Short description about dihedrals?

[vivaitalia1]

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

 

[Data]

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

 

[vivaitalia1]

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

 

[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.

 

[morphism]

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

 

[Data]

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

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

 

[matt grime]

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