Generalized autocommuting probability of a finite group relative to its subgroups

Abstract

Let $H \subseteq K$ be two subgroups of a finite group $G$  and $\mathrm{Aut}(K)$ the automorphism group of  $K$. In this paper, we consider the generalized autocommuting probability of $G$ relative to its subgroups $H$ and $K$, denoted by  ${Pr}_g(H,\mathrm{Aut}(K))$, which is the probability  that the autocommutator of a randomly chosen pair of elements, one from $H$ and the other from $\mathrm{Aut}(K)$, is equal to a given element $g \in K$. We study several properties as well as obtain several computing formulae of  this probability. As applications of the computing formulae, we also obtain several  bounds for ${Pr}_g(H,\mathrm{Aut}(K))$ and characterizations of some finite groups through ${Pr}_g(H,\mathrm{Aut}(K))$.

Keywords

• automorphism group,autocommuting probability,autoisoclinism

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