The dihedral group expresses the symmetries of a regular n-gons (n rotations and n reflections so it is O(2n)).

D2= symmetries of a rectangle = {e,a,b,c} where a = reflection, b= reflection through a different axis, and c=rotation by Pi.

a^2=e, b^2=e, c^2=e, ab=c=ba

All of the other relations between e,a,b,c should be derivable from this.

D3 is the simplest non-abelian group. It is the symmetries of a triangle. D3= {e,c,c^2,b,bc,bc^2}

c= rotation by Pi/3 b= reflection through a fixed axis of reflection.

When applying transformations, apply the leftmost one first.

To see non-abelian consider:

(bc)^2=e=bcbc

=> b=cbc (b^2=e) => c^2b=bc (c^3=e)

=> non-abelian

source

jl@crush.caltech.edu index

example

Dn