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

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

Section 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: Coset decomposition with respect to the translation subgroup

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


Let [{\cal G}] be a space group with translation subgroup [{\cal T} ].

  • (i) The right coset [\ispecialfonts{\cal T} {\sfi W}] of an operation [\ispecialfonts{\sfi W} \in {\cal G} ] with respect to [{\cal T}] is the set [\ispecialfonts\{ {\sfi t} {\sfi W} \mid {\sfi t} \in {\cal T} \} ].

    Analogously, the set [\ispecialfonts{\sfi W} {\cal T} = \{ {\sfi W} {\sfi t} \mid {\sfi t} \in {\cal T} \} ] is called the left coset of [\ispecialfonts{\sfi W}] with respect to [{\cal T}].

  • (ii) A set [\ispecialfonts\{ {\sfi W}_1, \ldots, {\sfi W}_m \} ] of operations in [{\cal G}] is called a system of coset representatives relative to [{\cal T}] if every operation [\ispecialfonts{\sfi W}] in [{\cal G}] is contained in exactly one coset [\ispecialfonts{\cal T} {\sfi W}_i ].

  • (iii) Writing [{\cal G}] as the disjoint union [ \ispecialfonts{\cal G} = {\cal T} {\sfi W}_1 \cup \ldots \cup {\cal T} {\sfi W}_m ]is called the coset decomposition of [{\cal G}] relative to [{\cal T}].

If the translation subgroup [{\cal T}] is a subgroup of index [i] in [{\cal G}], a set of coset representatives for [{\cal G} ] relative to [{\cal T}] consists of [i] operations [\ispecialfonts{\sfi W}_1, {\sfi W}_2, \ldots, {\sfi W}_{[i]}], where [\ispecialfonts{\sfi W}_1 ] is assumed to be the identity element [\ispecialfonts{\sfi e}] of [{\cal G} ]. The cosets of [{\cal G}] relative to [{\cal T}] can be imagined as columns of an infinite array with [i] columns, labelled by the coset representatives, as displayed in Table[link].

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Right-coset decomposition of [{\cal G}] relative to [{\cal T}]

[\ispecialfonts{\sfi W}_1 = {\sfi e}][\ispecialfonts{\sfi W}_2][ \ispecialfonts{\sfi W}_3][\ldots][\ispecialfonts{\sfi W}_{[i]}]
[\ispecialfonts{\sfi t}_1] [\ispecialfonts{\sfi t}_1 {\sfi W}_2] [\ispecialfonts{\sfi t}_1 {\sfi W}_3] [\ldots ] [\ispecialfonts{\sfi t}_1 {\sfi W}_{[i]}]
[\ispecialfonts{\sfi t}_2] [\ispecialfonts{\sfi t}_2 {\sfi W}_2] [\ispecialfonts{\sfi t}_2 {\sfi W}_3] [\ldots ] [\ispecialfonts{\sfi t}_2 {\sfi W}_{[i]}]
[\ispecialfonts{\sfi t}_3] [\ispecialfonts{\sfi t}_3 {\sfi W}_2] [\ispecialfonts{\sfi t}_3 {\sfi W}_3] [\ldots ] [\ispecialfonts{\sfi t}_3 {\sfi W}_{[i]}]
[\ispecialfonts{\sfi t}_4] [\ispecialfonts{\sfi t}_4 {\sfi W}_2] [\ispecialfonts{\sfi t}_4 {\sfi W}_3] [\ldots ] [\ispecialfonts{\sfi t}_4 {\sfi W}_{[i]}]
[\vdots] [\vdots] [\vdots]   [\vdots]

Remark: We can assume some enumeration [\ispecialfonts{\sfi t}_1, {\sfi t}_2, {\sfi t}_3, \ldots ] of the operations in [{\cal T}] because the translation vectors form a lattice. For example, with respect to a primitive basis, the coordinate vectors of the translations in [{\cal G}] are simply columns [\pmatrix{ l \cr m \cr n }] with integral components [l,m,n]. A straightforward enumeration of these columns would start with [\eqalign{&\pmatrix{ 0 \cr 0 \cr 0 }, \, \pmatrix{ 1 \cr 0 \cr 0 }, \, \pmatrix{ 0 \cr 1 \cr 0 }, \, \pmatrix{ 0 \cr 0 \cr 1 }, \, \pmatrix{ \bar{1} \cr 0 \cr 0 }, \, \pmatrix{ 0 \cr \bar{1} \cr 0 }, \, \pmatrix{ 0 \cr 0 \cr \bar{1} }, \cr & \pmatrix{ 1 \cr 1 \cr 0 }, \, \pmatrix{ 1 \cr 0 \cr 1 }, \, \pmatrix{ 0 \cr 1 \cr 1 } \ldots }]

Writing out the matrix–column pairs, the coset [{\cal T} ({\bi W}, {\bi w}) ] consists of the operations of the form [({\bi I}, {\bi t}) ({\bi W}, {\bi w}) = ({\bi W}, {\bi w} + {\bi t}) ] with [{\bi t}] running over the lattice translations of [{\cal T}]. 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.


Let [\ispecialfonts{\sfi W} = ({\bi W}, {\bi w})] and [\ispecialfonts{\sfi W}' = ({\bi W}', {\bi w}') ] be two operations of a space group [{\cal G}] with translation subgroup [{\cal T}].

  • (1) If [{\bi W} \neq {\bi W}'], then the cosets [\ispecialfonts{\cal T} {\sfi W} ] and [\ispecialfonts{\cal T} {\sfi W}'] are disjoint, i.e. their intersection is empty.

  • (2) If [{\bi W} = {\bi W}'], then the cosets [\ispecialfonts{\cal T} {\sfi W}] and [\ispecialfonts{\cal T} {\sfi W}'] are equal, because [\ispecialfonts{\sfi W} {\sfi W}'^{-1}] has linear part [{\bi I}] and is thus an operation contained in [{\cal T}].

The one-to-one correspondence between the point-group operations and the cosets relative to [{\cal T}] explicitly displays the isomorphism between the point group [{\cal P}] of [{\cal G}] and the factor group [{\cal G}/{\cal T}]. 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 [{\cal G}] relative to [{\cal T} ], which correspond to the first row of the array in Table[link]. As just explained, the other operations in [{\cal G} ] 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 [{\cal T}] makes it easy to find a system of coset representatives [\ispecialfonts\{ {\sfi W}_1, \ldots, {\sfi W}_m \}] of [{\cal G}] relative to [{\cal T}]. What is required is that the linear parts of the [\ispecialfonts{\sfi W}_i] are precisely the operations in the point group of [{\cal G}]. If [{\bi W}_1, \ldots, {\bi W}_m] are the different operations in the point group [{\cal P}] of [{\cal G}], then a system of coset representatives is obtained by choosing for every linear part [{\bi W}_i] a translation part [{\bi w}_i] such that [\ispecialfonts{\sfi W}_i = ({\bi W}_i, {\bi w}_i)] is an operation in [{\cal G} ].

It is customary to choose the translation parts [{\bi w}_i] 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 [{\bi o}].

Note that due to the fact that [{\cal T}] is a normal subgroup of [{\cal G}], a system of coset representatives for the right cosets is at the same time a system of coset representatives for the left cosets.

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