FINITE -GROUPS ALL OF WHOSE NONNORMAL SUBGROUPS HAVE BOUNDED NORMAL CORES

Abstract

Given a positive integer $m$, a finite $p$-group $G$ is called a $BC(p^{m})$-group if $|H_{G}|\leq p^{m}$ for every nonnormal subgroup $H$ of $G$, where $H_{G}$ is the normal core of $H$ in $G$. We show that $m+2$ is an upper bound for the nilpotent class of a finite $BC(p^{m})$-group and obtain a necessary and sufficient condition for a $p$-group to be of maximal class. We also classify the $BC(p)$-groups.