Generalized autocommuting probability of a finitegroup relative to its subgroups

Abstract

LetH⊆Kbe two subgroups of a finite groupGandAut(K)the automorphism groupofK. In this paper, we consider the generalized autocommuting probability ofGrelativeto its subgroupsHandK, denoted byPrg(H,Aut(K)), which is the probability thatthe autocommutator of a randomly chosen pair of elements, one fromHand the otherfromAut(K), is equal to a given elementg∈K. We study several properties as well asobtain several computing formulae of this probability. As applications of the computingformulae, we also obtain several bounds forPrg(H,Aut(K))and characterizations of somefinite groups throughPrg(H,Aut(K)).