Every group is a quotient of a free group

Statement
Suppose $$G$$ is a group and $$S$$ is a generating set for $$G$$. Then, $$G$$ is isomorphic to a quotient group of a free group of rank equal to the cardinality of $$S$$.

More specifically, we let $$T$$ be a set in bijection with $$S$$, and $$F(T)$$ be the free group on $$T$$, and we can construct a surjective homomorphism:

$$\! F(T) \to G$$

which is the unique homomorphism sending each element in $$T$$ to its image in $$S$$ under the bijection.

Particular cases

 * Every finitely generated group is a quotient of a finitely generated free group.
 * In general, every infinite group is a quotient of an infinite free group of the same cardinality.

Related facts

 * Free group of rank two is SQ-universal: This states that every finitely generated group is isomorphic to a subquotient of the free group of rank two.
 * Cayley's theorem: This states that every group is a subgroup of a symmetric group on itself as a set.
 * Every finite group is a subgroup of a finite simple group
 * Every finite group is a subgroup of a finite complete group
 * Every group is a subgroup of a complete group