A note on automorphisms of finite $p$-groups

Let $G$ be a finite non-cyclic $p$-group of order at least $p^3$. If $G$ has an abelian maximal subgroup, or if $G$ has an elementary abelian centre with $C_G(Z(Φ(G))) \ne Φ(G)$, then $|G|$ divides $|\text{Aut}(G)|$.

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