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

International Tables for Crystallography (2016). Vol. A, ch. 1.1, pp. 5-6

Section 1.1.4. Cosets

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

1.1.4. Cosets

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A subgroup allows us to partition a group into disjoint subsets of the same size, called cosets.

Definition. Let [\ispecialfonts{\cal H} = \{ {\sfi h}_1, {\sfi h}_2, {\sfi h}_3, \ldots \}] be a subgroup of [{\cal G}]. Then for [\ispecialfonts{\sfi g} \in {\cal G}] the set [\ispecialfonts {\sfi g}{\cal H}: = \{ {\sfi g} {\sfi h}_1, {\sfi g} {\sfi h}_2, {\sfi g} {\sfi h}_3, \ldots \} = \{ {\sfi g} {\sfi h} \mid {\sfi h} \in {\cal H} \} ]is called the left coset of [{\cal H}] with representative [\ispecialfonts{\sfi g}]. Analogously, the right coset with representative [\ispecialfonts{\sfi g}] is defined as [\ispecialfonts {\cal H}{\sfi g}: = \{ {\sfi h}_1 {\sfi g}, {\sfi h}_2 {\sfi g}, {\sfi h}_3 {\sfi g}, \ldots \} = \{ {\sfi h} {\sfi g} \mid {\sfi h} \in {\cal H} \}. ]The coset [\ispecialfonts{\sfi e} {\cal H} = {\cal H} = {\cal H} {\sfi e}] is called the trivial coset of [{\cal H}].

Remarks

  • (i) Since two elements [\ispecialfonts{\sfi g} {\sfi h}] and [\ispecialfonts{\sfi g} {\sfi h}'] in the same coset [\ispecialfonts{\sfi g} {\cal H}] can only be the same if [\ispecialfonts{\sfi h} = {\sfi h}'], the elements of [\ispecialfonts{\sfi g} {\cal H}] are in one-to-one correspondence with the elements of [{\cal H}]. In particular, for a finite subgroup [{\cal H}] the number of elements in each coset of [{\cal H}] equals the order [|{\cal H}|] of the subgroup [{\cal H}].

  • (ii) Every element contained in [\ispecialfonts{\sfi g} {\cal H}] may serve as representative for this coset, i.e. [\ispecialfonts{\sfi g}' {\cal H} = {\sfi g} {\cal H}] for every [\ispecialfonts{\sfi g}' \in {\sfi g} {\cal H}]. In particular, if an element [\ispecialfonts{\sfi g}''] is contained in the intersection [\ispecialfonts{\sfi g} {\cal H} \cap {\sfi g}' {\cal H}] of two cosets, one has [\ispecialfonts{\sfi g}'' {\cal H} = {\sfi g} {\cal H}] and [\ispecialfonts{\sfi g}'' {\cal H} = {\sfi g}' {\cal H}]. This implies that two cosets are either disjoint (i.e. contain no common element) or they are equal.

These two remarks have an important consequence: since an element [\ispecialfonts{\sfi g} \in {\cal G}] is contained in the coset [\ispecialfonts{\sfi g} {\cal H}], the cosets of [{\cal H}] partition the elements of [{\cal G}] into sets of the same cardinality as [{\cal H}] (which is of the order of [{\cal H}] in the case where this is finite).

Definition. If the number of different cosets of a subgroup [{\cal H} \leq {\cal G}] is finite, this number is called the index of [{\cal H}] in [{\cal G}], denoted by [[i]] or [[{\cal G}:{\cal H}]]. Otherwise, [{\cal H}] is said to have infinite index in [{\cal G}].

In the case of a finite group, the partitioning of the elements of [{\cal G}] into the cosets of [{\cal H}] shows that both the order of [{\cal H}] and the index of [{\cal H}] in [{\cal G}] divide the order of [{\cal G}]. This is summarized in the following famous result.

