Finite Groups with a Subnormality Condition

Abstract

Let $m$ and $n$ be positive integers. We study ${\mathcal{S}}(m,n)$, the class of all groups $G$ that for all subsets $M$ and $N$ of $G$ of sizes $m$ and $n$, there exist $x\in M$ and $y\in N$ such that $\langle x\rangle$ is subnormal in $\langle x,y\rangle$. In fact, if $G$ is a finite group, then we find some sharp bounds for $m$ and $n$ such that $G\in{\mathcal{S}}(m,n)$ implies that $G$ is a nilpotent group. Also we find a bound depending only on $m$ and $n$ for the order of all nonnilpotent finite groups in ${\mathcal{S}}(m,n)$.