Symmetry groups of plane figures

We now consider some interesting subgroups of I(R²).

Following the spirit of Klein's Erlangen program we look for properties preserved by these subgroups. In particular we look for subsets X of S = R² which are mapped to themselves by all the elements of these subgroups.
The subgroup is then the symmetry group of the subset X.

Examples

1. Subgroups generated by a rotation

   Rotation by 2π/n generates a subgroup isomorphic to C_n the cyclic group of order n.

   Note that C_n is the symmetry group of many figures in R².

   has symmetry group C₂ , has symmetry group C₃ , has symmetry group C₄ , etc.

2. Dihedral groups

   The Dihedral group D_n is the group of symmetries of a regular n-gon. It has order 2n.

   The group D₃ consists of rotations by 0, 2π/3 and 4π/3 and three reflections.

   The group D₄ consists of four rotations by multiples of π/2 and four reflections -- two through lines joining vertices and two through lines joining mid-points of sides.

   One may show that in terms of generators and relations, D_n= < a, b | aⁿ = b² = 1, bab = a ^-1 >.

   For n ≥ 3 it is a non-abelian group. The group D₃ S₃ (the symmetric group on three symbols).

   The group D₂ is the symmetry group of a "regular 2-gon" and consists of the identity, rotation by π and reflections in two perpendicular lines.
   It is called the Klein 4-group. It is sometimes written V and is isomorphic to C₂× C₂ .

   The group D₁ is the group consisting of the identity and reflection in a line:
   Note that as a group this is a cycle group of order 2 but that it is not the same subgroup as any of the subgroups C₂ of (1) above which are generated by rotations.
   

Note that for each point p of R² there is a subgroup in I(R²) isomorphic to C_n and many subgroups isomorphic to D_n (corresponding to the different directions that one could choose for the mirrors through p).

Theorem (Leonardo da Vinci about 1500)
The above examples C_n and D_n for some positive integer n are the only finite subgroups of I(R²).

Proof
Let G be a finite subgroup of I(R²). Then G cannot contain a translation or glide reflection since these elements have infinite orders.
It cannot contain rotations about two distinct points otherwise (See Exercises 3 Question 6) it would contain a translation.
If it contains a rotation about a point a, it cannot contain a reflection in any line not through a (See Exercises 3 Question 2).
It can only contain reflections in lines through a common point (Otherwise one could combine two of them to get a rotation about one of the meeting points and a reflection through a different point).
Hence there is a point of R² left fixed by all the elements of G.

Take this point as the origin and then G O(2).
H = G SO(2) contains only rotations and hence contains a smallest (non-zero) rotation. All the others must be multiples of this one (otherwise one could combine them to get a rotation whose magnitude was their hcf) and so we have H C_n .
If G ≠ H then H has two cosets in G (since SO(2) has two cosets in O(2)) and the reflections form a coset rH and it is easy to verify that if h is a generator of H then rhr = h ^-1 .
That is, G D_n .

Remark

In fact (See Exercises 4 Question 1) any two copies G₁ and G₂ of the groups C_n of rotations are conjugate in I(R²). That is, for some element g ∈ I(R²) we have g G₁ g^-1 = G₂. Similarly any two copies of the same dihedral group are also conjugate. Thus the classification of the finite subgroups of I(R²) is up to conjugacy in I(R²).