H a subgroup X(A) of H^{∗}, and a faithful, finite H^{∗}-set Ω^{∗} such that

(a) H ≤ ⟨h^{H^{∗}}⟩ for all 1 ≠ h ∈ H.

(b) X(A) ∩ H = A for all A ≤ H.

(c) If A ≤ B ≤ H, then A ⊴ B if and only if X(A) ⊴ X(B).

(d) X(H) ⊴ ⟨X(H)^{H^{∗}}⟩ but X(H) is not a normal subgroup of H^{∗}.

(e) Every non-trivial orbit for H on Ω^{∗} is isomorphic to an orbit for H on Ω.

(f) There exists a finite X(H)-set Λ so that

(fa) X(H)^{Λ} = Alt(Λ)

(fb) H acts faithfully on Λ.

(fc) Every non-trivial orbit for H on Λ is isomorphic to an orbit for H on Ω.

Proof: Let I be a faithful, finite H-set so that every non-trivial orbit for H on I is isomorphic to an orbit for H on Ω. Let S = Alt(I) and α : H → S be the monomorphism associated to the action of H on I.

Let T be any non trivial, finite, perfect group and J a faithful, finite T -set such that T acts transitively on J. We assume that 0 ∈ I and {0, 1} ⊆ J. Let K = H≀_{I}S. For i ∈ I, let β_{i} : H → K be the canonical isomorphism between H and the i-th component of the base group of K and let β be the canonical monomorphism from S to K. Let H^{∗} = K≀_{J}T and for j ∈ J, let γ_{j} : K → H^{∗} be the canonical isomorphism between K and the j-th component of the base group of H^{∗}. Define ρ : H → H^{∗} by ρ(h) = γ_{0}(β_{0}(h))γ_{1}(β(α(h))). Then ρ is clearly a monomorphism. For A ≤ H, let X(A) be the set of elements in the base group of H^{∗} such that the projection onto the 0-th component is contained in γ_{0}(∏_{i∈I} β_{i}(A)). Identifying H with ρ(H), we see immediately that (b) and (c) hold. Note that ⟨X(H)^{H^{∗}}⟩ is the base group of H^{∗} and so (d) holds. One easily checks that ⟨h^{H^{∗}}⟩ is the base group of H^{∗} for all 1 ≠ h ∈ H and so (a) holds.

Note that K = H ≀_{I} S acts faithfully on Ω × I and H^{∗} = K ≀_{J} T acts faithfully on Ω^{∗} := Ω × I × J. By definition of the embedding of H into H^{∗}, we see that

• Ω × {0} × {0} is isomorphic to Ω as an H-set.

• H acts trivially on Ω × {i} × {0} for all i ∈ I \ {0}.

• {ω} × I × {1} is isomorphic to I as an H-set for all ω ∈ Ω.

• H acts trivially on Ω × I × {j} for all j ∈ J \ {0, 1}.

By assumption, every non-trivial orbit for H on I is isomorphic to an orbit for H on Ω. So (e) holds.

Put Λ = {Ω × {i} × {1} | i ∈ I}. Then the base group of H^{∗} normalizes Λ. Hence Λ is an X(H)-set. Since γ_{1}(K) ≤ X(H) and S = Alt(I), we get X(H)^{Λ} = Alt(I). Also, Λ is isomorphic to I as an H-set. So (f) holds. □

Proof of Theorem ??:

Let G_{1} be a finite perfect group and Ω_{1} a faithful, finite G_{1}-set so that G_{1} has no regular orbits on Ω_{1} . Inductively, for i ≥ 1, let G_{i+1} = G_{i}^{∗} , X_{i} a function from the subgroups of G_{i} to the subgroups of G_{i+1}, Ω_{i+1} a faithful, finite G_{i+1}-set and Λ_{i+1} a finite X_{i}(G_{i})-set which fulfills ??. Let G = ⋃_{i=1}^{∞} G_{i}. Then by ??(a)(d), G is an infinite, locally finite simple group.

Put M_{1,1} = 1, M_{1,2} = G_{1} and inductively, for n ≥ 1, we put M_{n+1,j} = X_{n}(M_{n,j}) for 1 ≤ j ≤ 2n, M_{n+1,2n+1} = ⟨X_{n}(G_{n})^{G_{n+1}}⟩ and M_{n+1,2n+2} = G_{n+1}. Using induction on n and ??(c)(d), we get that for all n ≥ 1, M_{n,i} ⊲ M_{n,i+1} for 1 ≤ i ≤ 2n − 1 and M_{m,i} ∩ G_{n} = M_{n,i} for all m > n and 1 ≤ i ≤ 2n. For i ≥ 1, put M_{i} = ⋃_{n≥ \frac{i}{2}} M_{n,i}. Then G_{n} ≤ M_{2n} for all n ≥ 1 and so G = ⋃_{i=1}^{∞} M_{i}. Also, G_{n} ∩ M_{i} = M_{n,i} for all n ≥ 1 and 1 ≤ i ≤ 2n. By ??(d), M_{i} ⊲ M_{i+1} for all i ≥ 1.

Hence G is not absolutely simple. Suppose G is finitary. Then by [?], G is an alternating group and so absolutely simple. Therefore G is not finitary. For i ≥ 1, let H_{i+1} = X_{i}(G_{i}). Then G_{i} acts faithfully on Λ_{i+1} and so D = {(H_{i}, Λ_{i}) | i ≥ 2} is an alternating Kegel cover for G. By ??(e)(fc) and induction on i, each non-trivial orbit for G_{1} on Ω_{i} or Λ_{i} is isomorphic to a G_{1}-orbit on Ω_{1} for all i ≥ 2. Since G_{1} has no regular orbits on Ω_{1}, we conclude that G_{1} has no regular orbits on Λ_{i} for all i ≥ 2. So by ??, G_{1} is non-regular and G is of non-regular alternating type. □

References

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