Probability that an autocommutator element of a finite group equals to a fixed element

Abstract

In this article we introduce a formula for the probability which an autocommutator element of a finite group G, equals to a fixed element of G and derive some properties of this formula. Moreover, we obtain a lower bound and an upper bound for this probability in the special cases. This generalizes some results of Das et al. in 2010 and Moghaddam et al. in 2011.