Course MT3818 Topics in Geometry

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The point groups

Every element in the symmetry group G of Rn is of the form TaR with RO(n). If one composes two such elements, TaR1 and TbR2 then one gets an element whose linear part is the composite R1R2 .
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.

Examples

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 TaV 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 TaH 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 TaR 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.

In fact in general, the point group of G is a factor group of G

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 TaR with RO(n) and we may define a homomorphism θ : G O(n) by TaRR
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 fG we have f = TvA.
Then ATa(x) = A(a + x) = A(a) + A(x) = TA(a)A(x).
Now calculate fTaf -1 which is an element of G.
This is Tv ATa A -1T-v = Tv(TA(a)A)A -1T-v = TvTA(a)T-v = TA(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 rectangularrhomboid 21 pm pg cm D2 rectangularrhomboid 31 pmm pgg pmgcmm D3 equilateral 2 p3m1 p31m D4 square 2 p4m p4g D6 equilateral 1 p6m

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JOC February 2003