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 Ta comp R with R belongs O(n). If one composes two such elements, Ta comp R1 and Tb comp R2 then one gets an element whose linear part is the composite R1 comp 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.

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 iso the symmetry group of the motif iso the subgroup fixing a point of the lattice.]


  2. Rectangular lattice
    For this pattern, the symmetry group consists only of translations and compositions Tacomp V with V a vertical reflection through a lattice point. Hence the point group is generated by V and is D1
    [Again this is iso the symmetry group of the motif iso the subgroup fixing a point of the lattice.]


  3. Rhomboid lattice
    The symmetry group consists of translations and compositions Tacomp H with H a half turn. Hence the point group is generated by H and is C2
    [Again this is iso the symmetry group of the motif iso the subgroup fixing a point of the lattice.]


  4. Rectangular lattice
    The symmetry group includes a glide reflection which is of the form Ta comp 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.


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 Tacomp R with R belongs O(n) and we may define a homomorphism theta : G rarrow O(n) by Tacomp R goesto 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 belongs P subset O(2) and a belongs L then A(a) belongs L also.

Proof
Since A is in the point group for some f belongs G we have f = Tv comp A.
Then A comp Ta(x) = A(a + x) = A(a) + A(x) = TA(a) comp A(x).
Now calculate f comp Ta comp f -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 groupLattice#groupsNames of groups
C1general1p1
C2general1p2
C3equilateral1p3
C4square1p4
C6equilateral1p6
D1rectangular
rhomboid
2
1
pm pg
cm

D2rectangular
rhomboid
3
1
pmm pgg pmg
cmm

D3equilateral2p3m1 p31m
D4square2p4m p4g
D6equilateral1p6m


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JOC February 2003