Groups form a full subcategory of monoids

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The category of groups forms a full subcategory of the category of monoids, via the functor that sends a group to its underlying monoids forgetting the inverse map.

In other words, the forgetful functor from the category of groups to the category of monoids is full, faithful and injective.

Related facts



This follows from the fact that the inverse map on a group is completely determined by its binary operation and identity element. Further information: Equivalence of definitions of group


This follows from the fact that a group homomorphism is completely described by what it does at the set level, hence, it is completely described by the corresponding monoid homomorphism.


This follows from the fact that, to check that a set-theoretic map of groups is a group homomorphism, it suffices to check that it preserves the binary operation.