Finite totally k-closed groups

Churikov, Dmitry; Praeger, Cheryl E.

doi:10.21538/0134-4889-2021-27-1-240-245

“…Theorem 1.1 generalizes recent results of [2,3], where similar theorems were proved for 2-closures of nilpotent groups [2] and for k-closures of abelian groups [3]. It is worth mentioning that nilpotency of k-closure of nilpotent group is already known and it easily follows from [2, Thm.…”

supporting

confidence: 80%

Structure of $k$-closures of finite nilpotent permutation groups

Churikov¹

2021

Preprint

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Let G be a permutation group on a set Ω, and k a positive integer. The k-closure G (k) of G is the largest subgroup of Sym(Ω), with the same as G orbits of componentwise action on Ω k . We prove that the k-closure of a finite nilpotent permutation group is the direct product of k-closures of its Sylow subgroups. ItroductionLet G be a permutation group on a finite nonempty set Ω, and k a positive integer. Denote by Orb k (G) the set of the orbits of componentwise action of G on the Cartesian power Ω k . Elements of the set Orb k (G) are called k-orbits of G, and they form a partition of the setEquivalently, G (k) is the largest subgroup of the symmetric group Sym(Ω) such that Orb k (G) = Orb k (G (k) ). The following inclusions for closures of G are known [1, Thm. 5.8]:(1)

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“…In previous work it has been shown that k(G) = 2 if G is a cyclic group of prime order [13, Theorem 1.2], and for exactly six nonabelian simple groups [3, Theorem 1.1], namely the Janko groups J 1 , J 3 and J 4 , together with Ly, T h and the Monster M. By (1), k(G) ≥ 3 for all other finite simple groups. The problem of determining the closure number k(G) for the remaining simple groups was posed in [3, Question 3], and in particular for determining all those with closure number 3 (if any exist), see also [13,Problem 3] and the Kourovka Notebook [19,New Problems 20.2 and 20.3].…”

Section: Introductionmentioning

confidence: 99%

Total closure for permutation actions of finite nonabelian simple groups

Freedman

¹

,

Giudici

²

,

Praeger

³

2023

Monatsh Math

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For a positive integer k, a group G is said to be totally k-closed if for each set $$\Omega $$ Ω upon which G acts faithfully, G is the largest subgroup of $$\textrm{Sym}(\Omega )$$ Sym ( Ω ) that leaves invariant each of the G-orbits in the induced action on $$\Omega \times \cdots \times \Omega =\Omega ^k$$ Ω × ⋯ × Ω = Ω k . Each finite group G is totally |G|-closed, and k(G) denotes the least integer k such that G is totally k-closed. We address the question of determining the closure number k(G) for finite simple groups G. Prior to our work it was known that $$k(G)=2$$ k ( G ) = 2 for cyclic groups of prime order and for precisely six of the sporadic simple groups, and that $$k(G)\geqslant 3$$ k ( G ) ⩾ 3 for all other finite simple groups. We determine the value for the alternating groups, namely $$k(A_n)=n-1$$ k ( A n ) = n - 1 . In addition, for all simple groups G, other than alternating groups and classical groups, we show that $$k(G)\leqslant 7$$ k ( G ) ⩽ 7 . Finally, if G is a finite simple classical group with natural module of dimension n, we show that $$k(G)\leqslant n+2$$ k ( G ) ⩽ n + 2 if $$n \geqslant 14$$ n ⩾ 14 , and $$k(G) \leqslant \lfloor n/3 + 12 \rfloor $$ k ( G ) ⩽ ⌊ n / 3 + 12 ⌋ otherwise, with smaller bounds achieved by certain families of groups. This is achieved by determining a uniform upper bound (depending on n and the type of G) on the base sizes of the primitive actions of G, based on known bounds for specific actions. We pose several open problems aimed at completing the determination of the closure numbers for finite simple groups.

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“…In this paper we study the closure numbers for finite simple groups. In previous work it has been shown that k(G) = 2 if G is a cyclic group of prime order [12,Theorem 1.2], and for exactly six nonabelian simple groups [2, Theorem 1.1], namely the Janko groups J 1 , J 3 and J 4 , together with Ly, T h and the Monster M. By (1), k(G)…”

Section: Introductionmentioning

confidence: 99%

“…3 for all other finite simple groups. The problem of determining the closure number k(G) for the remaining simple groups was posed in [2, Question 3], and in particular for determining all those with closure number 3 (if any exist), see also [12,Problem 3] and the Kourovka Notebook [18,New Problems 20.2 and 20.3].…”

Section: Introductionmentioning

confidence: 99%

Total closure for permutation actions of finite nonabelian simple groups

Freedman¹,

Giudici²,

Praeger³

2022

Preprint

0

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For a positive integer k, a group G is said to be totally k-closed if for each set Ω upon which G acts faithfully, G is the largest subgroup of Sym(Ω) that leaves invariant each of the G-orbits in the induced action on Ω × • • • × Ω = Ω k . Each finite group G is totally |G|-closed, and k(G) denotes the least integer k such that G is totally k-closed. We address the question of determining the closure number k(G) for finite simple groups G. Prior to our work it was known that k(G) = 2 for cyclic groups of prime order and for precisely six of the sporadic simple groups, and that k(G) 3 for all other finite simple groups. We determine the value for the alternating groups, namely k(A n ) = n − 1. In addition, for all simple groups G, other than alternating groups and classical groups, we show that k(G) 7. Finally, if G is a finite simple classical group with natural module of dimension n, we show that k(G) n + 2 if n 14, and k(G) ⌊n/3 + 12⌋ otherwise, with smaller bounds achieved by certain families of groups. This is achieved by determining a uniform upper bound (depending on n and the type of G) on the base sizes of the primitive actions of G, based on known bounds for specific actions. We pose several open problems aimed at completing the determination of the closure numbers for finite simple groups.

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Finite totally k-closed groups

Cited by 5 publications

References 5 publications

Structure of $k$-closures of finite nilpotent permutation groups

Structure of $k$-closures of finite nilpotent permutation groups

Total closure for permutation actions of finite nonabelian simple groups

Total closure for permutation actions of finite nonabelian simple groups

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