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In the following post by DavidLHarden : See Here

He quoted the following claim: "There is a theorem that says that if $p$ is a prime and $|G|=p^n$ , then $|AutG|$ divides $\Pi_{k=0}^{n-1} (p^{n}-p^{k})$ " .

I can't find any reference for this theorem , and although Jr. Hall gives something similar as theorem 12.2, it still isn't in this form.

Does someone know of any reference for this fact?

He quoted the following claim: "There is a theorem that says that if $p$ is a prime and $|G|=p^n$ , then $|AutG|$ divides $\Pi_{k=0}^{n-1} (p^{n}-p^{k})$ " .