2-closures of primitive permutation groups of holomorph type

Xue Yu; Jiangmin Pan

doi:10.1515/math-2019-0063

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2-closures of primitive permutation groups of holomorph type

Xue Yu und Jiangmin Pan

Veröffentlicht/Copyright: 31. Juli 2019

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Aus der Zeitschrift Open Mathematics Band 17 Heft 1

Abstract

The 2-closure G⁽²⁾ of a permutation group G on a finite set Ω is the largest subgroup of Sym(Ω) which has the same orbits as G in the induced action on Ω × Ω. In this paper, the 2-closures of certain primitive permutation groups of holomorph simple and holomorph compound types are determined.

Keywords: 2-closure; primitive permutation group; holomorph type

MSC 2010: 05C25; 20B15; 20B30

1 Introduction

Let G be a permutation group on a finite set Ω. Then G has a natural induced action on Ω⁽²⁾ := Ω × Ω as follows:

(α,β)g=(αg,βg), where α,β∈Ω, g∈G.

Each orbit of G of the above action is called an orbital of G on Ω. We denote by Orbl(G, Ω) the set of all orbitals of G on Ω.

In 1969, Wielandt [1] defined the 2-closure of G to be the group

G(2):={x∈Sym(Ω)∣Ox=OforeachO∈Orbl(G,Ω)},

namely the largest subgroup of Sym(Ω) that has the same orbitals of G. Clearly, G⁽²⁾ ≥ G. If G = G⁽²⁾, then G is called 2-closed. See some basic properties of 2-closures in Wielandt [1, 2].

A transitive permutation group G ≤ Sym(Ω) is called primitive if the vertex stabilizer G_α := {g ∈ G | α^g = α} is maximal in G. Denote by soc(G) the socle of G, namely the product of all minimal normal subgroups of G. Then either soc(G) is a minimal normal subgroup of G, or soc(G) = M × N with M ≅ N being minimal normal subgroups of G. According to the structure and the action of soc(G), the well-known O’Nan-Scott theorem divided the finite primitive permutation groups into eight types, refer to [3, 4, 5]. The eight types are described as follows:

HA (Holomorph Affine): soc(G) ≅ Zpl is regular on Ω and X ≤ Zpl : GL(l, p), where p is a prime and l ≥ 1;

HS (Holomorph Simple): soc(G) = M × N, where M ≅ N ≅ T are nonabelian simple normal subgroups of X and regular on Ω;

HC (Holomorph Compound): soc(G) = M × N, where M ≅ N ≅ T^k are minimal normal subgroups of X and regular on Ω with T nonabelian simple and k ≥ 2;

AS (Almost simple): soc(G) = T is a nonabelian simple group and not regular on Ω, and T ⊲ G ≤ Aut(T);

SD (Simple Diagonal): soc(G) ≅ T^l is a minimal normal subgroup of G, with l ≥ 2 and T is nonabelian simple, (soc(G))_α = {(t, ⋯, t) | t ∈ T} ≅ T for α ∈ Ω, and G ≤ T^l.(Out(T) × S_l);

CD (Compound Diagonal): soc(G) ≅ T^l with l ≥ 4 and T nonabelian simple, and (soc(G))_α ≅ T^k with k ≥ 2 and k | l;

TW (Twisted Wreath Product): soc(G) ≅ T^l is regular on Ω and G_α is insoluble, where T is nonabelian simple and l ≥ 6;

PA (Product Action): soc(G) ≅ T^l is a minimal normal subgroup of X with T nonabelian simple and l ≥ 2, and G ≤ Aut(T) ≀ S_l; moreover, there exists a set Δ such that Ω = Δ^l, Aut(T) acts irregularly on Δ and G acts on Δ^l as follows: for (w₁, …, w_l) ∈ Ω^l, (h₁, …, h_l) ∈ Aut(T)^l and σ ∈ S_l,

(w1,⋯,wl)(h1,…,hl)σ=(w1σ−1h1σ−1,…,wlσ−1hlσ−1).

In 1988, Liebeck, Praeger and Saxl [6] studied the 2-closures of simply primitive (namely primitive but not 2-transitive) permutation groups of almost simple type, and proved that if G is a such group with socle T, then either T ⊴ G⁽²⁾ ≤ Aut(T) or (G, G⁽²⁾) is given in an explicit list. For more results regarding primitive groups and their 2-closures, see [7, 8, 9, 10, 11, 12, 13]. The main purpose of this paper is to determine the 2-closures of certain primitive permutation groups of HS and HC types; for convenience both types are called holomorph type.

