Title: Autocommuting probability of a finite group,
Abstract: Let $G$ be a finite group and $\Aut(G)$ the automorphism group of $G$. The
autocommuting probability of $G$, denoted by $\Pr(G, \Aut(G))$, is the
probability that a randomly chosen automorphism of $G$ fixes a randomly chosen
element of $G$. In this paper, we study $\Pr(G, \Aut(G))$ through a
generalization. We obtain a computing formula, several bounds and
characterizations of $G$ through $\Pr(G, \Aut(G))$. We conclude the paper by
showing that the generalized autocommuting probability of $G$ remains unchanged
under autoisoclinism.