Group theory and generalizations Example

Category of groups

Example

The category of groups $\mathbf{Grp}$ has the groups as objects and the group homomorphisms $f \colon G \to H$ as morphisms. Composition is the composition of maps, and the identity morphism of $G$ is the identity map $\id_G$ . This is a category. If $f \colon G \to H$ and $g \colon H \to K$ are homomorphisms, then $g(f(xy)) = g(f(x)f(y)) = g(f(x))\,g(f(y))$ , so $g \circ f$ is a homomorphism. The identity map is a homomorphism, and composition of maps is associative, so both axioms of a category hold.

Remarks

The trivial group is a zero object: from each group there is exactly one homomorphism to it, and from it exactly one homomorphism to each group. The product of $G$ and $H$ is the direct product $G \times H$ , and the coproduct is the free product $G * H$ . The monomorphisms are the injective homomorphisms, and the epimorphisms are the surjective homomorphisms. Forgetting the group structure gives a functor to the category of sets, whose left adjoint sends a set to the free group on it. $\mathbf{Grp}$ is not an abelian category: for example, the coproduct $\Z * \Z$ differs from the product $\Z \times \Z$ , since the first group is not abelian.

Reading path

To understand this page from the ground up, read these 3 pages in order. The foundations come first.

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Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.

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Group concept
Group homomorphism concept
Category of groups example

Tag	`2CX7RPPXA9`
URI	`m2://2CX7RPPXA9`
Kind	example
Writing	W0 stub
MSC	20J15, 18A05
Instance of	Category
Uses	Group, Group homomorphism
Same as	Wikidata Q3912012 Wikipedia Category of groups
Changed	2 November 2026

Category of groups

Example

Remarks

Reading path

Further reading

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