The notations used in this paper are standard. For example, for a positive integer n, we denote by ℤ_n the cyclic group of order n, and by S_n the symmetric group of degree n. Given two groups N and H, denote by N × H the direct product of N and H, by N.H an extension of N by H, and if such an extension is split, then we write N : H instead of N.H. Also, we denote by Gⁿ the direct product of the group G of n terms, and by H ≀ P the wreath product of groups H and P.

The main results of this paper are as follows.

Theorem 1.1

Let G = T : Aut(T) be a primitive permutation group of type HS on Ω, where T is a nonabelian simple group. Then the following statements hold:

If there exists t ∈ T which is not conjugate to t^–1 in Aut(T), then G⁽²⁾ = G.
If each t ∈ T is conjugate to t^–1 in Aut(T), then G⁽²⁾ = G.ℤ₂ is primitive of type SD.

Theorem 1.2

Let G = N : Aut(N) be a primitive permutation group of type HC on Ω, where N ≅ T^d with T a nonabelian simple group and d ≥ 2. Then the following statements hold:

If there exists t ∈ T which is not conjugate to t^–1 in Aut(T), then G⁽²⁾ = G.
If each t ∈ T is conjugate to t^–1 in Aut(T), then

G(2)=[T2.(Out(T)×Z2)]d.Sd

is primitive of type CD.

After this introductory section, we explain the actions of certain primitive permutation groups in Section 2, and complete the proofs of Theorems 1.1 and 1.2 in Section 3.

2 Actions of primitive permutation groups

In this section, we explain the actions of primitive permutation groups of several types that will be used in Section 3. The main contents of this section can be found in Giudici, Li and Praeger [14, Section 2] and Liebeck, Praeger and Saxl [15].

For a group G, denote by

G^={g^∣g^: x↦xg, for all g,x∈G},

Gˇ={g^∣g^: x↦g−1x, for all g,x∈G}

the right regular representation and the left regular representation of G, respectively. Clearly, both Ĝ and Ǧ can be viewed as regular (permutation) subgroups on G, and centralize other.

Let H be a subgroup of G, and denote by [G : H] the set of cosets of H in G. The coset action of G on [G : H] is defined as follows:

(Hg)x=Hgx, for g,x∈G.

It is well known each transitive action of a group X on a set Ω can be identified (permutation equivalent) with the coset action of X on [X : X_α] with α ∈ Ω. In particular, if X is regular on Ω, then X_α = 1, and the action of X on Ω can be identified with the right regular representation X̂ (and so Ω can be identified with X).

In the following, we give several remarks on the actions of primitive permutation groups of several types. Let G ≤ Sym(Ω) be a primitive permutation group, and let N = soc(G).

Remark 1

Assume that G is of type HS or HC. Then N = M₁ × M₂, where M₁ ≅ M₂ ≅ T^d with d ≥ 1 are nonabelian minimal normal subgroups of G and regular on Ω, and G ≤ N_Sym(Ω)(N) = N : Aut(N). Moreover, the action of G on Ω as follows:

xnσ=(xn)σ=xσnσ,

where x ∈ Ω, n ∈ N and σ ∈ Aut(N).

On the other hand, since M₁ ≅ T^d is regular on Ω, we may identify Ω = T^d and identify the action of M₁ on Ω with the right regular representation T̂^d, hence M₂ = C_G(M₁) can be identified with Ť^d. Consequently, we obtain

G≤(Td^×Tdˇ).(Out(T)d≀Sd),

and the action of G on Ω = T^d is as follows:

(t1,t2,…,td)(g1,g2,…,gd)^=(t1g1,t2g2,…,tdgd),(t1,t2,…,td)(g1,g2,…,gd)ˇ=(g1−1t1,g2−1t2,…,gd−1td),(t1,t2,…,td)(σ1,σ2,…,σd)=(t1σ1,t2σ2,…,tdσd),(t1,t2,…,td)τ=(t1τ−1,t2τ−1,…,tdτ−1),

where (t₁, t₂, …, t_d), (g₁, g₂, …, g_d) ∈ T^d, (σ₁, σ₂, …, σ_d) ∈ Out(T)^d and τ ∈ S_d.

Remark 2

Assume that G is of type SD and N = soc(G) = T², with T a nonabelian simple group. Then N_α = {(t, t, …, t) | t ∈ T} ≅ T for α ∈ Ω, and

G≤T2.(Out(T)×S2)≤Aut(T2).

