- Hacettepe Journal of Mathematics and Statistics
- Vol: 49 Issue: 1
- Generalized autocommuting probability of a finite group relative to its subgroups
Generalized autocommuting probability of a finite group relative to its subgroups
Authors : Parama Dutta, Rajat Nath
Pages : 389-398
Doi:10.15672/hujms.568258
View : 9 | Download : 2
Publication Date : 2020-02-06
Article Type : Research
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