Finite $p$-groups of class 2 have noninner automorphisms of order $p$

We prove that for any prime number $p$, every finite non-abelian $p$-group $G$ of class 2 has a noninner automorphism of order $p$ leaving either the Frattini subgroup $Φ(G)$ or $Ω_1(Z(G))$ elementwise fixed.

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