The
Dihedral group Dn is the
symmetry group
of the regular
n-gon
1.
This group has 2n elements. They are the rotations
given by the powers of r, rotation anti-clockwise through
2pi/n, and the n reflections given by reflection
in the line through a vertex (or the midpoint of an edge) and the centre of the polygon.
|
\ | /
\ ----|---- /
| \ | / |
| \ | / |
------------------
| / | \ |
| / | \ |
/ ----|---- \
/ | \
Let
s denote one of these reflections.
It's not hard to see that
Dn={1,r,r2,...,rn-1,s,rs,...,rn-1s}
and that we have the relations:
rn=1 s2=1 and
srs-1=r-1. In fact the
Dihedral group is given by these
generators and
relations.
If we label the vertices of the polygon by {1,2,...,n}
then each symmetry of the polygon gives rise to a permutation
of {1,2...,n}. This gives a homomorphism to the
Symmetric group Dn->Sn.
It' s easy to see that it is an injection.
In the case n=4 this is what we get.
2 1
--------
| |
| |
----------------
| |
| |
--------
3 4
Let
s be
reflection in the
x-axis and
let
r be as before. Then
s corresponds
to the permutation
(23)(14) and
t corresponds
to (1234). Thus D4 is a
Sylow 2-subgroup of S4.
1. There is some disagreement about notation here. Some
people write D2n for what I've called Dn