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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

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2 Answers 2

up vote 1 down vote accepted

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.

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Define "more concrete". Do you mean "a more natural way of thinking about it?" –  user1729 Jan 28 '13 at 10:36
    
@Andreas Caranti Thanks for answering. So finding the elements of $D_{2n}$ or composing it's elements is not as direct as the elements and the compositions of the elements in $Z_n$, for instance, right? There is no easy and direct way to learn these (unlike $Z_n$), so we always need to relate it something else? –  user58289 Jan 28 '13 at 10:52
    
@user1729 I am not claiming it is more natural, it is just that some may prefer this approach, at least when making first knowledge with dihedral groups. –  Andreas Caranti Jan 28 '13 at 11:00
    
@Artus "easy and direct" is a matter of opinion, familiarity or taste. Note that wou can write the maps above also in terms of matrices $\begin{bmatrix}\pm 1 & 0\\b & 1\end{bmatrix}$, which some my consider easy and direct. –  Andreas Caranti Jan 28 '13 at 11:03
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  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$

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Thanks. For question 1), let's say we are looking at $r^2$ (or any $r^k$ for that matter)...we cannot tell the angle of rotation unless we draw the picture, right? –  user58289 Jan 28 '13 at 10:45
    
@Artus, no pictures needed. If $r$ is a rotation of an angle $\alpha$, then $r^k$ is a rotation of an angle $k \alpha$. –  Andreas Caranti Jan 28 '13 at 11:04
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