3-central implies 9-abelian

Statement

Suppose $G$ is a 2-central group: its inner automorphism group is a group of exponent three, or equivalently, every cube element of $G$ is in the center.

Then, $G$ is a 9-abelian group: the ninth power map is an endomorphism, and hence a universal power endomorphism, of $G$. In symbols:

$(xy)^{9}=x^{9}y^{9}\ \forall \ x,y\in G$

Related facts

• 2-central implies 4-abelian
• 4-central implies 16-abelian
• 6-central implies 36-abelian