On the commuting probability and supersolvability of finite groups

Paul Lescot (1) , Hung Ngoc Nguyen (2) , Yong Yang (3)

1 LMRS - Laboratoire de Mathématiques Raphaël Salem
2 Department of Mathematics
3 Department of Mathematics [Madison]

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 $A_4$.

Domaines: Probabilités [math.PR]; Théorie des groupes [math.GR]

Paul Lescot, Hung Ngoc Nguyen, Yong Yang. On the commuting probability and supersolvability of finite groups. 2013. ⟨hal-00878343v2⟩