1Groups and homomorphisms

IA Groups

1.4 Dihedral groups

Definition (Dihedral groups

D

2n

). Dihedral groups are the symmetries of a

regular

n

-gon. It contains

n

rotations (including the identity symmetry, i.e.

rotation by 0

◦

) and n reflections.

We write the group as

D

2n

. Note that the subscript refers to the order of

the group, not the number of sides of the polygon.

The dihedral group is not hard to define. However, we need to come up with

a presentation of D

2n

that is easy to work with.

We first look at the rotations. The set of all rotations is generated by

r

=

360

◦

n

.

This r has order n.

How about the reflections? We know that each reflection has order 2. Let

s

be our favorite reflection. Then using some geometric arguments, we can show

that any reflection can be written as a product of

r

m

and

s

for some

m

. We also

have srs = r

−1

.

Hence we can define

D

2n

as follows:

D

2n

is a group generated by

r

and

s

,

and every element can be written as a product of

r

’s and

s

’s. Whenever we see

r

n

and s

2

, we replace it by e. When we see srs, we replace it by r

−1

.

It then follows that every element can be written in the form r

m

s.

Formally, we can write D

2n

as follows:

D

2n

= ⟨r, s | r

n

= s

2

= e, srs

−1

= r

−1

⟩

= {e, r, r

2

, · · · r

n−1

, s, rs, r

2

s, · · · r

n−1

s}

This is a notation we will commonly use to represent groups. For example, a

cyclic group of order n can be written as

C

n

= ⟨a | a

n

= e⟩.