On the probability that an automorphism of a group fixes an element of the group

摘要

Let G be a finite group and let P-A(G) denote the probability that a randomly chosen element from Aut(G) fixes a randomly chosen element from G. We classify all finite abelian groups G such that P-A(G) = 1/p in the cases when p is the smallest prime dividing vertical bar Aut(G)vertical bar, and when p is any prime. We also compute P-A(G) for some classes of finite groups. As a consequence of our results, we deduce that if G is a finite p-group having a cyclic maximal subgroup, then vertical bar G vertical bar divides vertical bar Aut(G)vertical bar.