A finite permutation group $G \leq Sym (Ω)$ $G \leq Sym (Ω)$ is called $2$ $2$ -closed if $G$ $G$ is the largest subgroup of $Sym (Ω)$ $Sym (Ω)$ which leaves invariant each of the $G$ $G$ -orbits for the induced action on $Ω \times Ω$ $Ω \times Ω$ . Introduced by Wielandt in 1969, the concept of $2$ $2$ -closure has developed as one of the most useful approaches for studying relations on a finite set invariant under a group of permutations of the set; in particular for studying automorphism groups of graphs and digraphs. The concept of total $2$ $2$ -closure switches attention from a particular group action, and is a property intrinsic to the group: a finite group $G$ $G$ is said to be totally $2$ $2$ -closed if $G$ $G$ is $2$ $2$ -closed in each of its faithful permutation representations. There are infinitely many finite soluble totally $2$ $2$ -closed groups, and these have been completely characterized, but up to now no insoluble examples were known. It turns out, somewhat surprisingly to us, that there are exactly $6$ $6$ totally $2$ $2$ -closed finite nonabelian simple groups: the Janko groups $J_{1}$ $J_{1}$ , $J_{3}$ $J_{3}$ and $J_{4}$ $J_{4}$ , together with $Ly$ $Ly$ , $Th$ $Th$ and the Monster $𝕄$ . Moreover, if a finite totally $2$ -closed group has no nontrivial abelian normal subgroup, then we show that it is a direct product of some (but not all) of these simple groups, and there are precisely $47$ examples.

In the course of obtaining this classification, we develop a general framework for studying $2$ -closures of transitive permutation groups, which we hope will prove useful for investigating representations of finite groups as automorphism groups of graphs and digraphs, and in particular for attacking the long-standing polycirculant conjecture. In this direction, we apply our results, proving a dual to a 1939 theorem of Frucht from Algebraic Graph Theory. We also pose several open questions concerning closures of permutation groups.

Communicated by Efim Zelmanov

Keywords:

AMSC: 20B05, 20E32, 05C20

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Vol. 14, No. 01

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Received 28 June 2022

Accepted 28 September 2022

Published: 18 March 2023

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This is an Open Access article published by World Scientific Publishing Company. It is distributed under the terms of the Creative Commons Attribution 4.0 (CC BY) License which permits use, distribution and reproduction in any medium, provided the original work is properly cited.

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Totally 22<math altimg="eq-00001.gif" display="inline"><mn>2</mn></math>-closed finite groups with trivial Fitting subgroup

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Totally $2$ $2$ -closed finite groups with trivial Fitting subgroup