Since there is an one to one correspondence between [N : N_α] to T via Nα(t1,t2)→t1−1t2, we may identify Ω with T. Now, for t ∈ Ω, we easily conclude that the action of G on Ω is as following (or see [15, P. 307]):

t(t1,t2)=t1−1tt2for(t1,t2)∈T2,t(α,α)=tαfor(α,α)∈Out(T),t(12)=t−1for(12)∈S2.

Remark 3

Assume that G is of type HC or CD and N = soc(G) = T^d, with T a nonabelian simple group and d ≥ 4. Then G can be built from a primitive permutation group of type HS or SD respectively by product action, see [14, P. 296]. Hence there exists a set Δ, and a primitive permutation group H of type HS or SD on Δ, such that Ω = Δ^k for some k | d and G ≤ H ≀ S_k = H^k : S_k, and G acts on Δ^k in product action.

3 Proofs of Theorems 1.1 and 1.2

In this section, we prove Theorem 1.1 and Theorem 1.2. We first give an observation.

Lemma 3.1

Let G = T^d : Aut(T^d) be a primitive permutation group of type HC on Ω. We may identify Δ = T^d (see Remark 1 in Section 2). Then the vertex stabilizer G₁ := {g ∈ G | 1^g = 1} = Aut(T^d), where 1 is the identity element of T^d.

Proof

Let gσ ∈ G, where g ∈ T^d and σ ∈ Aut(T^d). By Remark 1, we have 1^gσ = g^σ. Hence gσ ∈ G₁ if and only g^σ = 1, or equivalently g = 1. The lemma follows.□

Proof of Theorem 1.1

Suppose that G = T : Aut(T) is a primitive permutation group of type HS on Ω, where T is a nonabelian simple group. Since G⁽²⁾ is an overgroup of G, by Praeger [3, Proposition 8.1], either G⁽²⁾ = A_n or S_n with n = |Ω|, or soc(G⁽²⁾) = soc(G) = T² and G⁽²⁾ is of type HS or SD.

If G⁽²⁾ = A_n or S_n, then G⁽²⁾ is 2-transitive on Ω, hence G⁽²⁾ and so G has exactly two orbitals, namely G is 2-transitive, which is a contradiction as a well known theorem of Burnside (see [16, Theorem 4.1B]) states that a 2-transitive permutation group is either of type HA or AS.

Therefore, G⁽²⁾ is of type HS or of type SD and soc(G⁽²⁾) = soc(G) = T². Consequently, we always have G ≤ G⁽²⁾ ≤ G : S₂. Since T is regular on Ω, we may identify Ω = T.

Assume first that there exists an element t ∈ T which is not conjugate to t^–1 in Aut(T). Let O₁ = {(1, t)^G and O₂ = (1, t^–1)^G be two orbitals of G on Ω. If O₁ = O₂, then (1, t) = (1, t^–1)^x = (1^x, (t^–1)^x) for some x ∈ G. It follows that t = (t^–1)^x and 1^x = 1. Thus x ∈ G₁ = Aut(T) by Lemma 3.1, and t is conjugate to t^–1 in Aut(T), a contradiction.

Therefore, O₁ ≠ O₂. For each t ∈ T and (12) ∈ S₂, we have t⁽¹²⁾ = t^–1 by Remark 2 in Section 2, it follows

O1(12)=(1,t)G(12)=(1,t)(12)G=(1,t−1)G=O2,

that is, (12) ∉ G⁽²⁾. Hence G⁽²⁾ = G, part (i) of Theorem 1.1 holds.

Now assume that each element t ∈ T is conjugate to t^–1 in Aut(T). By the transitivity of G on Ω, each orbital of G on Ω can be expressed as (1, t)^G for some t ∈ T. By assumption, there is α ∈ Aut(T) ≤ G such that t^α = t^–1. By Remark 2, it follows that

((1,t)G)(12)=(1,t)G(12)=(1,t)(12)G=(1,t−1)G=(1,t)αG=(1,t)G,

that is, (12) ∈ G⁽²⁾. Hence G⁽²⁾ = G.2, part (ii) of Theorem 1.1 holds. This completes the proof of Theorem 1.1.□

Now, we complete the proof of Theorem 1.2.

