Finite $p$-groups in which the normalizer of each nonnormal subgroup is small

Abstract

Let $G$ be a finite non-Dedekindian $p$-group which satisfies $N_G(H)=HZ(G)$ for each nonnormal subgroup $H$, we call it an $NS$-group. In this paper, it is proved that an $NS$-group is the product of minimal nonabelian group and the center.