Finite p-groups in which the cores of all the nonnormal subgroups are in the center

Abstract

Let $G$ be a finite $p$-group. Then $G$ is said to be a $CZ$-group if $H_G\le Z(G)$ for every nonnormal subgroup $H$ of $G$. In this paper, we study the $CZ$-group $G$ and get $c(G)\le 3$. It is proved that $\mathrm{\text{exp}}(G^{\prime })=p$ if $c(G)=2$ and $\left|G\right|\le p^5$ if $c(G)=3$.