I was reading articles and books on automorphism group. It is always an interesting question to decide when the automorphism group is a $p$-group.

In this regard my question is

Let $\gamma(H)$ be the last non-trivial term of lower-central series of a finite $p$-group $H$. Suppose automorphism group of $G/\gamma(H)$ is a $p$-group. Then prove or disprove that the automorphism group of $H$ be a $p$-group

Sorry for not showing much effort from my side, I will appreciate any help.

Thanks in advance.

  • 2
    $\begingroup$ It's enough to prove that the subgroup of ${\rm Aut}(H)$ that induces the identity automorphism of $H/\gamma(H)$ is a $p$-group. Also note that any such automorphism induces the identity on $\gamma(H)$. $\endgroup$ – Derek Holt Sep 13 '18 at 6:23

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