# Category of groups

In mathematics, the category **Grp** (or **Gp**^{[1]}) 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.

There are two forgetful functors from **Grp**, M: **Grp** → **Mon** from groups to monoids and U: **Grp** → **Set** from groups to sets. M has two adjoints: one right, I: **Mon**→**Grp**, and one left, K: **Mon**→**Grp**. I: **Mon**→**Grp** is the functor sending every monoid to the submonoid of invertible elements and K: **Mon**→**Grp** the functor sending every monoid to the Grothendieck group of that monoid. The forgetful functor U: **Grp** → **Set** has a left adjoint given by the composite KF: **Set**→**Mon**→**Grp**, where F is the free functor; this functor assigns to every set *S* the free group on *S.*

The monomorphisms in **Grp** are precisely the injective homomorphisms, the epimorphisms are precisely the surjective homomorphisms, and the isomorphisms are precisely the bijective homomorphisms.

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

The notion of exact sequence is meaningful in **Grp**, and some results from the theory of abelian categories, such as the nine lemma, the five lemma, and their consequences hold true in **Grp**. The snake lemma however is not true in **Grp**.