Finite nilpotent groups that coincide with their 2-closures in all of their faithful permutation representations

Abstract

Let G be any group and G be a subgroup of Sym(Ω) for some set Ω. The 2-closure of G on Ω, denoted by G(2),Ω, is by definition, {θ ∈ Sym(Ω) | ∀α, β ∈ Ω, ∃g ∈ G, αθ = αg, βθ = βg}. The group G is called 2-closed on Ω if G = G(2),Ω. We say that a group G is a totally 2-closed group if H = H(2),Ω for any set Ω such that G ≈H ≤ Sym(Ω). Here we show that the center of any finite totally 2-closed group is cyclic and a finite nilpotent group is totally 2-closed if and only if it is cyclic or a direct product of a generalized quaternion group with a cyclic group of odd order. © 2018 World Scientific Publishing Company.