International
Tables for
Crystallography
Volume A
Space-group symmetry
Edited by M. I. Aroyo

International Tables for Crystallography (2016). Vol. A. ch. 1.3, pp. 29-31

## Section 1.3.3.2. Coset decomposition with respect to the translation subgroup

B. Souvigniera*

aRadboud University Nijmegen, Faculty of Science, Mathematics and Computing Science, Institute for Mathematics, Astrophysics and Particle Physics, Postbus 9010, 6500 GL Nijmegen, The Netherlands
Correspondence e-mail: souvi@math.ru.nl

#### 1.3.3.2. Coset decomposition with respect to the translation subgroup

| top | pdf |

The translation subgroup of a space group can be used to distribute the operations of into different classes by grouping together all operations that differ only by a translation. This results in the decomposition of into cosets with respect to (see Section 1.1.4 for details of cosets).

#### Definition

Let be a space group with translation subgroup .

 (i) The right coset of an operation with respect to is the set . Analogously, the set is called the left coset of with respect to . (ii) A set of operations in is called a system of coset representatives relative to if every operation in is contained in exactly one coset . (iii) Writing as the disjoint union is called the coset decomposition of relative to .

If the translation subgroup is a subgroup of index [i] in , a set of coset representatives for relative to consists of [i] operations , where is assumed to be the identity element of . The cosets of relative to can be imagined as columns of an infinite array with [i] columns, labelled by the coset representatives, as displayed in Table 1.3.3.3.

 Table 1.3.3.3| top | pdf | Right-coset decomposition of relative to

Remark: We can assume some enumeration of the operations in because the translation vectors form a lattice. For example, with respect to a primitive basis, the coordinate vectors of the translations in are simply columns with integral components . A straightforward enumeration of these columns would start with

Writing out the matrix–column pairs, the coset consists of the operations of the form with running over the lattice translations of . This means that the operations of a coset with respect to the translation subgroup all have the same linear part, which is also evident from a listing of the cosets as columns of an infinite array, as in the example above.

#### Proposition

Let and be two operations of a space group with translation subgroup .

 (1) If , then the cosets and are disjoint, i.e. their intersection is empty. (2) If , then the cosets and are equal, because has linear part and is thus an operation contained in .

The one-to-one correspondence between the point-group operations and the cosets relative to explicitly displays the isomorphism between the point group of and the factor group . This correspondence is also exploited in the listing of the general-position coordinates. What is given there are the coordinate triplets for coset representatives of relative to , which correspond to the first row of the array in Table 1.3.3.3. As just explained, the other operations in can be obtained from these coset representatives by adding a lattice translation to the translational part.

Furthermore, the correspondence between the point group and the coset decomposition relative to makes it easy to find a system of coset representatives of relative to . What is required is that the linear parts of the are precisely the operations in the point group of . If are the different operations in the point group of , then a system of coset representatives is obtained by choosing for every linear part a translation part such that is an operation in .

It is customary to choose the translation parts of the coset representatives such that their coordinates lie between 0 and 1, excluding 1. In particular, if the translation part of a coset representative is a lattice vector, it is usually chosen as the zero vector .

Note that due to the fact that is a normal subgroup of , a system of coset representatives for the right cosets is at the same time a system of coset representatives for the left cosets.