If I know that one presentation for the dihedral group $D_{2n}$ is
$$D_{2n} = \langle r,s\mid r^n = s^2 = 1, rs = sr^{-1}\rangle$$
And also, I know that $$D_{2n} = \{1,r,r^2, \dots , r^{n-1},s,sr,sr^2, \dots , sr^{n-1}\}.$$
How can I find the order of each element?
Any help with this will be greatly appreciated!
Best Answer
Yes. The answer is obvious for elements of the form $r^k$. For elements of the form $sr^k$, you can repeatedly use the relationship $rs=sr^{-1}$ to "walk" the interior $s$ all the way to the left. In particular, you can prove by induction a formula for $(sr^k)^m$.