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In this paper we study the longstanding conjecture of whether there exists a non-inner automorphism of order $p$ for a finite non-abelian $p$-group. we prove that if $G$ is a finite non-abelian $p$-group such that $G/Z(G)$ is powerful then $G$ has a non-inner automorphism of order $p$ leaving either $\Phi (G)$ or $\Omega_1(Z(G))$ elementwise fixed. We also recall a connection between the conjecture and a cohomological problem and we give an alternative proof of the latter result for add p, by showing that the Tate cohomology $H^n(G/N,Z(N))\neq 0$ for all $n\geqslant 0$ , where $G$ is a finite $p$-group, $p$ is odd, $G/Z(G)$ is $p$-central (i.e., elements of order $p$ are central) and $N\vartriangleleft G$ with $G/N$ non-cyclic.
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