Groups form a full subcategory of monoids
In other words, the forgetful functor from the category of groups to the category of monoids is full, faithful and injective.
- Groups form a full subcategory of semigroups: This is actually a somewhat stronger fact.
- Monoids do not form a full subcategory of semigroups
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.