Reference Request: Abelianization of free product is the direct sum of abelianizations

Abelianization of free product is the direct sum of abelianizations

I am interested in the result above, and would like to study a textbook that delves into it deeper.

However the standard algebraic topology textbooks (e.g. Hatcher/Munkres) and standard algebra books like Hungerford do not contain this result.

Is there any suitable book that I can consult? Thanks!

Solution 1:

Let $\{G_i:i\in I\}$ be a collection of groups, $F$ be the free product of the $G_i$ and $A$ the direct sum of the abelianizations of the $G_i$. Free products, direct sums, and abelianizations all have universal properties. The universal property for the direct sum $A$ is that, for any abelian group $B$, we have

$$\mathrm{Hom}(A,B)=\prod_{i\in I}\mathrm{Hom}(G_i/[G_i,G_i],B)\text{.} $$

On the other hand, applying the universal property of abelianization, the free product, and finally abelianization again, gives

$$\mathrm{Hom}(F/[F,F],B)=\mathrm{Hom}(F,B)=\prod_{i\in I}\mathrm{Hom}(G_i,B)=\prod_{i\in I}\mathrm{Hom}(G_i/[G_i,G_i],B)$$.

These equalities (really canonical isomorphisms) constitute an isomorphism of functors $\mathrm{Hom}(A,-)\simeq\mathrm{Hom}(F/[F,F],-)$ from the category of abelian groups to the category of sets. Yoneda's lemma tells us that such an isomorphism is induced by a unique isomorphism $A\simeq F/[F,F]$.