Generalized autocommuting probability of a finite group relative to its subgroups

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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Hacettepe Journal of Mathematics and Statistics-Cover

Yayın Aralığı: Yılda 6 Sayı
Başlangıç: 2002
Yayıncı: Hacettepe Üniversitesi Fen Fakultesi

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