The automorphism group for $p$-central $p$-groups

Abstract

A $p$-group $G$ is $p$-central if $G^{p}le Z(G)$‎, ‎and $G$ is‎ ‎$p^{2}$-abelian if $(xy)^{p^{2}}=x^{p^{2}}y^{p^{2}}$ for all $x,yin‎ ‎G$‎. ‎We prove that for $G$ a finite $p^{2}$-abelian $p$-central‎ ‎$p$-group‎, ‎excluding certain cases‎, ‎the order of $G$ divides the‎ ‎order of $text{Aut}(G)$‎.