[Math] Direct products, direct sums and coproducts in category of groups

abstract-algebracategory-theory

I have couple questions about terms I mentioned in the title.

  1. Why we don't define direct sum of non-abelian groups (subset of direct products which consists of elements with almost every component equal to identity element)?
  2. Why is direct sum used in definition of coproduct in category of Abelian groups but it is not "good" for category of groups in general (I know free groups are used)? Where is essential difference?

Best Answer

Let's take your proposed definition of coproduct, call it $G+H$. Then given two groups $G$ and $H$, there are homomorphisms $G\rightarrow G*H$, and $H\to G*H$. From the universal property, we should get a homomorphism $G+H\to G*H$ with appropriate commutativity properties. Now, if $g\in G$ and $h\in H$ are not identity elements we know that $gh\in G+H$ has to map to $gh\in G*H$. And $hg\in G+H$ has to map to $hg\in G*H$. But, $gh=hg$ in $G+H$, and $gh\neq hg$ in $G*H$.

Related Question