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.

KEYWORDS:

• Autocommuting probability
• autoisoclinism
• automorphism group

2010 MATHEMATICS SUBJECT CLASSIFICATION:

• 20D60
• 20P05
• 20F28

Acknowledgment

The first author would like to thank DST for the INSPIRE Fellowship.