Free group

Definition
Note that the notions of freely generating set described in formulations (2) and (3) of the definition are equivalent.

Category-theoretic formulation
We can consider the free group functor: the functor that associates to any set, the group generated freely by that set. This is a functor because any map of sets gives rise to a map of the corresponding free groups.

The free group functor can be defined as the left adjoint to the forgetful functor from groups to sets. In other words, if $$\mathcal{U}$$ denotes the forgetful functor from groups to sets (that sends a group to its underlying set) and $$\mathcal{F}$$ denotes the free group functor, then for any set $$A$$ and group $$G$$, there is a natural isomorphism of sets:

$$\operatorname{Hom}(\mathcal{F}(A),G) = \operatorname{Hom}(A,\mathcal{U}(G))$$

where the left side is the set of group homomorphisms and the right set is the set of set homomorphisms (i.e., all the set-theoretic maps).

Examples

 * The free group on the empty set is the trivial group (this isn't typically considered a free group).
 * The free group on a set of size one is isomorphic to the group of integers $$\mathbb{Z}$$, i.e., it is infinite cyclic. it is the only Abelian nontrivial free group).
 * The free group on a set of size two is an important free group. It is non-Abelian, finitely generated, and is SQ-universal: every finitely generated group is a subquotient of this group

Stronger properties
Group properties stronger than the property of being free are:

Facts
The cardinalities of any two freely generating sets of the same free group are equal. This result actually follows from the fact that the corresponding result is true for free Abelian groups.

This cardinality is termed the rank of the free group. It is further clear that any two free groups of the same rank are isomorphic.

Metaproperties
More information on these metaproperties:

Every subgroup of a free group is free. This result is fairly nontrivial. In general, however, the number of generators of the subgroup could be bigger or smaller than the number of generators of the whole group.

Quotient-closedness
The property of being free is far from quotient-closed -- in fact, the quotient-closure of the property of being free is the property of being any group, viz any group can be expressed as a quotient of a free group by a normal subgroup.

In fact, if we take an arbitrary group $$G$$, and a generating set $$S$$ for $$G$$, we can express $$G$$ as the quotient of the free group $$F(S)$$ by the normal subgroup comprising all words in elements of $$S$$ that reduce to the identity element in $$G$$.

This is the idea behind a presentation of a group, for instance.

Direct product-closedness
A direct product of free groups is not free. This follows from the fact that we get commutation relation between the elements in the two direct factors, but a free group is centerless.

Free product-closedness
A free product of free groups is free. Here, the freely generating set of the free product is the union of the freely generating sets of the free factors.