Let $G$ be a group generated by $x,y,z$ such that the product $xyz$ is in its centre; that is, $xyz$ commutes with any element $g\in G$.
- What is the "freest" example of such a group $G$? Is it isomorphic to any familiar group?
- Are there any other non-Abelian groups that satisfy this property?
I can prove that the inner homomorphism group of $G$ is isomorphic to the free group with two generators. Another fact that is not hard to deduce is that $xyz=yzx=zxy$. Not sure if this helps.