# Is there a canonical “inverse” of Abelianization?

We know that the abelianization of a free product is the direct sum, for example $$Ab(\mathbb{Z}*\mathbb{Z})=\mathbb{Z}\oplus\mathbb{Z}$$.

Is there a “canonical” (or even non-canonical) operator that reverses the process of Abelianization? (For instance, it may goes from direct sum to free product?)

Thanks.

• If you want a group with a given abelianization $A$, one choice is $A$. That is boring, but you're unlikely to get an interesting answer in a canonical way. – KCd Apr 22 at 6:08

The abelianization functor, from the category of groups to the category of abelian groups, has a right adjoint, which is the forgetful functor, in other words the functor that sends an abelian group $$A$$ to the group $$A$$ itself. I guess this is as close to an "inverse" as you can reasonably hope.