Proof of Theorem 1.2

Suppose that G = N : Aut(N) is a primitive permutation group of type HC on Ω, where N = T^d with T a nonabelian simple group and d ≥ 2. Since G⁽²⁾ is an overgroup of G and soc(G) = T^2d with 2d ≥ 4, by Praeger [3, Proposition 8.1], either G⁽²⁾ = A_n or S_n with n = |Ω|, or soc(G⁽²⁾) = soc(G) and G⁽²⁾ is a primitive permutation group of type HC or CD.

If G⁽²⁾ = A_n or S_n, then G is 2-transitive on Ω, and so is G since G and G⁽²⁾ have the same orbitals, by a theorem of Burnside (see [16, Theorem 4.1B]), G is of type HA or AS, a contradiction.

Thus G⁽²⁾ is a primitive permutation group of type HC or CD, and soc(G⁽²⁾) = soc(G) = T^2d. By Remark 3, it easily follows that there exists a set Δ and a primitive permutation group H of type SD on Δ with soc(H) = T², such that Ω = Δ^d, and G ≤ G⁽²⁾ ≤ H ≀ S_d = H^d : S_d, and G⁽²⁾ acts on Δ^d in product action. Notice that T² ⊲ H ≤ T².(Out(T) × S₂).

Notice that H is a primitive permutation group of type SD on Δ with socle T², (soc(H))_δ = {(t, t) | t ∈ T} ≅ T for each δ ∈ Δ, soc(H) has a normal subgroup T which is regular on Δ. Thus we may identify Δ = T, and hence identify Ω = T^d. Let O be an orbital of G, and let 1 = (1, 1, …, 1) ∈ T^d = Ω. By the vertex transitivity of G, we may assume O = (1, γ)^G, where γ = (t₁, t₂, …, t_d) ∈ T^d = Ω.

Now, we prove that part (i) and part (ii) of Theorem 1.2 are true respectively.

Suppose G⁽²⁾ > G. Then there is x ∈ G⁽²⁾ ∖ G. Since G⁽²⁾ ≤ H ≀ S_d = H^d : S_d, we may set x = (h₁, h₂, …, h_d)τ, where h_i ∈ H_i with 1 ≤ i ≤ d and τ ∈ S_d. Since H is primitive of type SD with socle T², we may write h_i = ξ_i(u_i1, u_i2)σ_i, where (u_i1, u_i2) ∈ T², σ_i ∈ Out(T) and ξ_i ∈ S₂. By Remarks 2 and 3, we conclude

(1,γ)x=(1x,γx)=(1(h1,h2,...,hd)τ,(t1,t2,...,td)(h1,h2,...,hd)τ)=(1,(t1,t2,...,td))(ξ1(u11,12)σ1,ξ2(u21,22)σ2,...,ξd(ud1,d2)σd)τ.

If ξ_i = 1 for each i ∈ {1, 2, …, d}, then x ∈ G, a contradiction. Hence without loss of the generality, we may assume ξ₁ ≠ 1. Then

(1,γ)x=(1,(t1ξ1,t2ξ2,...,tdξd))((u11,u12)σ1,(u21,u22)σ2,...,(ud1,ud2)σd)τ=(1,(t1−1,t2ϵ2,...,tdϵd))((u11,u12)σ1,(u21,u22)σ2,...,(ud1,ud2)σd)τ,

where ϵ_i = 1 if ξ_i = 1, and ϵ_i = –1 if ξ_i ≠ 1 with 1 ≤ i ≤ d.

Let g₀ := ((u₁₁, u₁₂)σ₁, (u₂₁, u₂₂)σ₂, …, (u_d1, u_d2)σ_d)τ ∈ G. Then

(1,(t1−1,t2ϵ2,...,tdϵd))=(1,(t1−1,t2ϵ2,...,tdϵd))g0g0−1∈(1,γ)G

and (1, γ)^x ∈ (1, γ)^G. Hence

(1,(t1−1,t2ϵ2,...,tdϵd))=(1,(t1,t2,...,td))y=(1y,(t1,t2,...,td)y)

for some y ∈ G. So 1^y = 1, and hence y ∈ G₁ = Aut(T)^d by lemma 3.1. It follows that y = (η₁, η₂, …, η_d), where η_i ∈ Aut(T)ⁱ with 1 ≤ i ≤ d. Thus (t1−1,t2ϵ2,...,tdϵd)=(t1,t2,...,td)y=(t1η1,t2η2,...,tdηd). Hence t1−1=t1η1. That is, t₁ is conjugate to t1−1 in Aut(T), part (i) of Theorem 1.2 holds.
Suppose that each t ∈ T is conjugate to t^–1 in Aut(T). Note that G⁽²⁾ ≤ H ≀ S_d ≤ (T²(Out(T) × S₂)) ≀ S_d, we only need to prove T²(Out(T) × S₂)] ≀ S_d ≤ G. For any

