Group Presentation

set, where is the Free Group.
Let be the smallest normal subgroup of containing
group with generator and relative
Data is called a presentation of the group (finite presentation if and are finite sets)

Universal property of group presentations

For any group , the function

is bijective.
*where

(Note that above is the natural homomorphism induced by in Universal property of free groups)

Proof

Suppose , give
By the universal property of free groups then also , so:

(Consider )
As surjective have .

Now assume st the associated is st
for all . Then
, so (by minimality)
Then descends to a well defined

Corollary

Every group has a presentation.

Proof

Let be a group and the identity map.
Find the induced homomorphism by Universal property of free groups.
Let .
Then induces a homomorphism which is an isomorphism by the The isomorphism theorems and noting that was surjective.

Examples