A criterion for a group to be nilpotent

doi:10.1285/i15900932v20n1p33

A criterion for a group to be nilpotent

Neil Flowers

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 $p \in \pi(G)$ .