Groups form a full subcategory of semigroups

Statement
The category of groups forms a full subcategory of the category of semigroups, via the forgetful functor that sends a group to its underlying semigroup (forgetting the identity element and inverse map).

In other words, the forgetful functor from groups to semigroups, that sends a group to its underlying semigroup, is full, faithful, and injective.

Related facts

 * Groups form a full subcategory of monoids
 * Monoids do not form a full subcategory of semigroups

Facts used

 * 1) uses::Equivalence of definitions of group
 * 2) uses::Equivalence of definitions of homomorphism of groups

Injectivity
This follows from Fact (1), which says that that the identity element and inverse map in a group are completely determined by the binary operation.

Faithfulness
This follows from the fact that a homomorphism of groups is completely described by what it does at the set level, and hence is completely described by the corresponding homomorphism of semigroups.

Fullness
The follows from Fact (2), which says that, to check that a map of groups is a homomorphism, it suffices to check that it preserves the group multiplication.