Why is the coproduct in $\mathsf{Grp}$ so different from the coproduct in $\mathsf{Ab}$?

Question

Why is the coproduct in $\mathsf{Grp}$ so different from the coproduct in $\mathsf{Ab}$? What about $\mathsf{Grp}$ makes for a seemingly far-more-complicated coproduct? If your answer revolves around "$\mathsf{Grp}$ is not an abelian category", please explain in detail.

Answer 1

One quick answer is that the study of abelian groups is the same as the study of $\mathbb Z$-modules, while the study of groups is not at all the study of modules over a principal ideal domain (or any other ring). The theory of modules over a principal ideal domain is very down-to-earth, with a very nice representation theory, rank theory, etc.. Thus, the study of abelian groups is quite simpler than that of all groups.

A more direct, and still algebraic, answer is the following. Quite a lot of the difference between the categories can be seen just by examining free objects. The free abelian group on a set $S$ of generators is isomorphic to the direct product of $\mathbb Z$, with one copy for each generator $s\in S$. This is a very simple description of free objects, and the free objects are not very complicated. Note that the fact that these are indeed the free objects in $Ab$ strongly relies on the commutative property. The relation $ab=ba$ for all generators immediately makes the resulting free object rather small, since many formal elements are identified at once. The situation is very different with groups. Free groups are much larger, and far more complicated already for two generators. As free objects govern, or hint at, many properties in a category, it follows that $Grp$ is much more complicated than $Ab$.

Lastly, an answer with a geometric flavour. Through the theory of fundamental groups and covering spaces, a strong relationship is made between groups and spaces. Not every space has an abelian fundamental group and thus the difference between $Ab$ and $Grp$ is somewhat quantified by the difference between studying just a small portion of spaces (those with abelian fundamental groups) and that of all spaces (remembering that for every group there exists a space with that group as fundamental group).