On the commuting probability and supersolvability of finite groups

Abstract

For a finite group $G$, let $d(G)$ denote the probability that a randomly chosen pair of elements of $G$ commute. We prove that if $d(G)>1/s$ for some integer $s>1$ and $G$ splits over an abelian normal nontrivial subgroup $N$, then $G$ has a nontrivial conjugacy class inside $N$ of size at most $s-1$. We also extend two results of Barry, MacHale, and N\'{\i} Sh\'{e} on the commuting probability in connection with supersolvability of finite groups. In particular, we prove that if $d(G)>5/16$ then either $G$ is supersolvable, or $G$ isoclinic to $A_4$, or G/\Center(G) is isoclinic to $A_4$