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 Free Group > 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 Free Group > 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