On finite totally 2-closed groups

2020/01/27 by Aliréza Abdollahi, Abdollahi, Alireza, Majid Arezoomand +3
Mathematics · #Finite Group Theory Research #Advanced Topology and Set Theory #Rings, Modules, and Algebras

paper · pdf · doi:10.48550/arxiv.2001.09597

Abstract

An abstract group G is called totally 2-closed if H=H^(2),Ω for any set Ω with G≅ H≤\rm Sym(Ω), where H^(2),Ω is the largest subgroup of \rm 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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