Totally 2-closed finite groups with trivial Fitting subgroup

Arezoomand, Majid; Iranmanesh, Mohammad A.; Praeger, Cheryl E.; Tracey, Gareth

doi:10.1142/s1664360723500042

Cited by 3 publications

(4 citation statements)

References 31 publications

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“…However it is not difficult to see that it is totally 3-closed, since in every faithful permutation representation of G = S 3 on a set Ω there must be a G-orbit of length 3 or 6, and the stabiliser in G of two points α, β from such an orbit is trivial. Hence by [8,Theorem 5.12], G = G (3),Ω . As a first step it would be interesting to know which other non-nilpotent soluble groups are totally 3-closed.…”

Section: Introductionmentioning

confidence: 99%

“…For some time it was believed that all finite totally 2-closed groups would be soluble, and it was somewhat surprising to the authors of [3] to discover that exactly six or the sporadic simple groups are totally 2-closed, namely J 1 , J 3 , J 4 , Ly, T h, M. Problem 3. Find all the the totally 3-closed sporadic simple groups.…”

Section: Introductionmentioning

confidence: 99%

“…The classification of the finite nonabelian simple totally 2-closed groups is still not complete, and we refer the reader to [3] for details of the status of this problem and other open questions about total 2-closure.…”

Section: Introductionmentioning

confidence: 99%

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Finite totally k-closed groups

Churikov¹,

Praeger²

2021

Trudy IMM UrO RAN

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For a positive integer k, a group G is said to be totally k-closed if in each of its faithful permutation representations, say on a set Ω, G is the largest subgroup of Sym(Ω) which leaves invariant each of the Gorbits in the induced action on Ω × • • • × Ω = Ω k . We prove that every abelian group G is totally (n(G) + 1)-closed, but is not totally n(G)closed, where n(G) is the number of invariant factors in the invariant factor decomposition of G. In particular, we prove that for each k ≥ 2 and each prime p, there are infinitely many finite abelian p-groups which are totally k-closed but not totally (k − 1)-closed. This result in the special case k = 2 is due to Abdollahi and Arezoomand. We pose several open questions about total k-closure.

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Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Finite totally k-closed groups

Churikov¹,

Praeger²

2021

Trudy IMM UrO RAN

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“…The following corollary of Theorem 4.1 summarises when this occurs. Here, δ again denotes the Kronecker delta, and O is as in (3). that k(G) ≤ 6.…”

Section: Gives An Upper Bound For B(g )mentioning

confidence: 99%

Total closure for permutation actions of finite nonabelian simple groups

2023

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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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