Category of groups

This article describes a category (in the mathematical sense) where the notion of "object" is groupand the notion of morphism is homomorphism of groups. In other words, it gives a category structure to the collection of all groups.
View other category structures on groups

This article describes a way of viewing the collection of groups as a structure in its own right

Definition

The category of groups is defined as follows:

Aspect	Name	Definition/description
objects	groups	A group is a set with associative binary operation admitting an identity element and inverse map.
morphisms	homomorphisms of groups	A homomorphism between groups $G$ and $H$ is a set map $\phi :G\to H$ such that $\phi (xy)=\phi (x)\phi (y)$ for all $x,y\in G$. Note that this also forces that it preserves the identity element and the inverse map, and some definitions include these additional (redundant) conditions.
composition of morphisms	compose the homomorphisms as set maps	Given homomorphisms $\alpha :G\to H$ and $\beta :H\to K$, the composite is the set composition $\beta \circ \alpha$, a homomorphism $G\to K$.
identity morphism	identity map from a group to itself.

Categorical constants and constructs

Construct	Name in this category	Definition/description
isomorphism	isomorphism of groups	A bijective homomorphism; equivalently, a homomorphism whose inverse map is also a homomorphism.
monomorphism	injective homomorphism	The kernel of the mapping is trivial. Alternatively, it can be identified with a subgroup inclusion mapping. This is relatively straightforward to prove; see monomorphism iff injective in the category of groups
epimorphism	surjective homomorphism	The mapping is surjective. This is not immediately obvious, see epimorphism iff surjective in the category of groups
zero object	trivial group	The group with one element, namely its identity element.
categorical product	the usual external direct product	We take the external direct product with the coordinate-wise projection maps.
categorical coproduct	the usual external free product	We take the external free product with the natural inclusions.

Important functors

For lists of important functors involving the category of groups, refer the following:

Forgetful functor to sets

This functor takes a group and outputs the underlying set of the group. The forgetful functor is a faithful functor, because any group homomorphism is completely determined by the corresponding map at the set-theoretic level. Thus, the category of groups is a concrete category. This in particular shows that the category of groups is a locally small category.

Facts

• Category of groups has no nontrivial auto-equivalences