1. Apr 13, 2007

vivaitalia1

Can someone give me a short description about dihedrals? for example what would be the elements of D10? or D4?

2. Apr 13, 2007

Data

Well, do you know the geometric interpretation of a dihedral group? The elements are (in the geometric interpretation) rotations and reflections.

3. Apr 13, 2007

vivaitalia1

Yes I know the geometric interpretation. Is it the same for abstract algebra?

4. Apr 13, 2007

Data

Yes, the geometric interpretation is the motivation for dihedral groups. As I said, the elements are analogous to rotations and reflections. In general a group presentation for $D_n$ is

$$\langle s, r| s^n = 1, r^2 = 1 \rangle.$$

In that presentation, s corresponds to a rotation and r to a particular reflection.

Last edited: Apr 13, 2007
5. Apr 13, 2007

morphism

You also need $sr = rs^{-1}$ in that presentation.

6. Apr 13, 2007

Data

Indeed! I should be more careful. The group in my post isn't even finite for n>1. :yuck:

Last edited: Apr 14, 2007
7. Apr 14, 2007

matt grime

It is the free product of C_n and C_2, in fact.