Every group is a quotient of a free group

This article gives the statement, and possibly proof, of a group property (i.e., free group) satisfying a group metaproperty (i.e., quotient-universal group property)
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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