# How to convert one presentation into another? Please explain using a dihedral group as an example.

How can we convert a given presentation of a group $$G$$ into an another presentation?

Would anyone please explain to me by converting two different presentations of a dihedral group?

• – Derek Holt Nov 20 '16 at 19:32

One way to change a presentation of a group $$G$$ into another presentation of the same group is via Tietze transformations. They introduce or delete either generators or relations.

To use the example of the dihedral group $$D_n$$ of $$2n$$ elements, let's start with

$$P=\langle a, b\mid a^2, b^n, (ab)^2\rangle.$$

Let $$x=ab$$ be an element of $$D_n$$. Then $$a\stackrel{(1)}{=}xb^{-1}$$, so, introducing $$x$$ as a generator gives

$$P\cong\langle a, b, x\mid a^2, b^n, (ab)^2, x=ab\rangle,$$

which is then isomorphic to

$$Q=\langle b, x\mid (xb^{-1})^2, b^n, x^2\rangle$$

by eliminating $$a$$ (since $$(1)$$ tells us that it can be written as a product of the other generators, not including $$a$$).

Then $$Q$$ is a "new" presentation of $$D_n$$, although not entirely different from $$P$$.