Prof.
Alexandru Suciu
MATH 3175 Group Theory Fall 2010
The dihedral groups
The general setup. The dihedral group Dn is the group of symmetries of a regular
polygon with n vertices. We think of this polygon as having vertices on the unit circle,
with vertices labeled 0, 1, . . . , n − 1 starting at (1, 0) and proceeding counterclockwise
at angles in multiples of 360/n degrees, that is, 2π/n radians.
There are two types of symmetries of the n-gon, each one giving rise to n elements
in the group Dn :
• Rotations R0 , R1 , . . . , Rn−1 , where Rk is rotation of angle 2πk/n.
• Reflections S0 , S1 , . . . , Sn−1 , where Sk is reflection about the line through the
origin and making an angle of πk/n with the horizontal axis.
The group operation is given by composition of symmetries: if a and b are two
elements in Dn , then a · b = b ◦ a. That is to say, a · b is the symmetry obtained by
applying first a, followed by b.
The elements of Dn can be thought as linear transformations of the plane, leaving
the given n-gon invariant. This lets us represent the elements of Dn as 2 × 2 matrices,
with group operation corresponding to matrix multiplication. Specifically,
cos(2πk/n) − sin(2πk/n)
Rk = ,
sin(2πk/n) cos(2πk/n)
cos(2πk/n) sin(2πk/n)
Sk = .
sin(2πk/n) − cos(2πk/n)
It is now a simple matter to verify that the following relations hold in Dn :
Ri · Rj = Ri+j
Ri · Sj = Si+j
Si · Rj = Si−j
Si · Sj = Ri−j
where 0 ≤ i, j ≤ n − 1, and both i + j and i − j are computed modulo n.
The Cayley table for Dn can be readily computed from the above relations. In
particular, we see that R0 is the identity, Ri−1 = Rn−i , and Si−1 = Si .
MATH 3175 Handout 2 Fall 2010
The group D3 . This is the symmetry group of the equilateral triangle, with vertices
on the unit circle, at angles 0, 2π/3, and 4π/3. The matrix representation is given by
√ ! √ !
1 3 1 3
1 0 − −2 −√2
R0 = , R1 = √32 , R 2 = 2 ,
0 1 − 12 − 23 − 12
2
√ ! √ !
1 3 1 3
1 0 − − −
S0 = , S1 = √32 21 , S2 = √2 2 .
0 −1 − 23 1
2 2 2
while the Cayley table for D3 is:
R0 R1 R2 S0 S1 S2
R0 R0 R1 R2 S0 S1 S2
R1 R1 R2 R0 S1 S2 S0
R2 R2 R0 R1 S2 S0 S1
S0 S0 S2 S1 R0 R2 R1
S1 S1 S0 S2 R1 R0 R2
S2 S2 S1 S0 R2 R1 R0
The group D4 . This is the symmetry group of the square with vertices on the unit
circle, at angles 0, π/2, π, and 3π/2. The matrix representation is given by
1 0 0 −1 −1 0 0 1
R0 = , R1 = , R2 = , R3 = ,
0 1 1 0 0 −1 −1 0
1 0 0 1 −1 0 0 −1
S0 = , S1 = , S2 = , S3 = .
0 −1 1 0 0 1 −1 0
while the Cayley table for D4 is:
R0 R1 R2 R3 S0 S1 S2 S3
R0 R0 R1 R2 R3 S0 S1 S2 S3
R1 R1 R2 R3 R0 S1 S2 S3 S0
R2 R2 R3 R0 R1 S2 S3 S0 S1
R3 R3 R0 R1 R2 S3 S0 S1 S2
S0 S0 S3 S2 S1 R0 R3 R2 R1
S1 S1 S0 S3 S2 R1 R0 R3 R2
S2 S2 S1 S0 S3 R2 R1 R0 R3
S3 S3 S2 S1 S0 R3 R2 R1 R0
