Groups form a full subcategory of semigroups

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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

Facts used

  1. Equivalence of definitions of group
  2. Equivalence of definitions of homomorphism of groups

Proof

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.