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.

Question

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!!

Answer 1

The paper you are looking for is this: "Central automorphisms that are almost inner", by Curran and McCaughan. Can you provide an e-mail address? I'll be happy to send you a copy.