ON THE PROBABILITY OF GENERATING NILPOTENT SUBGROUPS IN A FINITE GROUP

Abstract

Let $G$ be a finite group. We denote by ${\it\nu}(G)$ the probability that two randomly chosen elements of $G$ generate a nilpotent subgroup and by $\text{Nil}_{G}(x)$ the set of elements $y\in G$ such that $\langle x,y\rangle$ is a nilpotent subgroup. A group $G$ is called an ${\mathcal{N}}$ -group if $\text{Nil}_{G}(x)$ is a subgroup of $G$ for all $x\in G$ . We prove that if $G$ is an ${\mathcal{N}}$ -group with ${\it\nu}(G)>\frac{1}{12}$ , then $G$ is soluble. Also, we classify semisimple ${\mathcal{N}}$ -groups with ${\it\nu}(G)=\frac{1}{12}$ .

Locations

• Bulletin of the Australian Mathematical Society - View - PDF