A remark on autocommuting probability of finite groups

Abstract

Let G be a finite group and let Aut(G) be its automorphism group.
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 characterize all finite groups G
such that Pr(G, Aut(G)) = p+q−1/pq, where p, q are the smallest primes
dividing | Aut(G)|, |G| respectively. We shall also show that there is no
finite group G such that Pr(G, Aut(G)) = q²+p−1/pq² , where p, q are primes
as mentioned above.