Nilpotent automorphism group implies nilpotent of class at most one more

Statement
Suppose $$G$$ is a group whose automorphism group is nilpotent, i.e., it has a nilpotent automorphism group. Then, $$G$$ is itself a fact about::nilpotent group. Moreover, the fact about::nilpotency class of $$G$$ is at most one more than the nilpotency class of $$\operatorname{Aut}(G)$$.

Similar facts

 * Aut-abelian implies class two: This is a special case where the nilpotency class of the automorphism group is one.
 * Trivial automorphism group implies trivial or order two: This is a stronger statement in the special case where the nilpotency class of the automorphism group is zero.
 * Aut-solvable implies solvable of derived length at most one more: A corresponding statement for solvability.

Converse
The converse is not true: nilpotent not implies nilpotent automorphism group and nilpotency class of group gives no upper bound on nilpotency class of automorphism group.

Facts used

 * 1) uses::Group acts as automorphisms by conjugation
 * 2) uses::Nilpotency is subgroup-closed (and the class of the subgroup is at most the class of the whole group).

Proof
Given: A group $$G$$ whose automorphism group is nilpotent of class $$c$$.

To prove: $$G$$ is nilpotent of class at most $$c + 1$$.

Proof: We need to show that $$G$$ is nilpotent and the upper central series of $$G$$ has length at most $$c + 1$$. To do this, it suffices to show that the upper central series of the quotient $$G/Z(G)$$ (where $$Z(G)$$ is the center of $$G$$) has length at most $$c$$.

By fact (1), there is a homomorphism $$G \to \operatorname{Aut}(G)$$ given by conjugation action. The kernel of this homomorphism is $$Z(G)$$ and the image is a subgroup of $$\operatorname{Aut}(G)$$ known as the inner automorphism group, and denoted $$\operatorname{Inn}(G)$$. Thus, $$G/Z(G)$$ is isomorphic to the subgroup $$\operatorname{Inn}(G)$$ of $$\operatorname{Aut}(G)$$. By fact (2), $$\operatorname{Inn}(G)$$, and hence $$G/Z(G)$$, is nilpotent of class at most $$c$$, completing the proof.