Supersoluble groups and the probability of generating a supersoluble subgroup

Abstract

Let $G$ be a finite group and let ${\mathrm{P}}_{\mathcal{U}}(G)$ denote the probability that two randomly chosen elements of $G$ generate a supersoluble subgroup. We prove that if ${\mathrm{P}}_{\mathcal{U}}(G)\ge 16/25$ then $G$ is supersoluble, and that the bound $16/25$ is sharp, being attained by the group $G=(C_5\times C_5)⋊Q_8$, where $Q_8$ acts faithfully and irreducibly on $C_5\times C_5$.