Free Group - Universal Property

Universal Property

The free group F_S is the universal group generated by the set S. This can be formalized by the following universal property: given any function ƒ from S to a group G, there exists a unique homomorphism φ: F_S → G making the following diagram commute:

That is, homomorphisms F_S → G are in one-to-one correspondence with functions S → G. For a non-free group, the presence of 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_S as sending each symbol to a word consisting of that symbol. To construct φ for given ƒ, first note that φ sends the empty word to identity of G and it has to agree with ƒ 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_S. The basis for a free group is not uniquely determined.

Being characterized by a universal property is the standard feature of free objects 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.