A criterion for a group to be nilpotent

Abstract

Let $G$ be a group with $|\pi(G)| \geq 3$. In this paper it is shown that $G$ is nilpotent if and only if for every subgroup $H$ of $G$ with$|\pi(H)| \geq 2$ we have $P \cap H \in \mbox{Syl}_{p}(H)$ for each $P \in \mbox{Syl}_{p}(G)$ and for every $p \in \pi(G)$.