Abelian automorphism group implies class two

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., group whose automorphism group is abelian) must also satisfy the second group property (i.e., group of nilpotency class two)
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This article gives a result about how information about the structure of the automorphism group of a group (abstractly, or in action) can control the structure of the group
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Statement

Let $G$ be a group whose automorphism group is abelian, i.e., the automorphism group of $G$ is an abelian group. Then, $G$ is a group of nilpotency class two

Related facts

Similar facts

• Nilpotent automorphism group implies nilpotent of class at most one more
• Trivial automorphism group implies trivial or order two

Other related facts

• Cyclic implies abelian automorphism group
• Abelian automorphism group not implies abelian
• Locally cyclic implies abelian automorphism group
• Abelian and abelian automorphism group not implies locally cyclic
• Finite abelian and abelian automorphism group implies cyclic

Facts used

1. Group acts as automorphisms by conjugation
2. Abelianness is subgroup-closed

Proof

Since $\operatorname {Aut} (G)$ is an abelian group, the subgroup $\operatorname {Inn} (G)$, the inner automorphism group is also an abelian group. But $\operatorname {Inn} (G)\cong G/Z(G)$, where $Z(G)$ is the center of $G$. Thus, $G/Z(G)$ is abelian, so $G$ has class two.