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?

Thanks in advance.


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 $P$.


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