2-central implies 4-abelian

Statement
Suppose $$G$$ is a 2-central group: its inner automorphism group is a group of exponent two (i.e., an elementary abelian 2-group), or equivalently, every square element of $$G$$ is in the center.

Then, $$G$$ is a 4-abelian group: the fourth power map is an endomorphism, and hence a universal power endomorphism, of $$G$$. In symbols:

$$(xy)^4 = x^4y^4 \ \forall \ x,y \in G$$

Related facts

 * 3-central implies 9-abelian
 * 4-central implies 16-abelian
 * 6-central implies 36-abelian

Facts used

 * 1) uses::Exponent two implies abelian
 * 2) uses::Class two implies commutator map is endomorphism
 * 3) uses::Formula for powers of product in group of class two

Straightforward proof with symbol manipulation
Given: A group $$G$$ such that every square element is in the center of $$G$$. Elements $$x,y \in G$$.

To prove: $$(xy)^4 = x^4y^4$$.

Proof: We start with $$(xy)^4$$ and simplify it in stages

Proof using given facts
Given: A group $$G$$ such that the inner automorphism group of $$G$$ has exponent (dividing) two, i.e., every square element is in the center of $$G$$. Elements $$x,y \in G$$.

To prove: $$(xy)^4 = x^4y^4$$.

Proof: