Théorie des groupes
On finite totally 2-closed groups
Comptes Rendus. Mathématique, Tome 360 (2022) no. G9, pp. 1001-1008.

An abstract group G is called totally 2-closed if H=H (2),Ω for any set Ω with GHSym(Ω), where H (2),Ω is the largest subgroup of Sym(Ω) whose orbits on Ω×Ω are the same orbits of H. In this paper, we classify the finite soluble totally 2-closed groups. We also prove that the Fitting subgroup of a totally 2-closed group is a totally 2-closed group. Finally, we prove that a finite insoluble totally 2-closed group G of minimal order with non-trivial Fitting subgroup has shape Z·X, with Z=Z(G) cyclic, and X is a finite group with a unique minimal normal subgroup, which is nonabelian.

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DOI : 10.5802/crmath.355
Classification : 20B05, 20D10, 20D25
Abdollahi, Alireza 1 ; Arezoomand, Majid 2 ; Tracey, Gareth 3

1 Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan 81746-73441, Iran
2 University of Larestan, Larestan 74317-16137, Iran
3 School of Mathematics, University of Birmingham, Edgbaston, Birmingham, B15 2TT, United Kingdom
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Abdollahi, Alireza; Arezoomand, Majid; Tracey, Gareth. On finite totally $2$-closed groups. Comptes Rendus. Mathématique, Tome 360 (2022) no. G9, pp. 1001-1008. doi : 10.5802/crmath.355. http://www.numdam.org/articles/10.5802/crmath.355/

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