Non-inner automorphisms of order $p$ in finite $p$-groups admitting cyclic center

Abstract

Let $G$ be a finite non-abelian $p$-group admitting cyclic center and $p$ be an odd prime. In this paper, we prove that if $C_{G}(Z(\gamma_{3}(G)G^{p}))\nleqslant\gamma_{3}(G)G^{p}$, then $G$ has a non-inner automorphism of order $p$.