The point groups

Every element in the symmetry group G of Rⁿ is of the form T_a ∘ R with R ∈ O(n). If one composes two such elements, T_a ∘ R₁ and T_b ∘ R₂ then one gets an element whose linear part is the composite R₁ ∘ R₂ .
This is the motivation for the following:

Definition

The set of all linear parts of elements of a symmetry group G is called the point group P of G.

Remarks

1. The above remark shows that this is indeed a group.

2. It often seems that the point group associated with the symmetry group of a pattern is the group of symmetries of the motif and that the point group is then a subgroup of the group of symmetries.
   Alas, this is not always true.
   

1. General lattice
   For this pattern, the symmetry group consists only of translations. That is G = L and so the point group is {I}.
   [This is ≅ the symmetry group of the motif ≅ the subgroup fixing a point of the lattice.]
   
   
   
2. Rectangular lattice
   For this pattern, the symmetry group consists only of translations and compositions T_a∘ V with V a vertical reflection through a lattice point. Hence the point group is generated by V and is D₁
   [Again this is ≅ the symmetry group of the motif ≅ the subgroup fixing a point of the lattice.]
   
   
   
3. Rhomboid lattice
   The symmetry group consists of translations and compositions T_a∘ H with H a half turn. Hence the point group is generated by H and is C₂
   [Again this is ≅ the symmetry group of the motif ≅ the subgroup fixing a point of the lattice.]
   
   
   
4. Rectangular lattice
   The symmetry group includes a glide reflection which is of the form T_a ∘ R with R a horizontal reflection. Hence the point group is generated by R and is D₁
   In this case the symmetry group of the motif and the subgroup fixing a point of the lattice are both trivial and so are not the same as the point group. The point group is not a subgroup of the symmetry group.
   
   

Theorem
The point group of a symmetry group G is the factor group G/L where L is the normal subgroup of translations in G.

Proof
Given a symmetry f, we may write it as T_a∘ R with R ∈ O(n) and we may define a homomorphism θ : G O(n) by T_a∘ R ↦ R
It is easy to verify that this is a group homomorphism with kernel the lattice L of translations and image the point group. The result then follows from the first isomorphism theorem.


Although the elements of the point group P are not elements of the symmetry group G, they do act on the lattice.

Theorem
If A ∈ P ⊆ O(2) and a ∈ L then A(a) ∈ L also.

Proof
Since A is in the point group for some f ∈ G we have f = T_v ∘ A.
Then A ∘ T_a(x) = A(a + x) = A(a) + A(x) = T_A(a) ∘ A(x).
Now calculate f ∘ T_a ∘ f ^-1 which is an element of G.
This is T_v AT_a A ^-1T_-v = T_v(T_A(a)A)A ^-1T_-v = T_vT_A(a)T_-v = T_A(a) and so A(a) is in the lattice.


This means that (notwithstanding the fact that P is not a subgroup of G) that the elements of P act on the lattice and so satisfy the crystallographic restriction. This limits the possible subgroups of O(2) which P can be.

Here are the possible choices for the groups P and the lattices on which they can act. We will see later how the final two columns can be filled in.

Point group	Lattice	#groups	Names of groups
C1	general	1	p1
C2	general	1	p2
C3	equilateral	1	p3
C4	square	1	p4
C6	equilateral	1	p6
D1	rectangular rhomboid	2 1	pm pg cm
D2	rectangular rhomboid	3 1	pmm pgg pmg cmm
D3	equilateral	2	p3m1 p31m
D4	square	2	p4m p4g
D6	equilateral	1	p6m