# dihedral group presentation proof

• Aug 30th 2010, 04:31 PM
hatsoff
dihedral group presentation proof
I'm given the following:

$\displaystyle D_{2n}=\langle r,s|r^n=s^2=e,rs=sr^{-1}\rangle$

This is a group presentation whose rigorous definition has something to do with quotient groups of free groups, or some such. However, I am only given the following non-rigorous definition:

$\displaystyle D_{2n}$ is the group generated by the objects $\displaystyle r$ and $\displaystyle s$, which are known to satisfy the following relations: $\displaystyle r^n=s^2=e$ and $\displaystyle rs=sr^{-1}$ (where $\displaystyle e$ is the identity).

I'm then asked to find the order of the cyclic subgroup generated by $\displaystyle r$. In other words, what is the order of $\displaystyle r$?

I just don't know how to make this work. Obviously the answer is $\displaystyle n$, but I don't know how to prove that from the available information. In fact, I suspect I may have somehow missed something. You can download a pdf of the textbook chapter here (the entire book can be had here, but it's in an unusual format called *.djvu). The exercise in question is #8 from that section (1.2).

if you look at the page 25, there are 6 facts about $\displaystyle D_{2n}.$ the exercise want you to prove the first one, i.e. $\displaystyle 1, r, \cdots , r^{n-1}$ are distinct and $\displaystyle r^n = 1,$ which means the order of $\displaystyle r$ is $\displaystyle n.$
to prove this, you can only use the geometric description of $\displaystyle r,$ i.e. a rotaion of a n-gon, which is fully explained at the beginning of the section.