Free group

Free groups are the free objects in . Let be a set. Then has the Group presentation , i.e. it is the minimal completion of so that it becomes a group. #m/def/group Concretely, is constructed by

Inserting the identity
Adding an inverse for each
Words (expressions made of group members) are only considered equal if the group laws demand so.

Likewise for any there exists a unique , which is just the homomorphic extension of mapping each single-element to the corresponding .

Universal property

The free group has a unique extension to a functor so that the natural injection becomes a natural transformation (thus creäting a Free-forgetful adjunction). This is enabled by characterising with the following universal property:

If is a group and is a function there exists a unique so that , i.e. the following diagram commutes

Proof

Let denote the product in the free group. Then for any , with and . It follows that

So is already determined by . Thus fulfils the universal property. If also satisfies the universal property than the following diagram commutes:

giving the required unique isomorphism.

#state/tidy | #lang/en | #SemBr