On the nilpotent probability and supersolvability of finite groups

Abstract

Let G be a finite group. We denote by $Nil_G(x)$ the set of elements $y\in G$ such that $\langle x,y\rangle$ is a nilpotent subgroup and by $\nu _1(G)$ and $\nu (G)$ the probability that two randomly chosen elements of G respectively generate an abelian subgroup and a nilpotent subgroup. A group G is called an ${\mathcal {N}}$-group if $Nil_G(x)$ is a group for all $x\in G$. It is proved that G is supersolvable if there exists a normal subgroup H such that either $\nu _1(H)>\frac{1}{3}|G:H|$ or G is an ${\mathcal {N}}$-group and $\nu (H)>\frac{1}{3}|G:H|$.