Part 16: Point groups and crystallographic space groups

Point groups

 Let P be a point in E2 and let F in Sym(2), the group of symmetries (= bijections = permutations) of E2. We first show that F has a unique factorisation as F = RP.S where S is a translation and RP is an isometry fixing P. 

Lemma 16.1

  1. F in Sym(2) can be factorised uniquely as F = RP.S where S is a translation and RP is an isometry fixing P.
  2. The map RP.S -> RP: Iso(2) -> Iso(2)P is an epimorphism whose kernel is the group T of translations.
Proof 1. Let S be the translation that maps P to PF. Then RP = F.S-1 is an isometry fixing P such that F = RP.S. 

If also F = R'P.S' where S' is a translation and R'P fixes P, then S and S' are translations mapping P to PF, so S = S'. Hence RP = R'P.

 2. Suppose E = VP.U and F = RP.S in Iso(2). Then E.F = VP.U.S = VP.F.F-1.U.F = VP.RP.S.F-1.U.F.

 Now VP.RP is an isometry fixing P and S.F-1.U.F is a translation (since T is normal in Iso(2)). Hence VP.RP is the unique P-fixing isometry which is the image of E.F under the map, so the map is a homomorphism.

 The map is onto, since any RP in Iso(2)P is the image of itself, and it clearly has kernel T , the group of translations.

 Note

  1. In group theory, if A is a normal subgroup of G and B is a subgroup of G such that G is generated by A and B and A 0 B = 1, it follows that each element of G has a unique factorisation x = yz with y in B and z in A. G is called the split product of A and B. So we have Sym(2) is the split product of SymP(2) and T.
  2. The same proof works for En for any n.
  3. If P and Q are elements of E2, and S is the translation mapping P to Q, then Sym(2)Q = S-1Sym(2)PS so these groups are conjugate and hence isomorphic.
  4. Sym(2)P is called the point group of Sym(2).
  5. An epimorphism of a group onto a subgroup is called a split product. It is not a direct product unless the subgroup is normal.
We now extend this definition to arbitrary subgroups of Sym(2).

 Theorem 16.2 Let G be a subgroup of Iso(2) and let P be a point in E2. Let F in G and write F = RP.S where S is a translation and RP a symmetry fixing P.
 

  1. The map RP.S -> RP : G -> Iso(2)P is a homomorphism with kernel G 0 T. Call the image HP.
  2. For all P and Q in E2, HP is isomorphic to HQ.
  3. GP is a subgroup of HP which is a subgroup of Sym(2) and GP = HP if and only if the G-orbit of P is the (G 0 T)-orbit of P.
Proof By Lemma 16.1 the map is a homomorphism with kernel G 0 T. Clearly GP is a subgroup of HP which is a subgroup of Iso(2). The various HP are conjugate in Iso(2) and hence isomorphic. (In fact if S is a translation mapping P to Q, then HQ = S-1.HP.S)

 If the G-orbit of P is the (G 0 T)-orbit of P then for all F in G, the translation S: P -> PF is in G so RP = F. S-1 is in GP and hence HP is contained in GP. Conversely, if HP = GP and F 1 G, then F = S.RP implies RP 1 G. Hence S 1 G 0 T so PF is in the (G 0 T)-orbit of P. Thus the G-orbit is contained in the (G 0 T)-orbit. The converse is obvious.

 Note

  1. The group HP defined in Theorem 16.2 is called the point group of G. It is the subgroup GP of G if and only if the condition of Theorem 16.2 is satisfied. 
  2. For example, Iso(2) contains all translations so satisfies the conditions. If G is a plane crystallographic group, then G 0 T is the translation lattice, so the G 0 T-orbit of a point P is the point lattice based on P. So for a fixed pattern HP = GP if and only if P is a point in the pattern moved only by translations. 

    3. The point groups of the plane symmetry groups are the cyclic groups C2, C3, C4 and C6 and the dihedral groups D3, D4 and D6.
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Crystallographic Space Groups

 Everything we have done for Iso(2) works equally well for Iso(3). A space crystallographic group is a subgroup of Iso(3) containing three independent translations of minimal length. The images of a point P under these translations forms a lattice of points in E3 which are the corners of parallelopipeds which tile the space.

 There are seven different lattices which are named after the seven types of crystal systems in Chemistry. Mathematically, these types are distinguished by the symmetries which they admit, which in turn are determined by the lengths a, b and c of the sides and angles a, b and g between the translations.

 Here is a table describing the seven types of space lattices:

Space lattices for crystal systems

Crystal system Translation lengths Translation angles
triclinic a, b, c distinct a, b, g distinct
monoclinic a, b, c distinct a = b = p/2 not= g
orthorhombic a, b, c distinct a = b= g = p/2
tetragonal a = b not= c a  = b = g = p/2
hexagonal a = b not= c a = b = p/2, 
g = 2p/3
trigonal a = b = c a = b = g < 2p/3
cubic a = b = c a = b = g = p/2

And here is a site with pictures of crystals realising most of these groups of symmetries.

A subgroup of Iso(3) is a crystallographic point group if it leaves a point P and a point lattice fixed. Every subgroup G has a point group H = HP isomorphic to G/T where T is the group of translations in G. 

It turns out, though I shall not prove it here, that there are exactly 32 crystallographic point groups which are all the finite subgroups of Iso(3) which satisfy the Crystallographic Restriction. They are built up from the cyclic and dihedral groups of degrees 2, 3, 4 and 6 and the tetrahedral and octahedral groups. For gory details, see 

Paul B Yale, Geometry and Symmetry, 1968.

 Finally, by associating each of the 32 point groups with whichever of the 7 lattice systems admit them, some in more than one way, we get 230 crystallographic space groups. This was first shown in the 1890's independently by W. Barlow (Britain), E S Fodorov (Russia) and A. Schoenflies (Germany).
 

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 For some applications to chemistry, see this page, and for some applications to mineralogy, click here.

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 Last update Dec 10, 1999

Author: Phill Schultz, schultz@maths.uwa.edu.au

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