[[Group theory MOC]]
# Category of groups

The **category of groups** $\Grp$ is the [[category]] where each object is a [[group]] and each morphism is a [[group homomorphism]]. #m/def/group 

## Properties

- The [[Initial and terminal objects]] are both the [[Trivial group]] $\{ e \}$
- The [[Products and coproducts]] are the [[Direct product of groups]] and the [[Free product of groups]] respectively
- A full subcategory is [[Category of abelian groups]].

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