On finite totally 2-closed groups

Abstract

An abstract group G is called totally 2-closed if H = H(2),Ω for any set Ω with G ∼= H ≤ Sym(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.