x=(ξ1(u11,u12)σ1,ξ2(u21,u22)σ2,...,ξd(ud1,ud2)σd)∈[T2.(Out(T)×S2)]d,

where ξ_i ∈ S₂, (u_i1, u_i2) ∈ T² and σ_i ∈ Out(T) with 1 ≤ i ≤ d, it is easy to see that O^x = O if and only if (1, γ)^gx ∈ O for each g ∈ G. Notice that

gx=g(ξ1(u11,u12)σ1,ξ2(u21,u22)σ2,...,ξd(ud1,ud2)σd)=g(ξ1,ξ2,...,ξd)((u11,u12)σ1,(u21,u22)σ2,...,(ud1,ud2)σd)=g((u11″,u12″)σ1″,(u21″,u22″)σ2″,...,(ud1″,ud2″)σd″)(ξ1,ξ2,...,ξd)

for some (ui1″,ui2″)σi″∈T2. Out(T) with 1 ≤ i ≤ d. Set

g′:=g((u11″,u12″)σ1″,(u21″,u22″)σ2″,...,(ud1″,ud2″)σd″).

Then g′ ∈ G. It follows that gx = g′(ξ₁, ξ₂, …, ξ_d). Note that g′ ∈ G = K^d.S_d = (T².Out(T))^d.S_d, then g = (τ₁, τ₂, …, τ_d)π, where τ_i ∈ K, π ∈ P with 1 ≤ i ≤ d. Thus

(1,γ)gx=(1,γ)g′(ξ1,ξ2,...,ξd)=(1,(t1,t2,...,td))g′(ξ1,ξ2,...,ξd)=(1,(t1τ1,t2τ2,...,tdτd)π)(ξ1,ξ2,...,ξd)=(1,(t1π−1τ1π−1,t2π−1τ2π−1,...,tdπ−1τdπ−1))(ξ1,ξ2,...,ξd)={1,((t1π−1τ1π−1)ξ1,(t2π−1τ2π−1)ξ2,...,(tdπ−1τdπ−1)ξd)}=(1,((t1π−1ξ1)τ1π−1,(t2π−1ξ2)τ2π−1,...,(tdπ−1ξd)τdπ−1))=(1,((t1π−1ϵ1)τ1π−1,(t2π−1ϵ2)τ2π−1,...,(tdπ−1ϵd)τdπ−1))=(1,((t1ϵ1)τ1,(t2ϵ2)τ2,...,(tdϵd)τd)π)=(1,((t1ϵ1)τ1,(t2ϵ2)τ2,...,(tdϵd)τd))π=(1,(t1ϵ1,t2ϵ2,...,tdϵd))(τ1,τ2,...,τd)π,

where (tiπ−1τiπ−1)ξi=tiπ−1τiπ−1 if ξi=1, and (tiπ−1τiπ−1)ξi=(tiπ−1τiπ−1)−1=(tiπ−1−1)τiπ−1 if ξi≠1. Since t is conjugate to t^–1 in Aut(T) for each t ∈ T, it follows that tiηi=tiϵi for some η_i ∈ Aut(T) with 1 ≤ i ≤ d. Then

(1,γ)gx=[1,(t1η1,t2η2,...,tdηd)]g′=[1,(t1,t2,...,td)](η1,η2,...,ηd)g′=(1,γ)(η1,η2,...,ηd)g′.

Note that (η₁, η₂, …, η_d)g′ ∈ G₁ = Aut(T)^d ≤ G, it follows that (1, γ)^gx ∈ (1, γ)^G. Thus x ∈ G and hence G⁽²⁾ = [T².(Out(T) × S₂)]^d.S_d. This completes the proof of Theorem 1.2.□

Acknowledgements

The authors thank Professor Cai Heng Li for some helpful discussions, and thank the referee for helpful comments.

This work was supported by the NSFC (11231008, 11861076), the Scientific Research Fund of Yunnan Education Department (2018Y008) and the NSF of Yunnan Province (2019FB-2).

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Received: 2018-04-18

Accepted: 2019-05-14

Published Online: 2019-07-31

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