Suppose $G$ is a finite $p$-group with odd $p$. Is it true, that $Aut(G)$ is abelian iff $G$ is cyclic?
When $G$ is cyclic, $Aut(G)$ is indeed abelian.
However, I do not know how to prove the statement that if $Aut(G)$ is abelian, then $G$ is cyclic. Nor do I possess any counterexamples.
The answer to the similar question about nilpotent groups is negative: When is the automorphism group of a finite $p$-group nilpotent?
Also, as any group with abelian automorphism group is metabelian, any $p$-group with abelian automorphism group is nilpotent of degree 2.
Any help will be appreciated.