Hi Everyone,
I was wondering if anyone could explain to me what Dihedral Groups are and perhaps list some properties of them, or distinctive characteristics (if any), ect.
Thanks in advance.
Dihedral Groups are groups that deal with the symmetries of the n-gon.
If we have the group $\displaystyle \left(D_n,\circ\right)\,\, (\forall\,n\geqslant3)$ (dihedral group under operation of composition), the only elements in the group are the rotation elements and the reflection elements about the axes.
Some interesting characteristics are:
1) if you compose one reflection with another reflection, you get a rotation
2) if you compose one rotation with another rotation, you get a rotation.
3) if you compose a rotation with a reflection (or vice versa), you get a reflection.
4) $\displaystyle \left(D_n,\circ\right)$ has a commutative subgroup (rotations). I.e. in $\displaystyle D_3$ (the symmetry group of the equilateral triangle), the rotation subgroup is $\displaystyle \left\lbrace R_0,R_{120},R_{240}\right\rbrace$. It is apparent that $\displaystyle R_{120}\circ R_{240}=R_0=R_{240}\circ R_{120}$
5) $\displaystyle \left(D_n,\circ\right)$ has an order of $\displaystyle 2n$ (order refers to the number of elements the group has [i.e. $\displaystyle D_3$ has 6 elements $\displaystyle \left\lbrace R_0,R_{120},R_{240},D,D^{\prime},D^{\prime\prime}\ right\rbrace$]).
There are probably more things, but that's all I can think of at the moment.
Does this shed enough light on what Dihedral Groups are? You can always find more information on the web. You can find more info here.
This is just gonna be pretty informal. Basically, it is a non abelian finite group of symmetries of polygons. So you usually denote it $\displaystyle D_{2n}$ or $\displaystyle D_n$ depending on what kind of person you are. I prefer the former because it tells you the order of the group, the latter tells you you are looking at a regular n-gon.
The group is generated by two elements r and s, the rotation and the flip. r has order n and s has order 2. This makes sense geometrically, if you rotate a square 4 times it comes back to the original position, similarly for a flip. They interact as follows, $\displaystyle rs=sr^{-1}$. This gives you a complte presentation for the group $\displaystyle D_{2n}=\{r^id^j|i,j \in \mathbb{Z} r^n=1=s^2, rs=sr^{-1}\}$.
If n is odd it has trivial center, if it is even it has center $\displaystyle Z(D_{2n})=\{1,r^{n/2}\}$
It always has a cyclic subgroup of order n, the group of rotations.
You can find the representation in terms of permutations by drawing the polygon and labeling the vertex and writing down the permutations for a rotation (vertex 1 goes to 2, 2 to 3, ... n to 1) and a flip (you would have to think about that one, I dunno offhand).
I dunno, that is all that comes to mind.
Hope it helped