# Proof of theorem: If $G$ is a finite nonabelian $p$-group, then $\mathrm{Aut}_c(G)=\mathrm{Inn}(G)$ if and only if $G′=Z(G)$ and $Z(G)$ is cyclic.

Consider the following theorem:

If $G$ is a finite nonabelian $p$-group, then $\operatorname{Aut}_c(G)=\operatorname{Inn}(G)$ if and only if $G′=Z(G)$ and $Z(G)$ is cyclic.

Notation

• $p$ is a prime number
• $G'$ is the commutator subgroup of $G$
• $Z(G)$ is the center of $G$
• $\operatorname{Aut}(G)$ is the automorphism group of $G$
• $\operatorname{Inn}(G)$ is the group of inner automorphisms of $G$
• $\operatorname{Aut}_c(G)$ is the group of central automorphisms of $G$. These are automorphisms which commute with every element of $\operatorname{Inn}(G)$.

I am looking for a proof which is available online free of charge. There is a reference in Centralizer of $Inn(G)$ in $Aut(G)$ but it is not free of charge.

This is not any form of homework or other assignment.

Thank you very much for any help, ideas or references!!

• Is this for general interest, or an assignment? Have you tried it yourself? It's also good practise to put the main question (in this case, the theorem in your title) in the body of your question as well. – Bill Wallis May 6 '18 at 15:34
• @Bill Wallis: This is for my personal interest. I have no assignments. Thanky you for the advice! – Moritz May 6 '18 at 17:09