Category of groups - Definition and Overview

In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory.

The category Grp is both complete and co-complete. The product is just the direct product while the coproduct is the free product. The zero objects in Grp are the trivial groups (consisting of just an identity element).

The category of abelian groups, Ab, is a full subcategory of Grp.