Are dihedral groups well defined by their generating groups?

Question

It is well known that $D_n=<r,s | r^n=s^2=id, srs=r^{-1}>$. Now, given a group of 2n elements and a generating group of two elements satisfying the above relations is it isomorphic to $D_n$?

Answer 1

Let $F$ be a free group on a finite set $X$ and let $R$ be a subset of $F$. The normal closure of $\left\langle \bar{R} \right\rangle$ of $R$ in $F$ is a the intersection of all normal subgroups of $F$ containing $R$ and as you see it is uniquely determined. Then the group $F / \left\langle \bar{R} \right\rangle$ is denoted by $\left\langle X|R \right\rangle$. Therefore two groups with the same presentation are isomorphic. In your example, $X = \{r,s\}$ and $R = \{r^n, s^2, (sr)^2\}$.