suppose $G$ is group,if $Aut(G)$ is abelian then $G$ is nilpotent from nilpotency class of at most 2.

Question

suppose $G$ is group,if $Aut(G)$ is abelian then $G$ is nilpotent from nilpotency class of at most 2.

it will be great if you help me how should I prove this.any note or reference will be great.thank you so much.

Answer 1

Hint: look at the inner automorphisms, a subgroup of $Aut(G)$, isomorphic to $G/Z(G)$, where $Z(G)$ is the center of $G$.