Dihedral Groups…

Question

I have two questions about dihedral groups:

1) Is there a way of knowing all elements of $D_{2n}$? In other words, to find the elements of $D_{2n}$ for any n, do we need to draw the picture and then look at what the elements are...or is there a way of knowing the elements without actually having to draw the picture?

2) When computing compositions of the elements do we need to draw the picture to see what the result is, or is there a way to compute the compositions from your head?

Thanks in advance

Answer 1

One way to compute in the dihedral group of order $2 n$ is to regard it as the group of maps $x \mapsto \pm x + b$ on $\mathbf{Z}_n$, for $b \in \mathbf{Z}_n$.

You will also find it described in terms of generators and relations, which is of course a very important point of view; the approach above is possibly more concrete.

Answer 2

1. The dihedral group of order $2n$ is usually comprised of the elements $$\{1,r,r^2,\ldots,r^{n-1},s ,sr,sr^2,\ldots, sr^{n-1}$$ where $r$ represent the usual notion of a rotation while $s$ the operation of relefction.

2. The most useful rule for making computations with the dihedral group is the following $$sr^k = r^{-k}s$$ for all $ 0 \leq k \leq n$