Group theory MOC

Free group

Free groups are the free objects in Category of groups. Let be a set. Then has the Group presentation , i.e. it is the minimal completion of so that it becomes a group. 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.

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