Editing Free group (section)

== Universal property ==
The free group ''F<sub>S</sub>'' is the [[Universal (mathematics)|universal]] group generated by the set ''S''.  This can be formalized by the following [[universal property]]: given any function {{mvar|f}} from ''S'' to a group ''G'', there exists a unique [[group homomorphism|homomorphism]] ''φ'':&nbsp;''F<sub>S</sub>''&nbsp;→&nbsp;''G'' making the following [[commutative diagram|diagram]] commute (where the unnamed mapping denotes the [[Inclusion map|inclusion]] from ''S'' into ''F<sub>S</sub>''):
[[Image:Free Group Universal.svg|center|100px]]
That is, homomorphisms ''F<sub>S</sub>''&nbsp;→&nbsp;''G'' are in one-to-one correspondence with functions ''S''&nbsp;→&nbsp;''G''. For a non-free group, the presence of [[group presentation|relations]] would restrict the possible images of the generators under a homomorphism.

To see how this relates to the constructive definition, think of the mapping from ''S'' to ''F<sub>S</sub>'' as sending each symbol to a word consisting of that symbol. To construct ''φ'' for the given {{mvar|f}}, first note that ''φ'' sends the empty word to the identity of ''G'' and it has to agree with {{mvar|f}} on the elements of ''S''. For the remaining words (consisting of more than one symbol), ''φ'' can be uniquely extended, since it is a homomorphism, i.e., ''φ''(''ab'') = ''φ''(''a'') ''φ''(''b'').

The above property characterizes free groups up to [[isomorphism]], and is sometimes used as an alternative definition.  It is known as the [[universal property]] of free groups, and the generating set ''S'' is called a '''basis''' for ''F<sub>S</sub>''.  The basis for a free group is not uniquely determined.

Being characterized by a universal property is the standard feature of [[free object]]s in [[universal algebra]].  In the language of [[category theory]], the construction of the free group (similar to most constructions of free objects) is a [[functor]] from the [[category of sets]] to the [[category of groups]].  This functor is [[left adjoint]] to the [[forgetful functor]] from groups to sets.