Lagrange's theorem

For a finite group [{\cal G}] and a subgroup [{\cal H}] of [{\cal G}] one has [ |{\cal G}| = |{\cal H}| \cdot [{\cal G}:{\cal H}], ]i.e. the order of a subgroup multiplied by its index gives the order of the full group.

For example, a group of order n cannot have a proper subgroup of order larger than [n/2].

Whether or not two cosets of a subgroup [{\cal H}] are equal depends on whether the quotient of their representatives is contained in [{\cal H}]: for left cosets one has [\ispecialfonts{\sfi g} {\cal H} = {\sfi g}' {\cal H}] if and only if [\ispecialfonts{\sfi g}^{-1} {\sfi g}' \in {\cal H}] and for right cosets [\ispecialfonts{\cal H} {\sfi g} = {\cal H} {\sfi g}'] if and only if [\ispecialfonts{\sfi g}' {\sfi g}^{-1} \in {\cal H}].

Definition. If [{\cal H}] is a subgroup of [{\cal G}] and [\ispecialfonts{\sfi g_1}, {\sfi g_2}, {\sfi g_3}, \ldots \in {\cal G}] are such that [\ispecialfonts{\sfi g_i} {\cal H} \neq {\sfi g_j} {\cal H}] for [i \neq j], and every [\ispecialfonts{\sfi g} \in {\cal G}] is contained in some left coset [\ispecialfonts{\sfi g_i} {\cal H}], then [\ispecialfonts{\sfi g_1}, {\sfi g_2}, {\sfi g_3}, \ldots] is called a system of left coset representatives of [{\cal G}] relative to [{\cal H}]. It is customary to choose [\ispecialfonts{\sfi g_1} = {\sfi e}] so that the coset [\ispecialfonts{\sfi g_1} {\cal H} = {\sfi e} {\cal H} = {\cal H}] is the subgroup [{\cal H}] itself. The decomposition [ \ispecialfonts{\cal G} = {\cal H} \cup {\sfi g_2} {\cal H} \cup {\sfi g_3} {\cal H} \ldots ]is called the coset decomposition of [{\cal G}] into left cosets relative to [{\cal H}].

Analogously, [\ispecialfonts{\sfi g_1}', {\sfi g_2}', {\sfi g_3}', \ldots \in {\cal G}] is called a system of right coset representatives if [\ispecialfonts{\cal H} {\sfi g_i}' \neq {\cal H} {\sfi g_j}'] for [i \neq j] and every [\ispecialfonts{\sfi g} \in {\cal G}] is contained in some right coset [\ispecialfonts{\cal H} {\sfi g_i}]. Again, one usually chooses [\ispecialfonts{\sfi g_1}' = {\sfi e}] and the decomposition [\ispecialfonts {\cal G} = {\cal H} \cup {\cal H} {\sfi g_2}' \cup {\cal H} {\sfi g_3}' \ldots ]is called the coset decomposition of [{\cal G}] into right cosets relative to [{\cal H}].

To obtain the coset decomposition one starts by choosing [{\cal H}] as the first coset (with representative [\ispecialfonts{\sfi e}]). Next, an element [\ispecialfonts{\sfi g_2} \in {\cal G}] with [\ispecialfonts{\sfi g_2} \not\in {\cal H}] is selected as representative for the second coset [\ispecialfonts{\sfi g_2} {\cal H}]. For the third coset, an element [\ispecialfonts{\sfi g_3} \in {\cal G}] with [\ispecialfonts{\sfi g_3} \not\in {\cal H}] and [\ispecialfonts{\sfi g_3} \not\in {\sfi g_2} {\cal H}] is required. If at a certain stage the cosets [\ispecialfonts{\cal H}, {\sfi g_2} {\cal H}, \ldots, {\sfi g_m} {\cal H}] have been defined but do not yet exhaust [{\cal G}], an element [\ispecialfonts{\sfi g_{m+1}}] not contained in the union [\ispecialfonts{\cal H} \cup {\sfi g_2} {\cal H} \cup \ldots \cup {\sfi g_m} {\cal H}] is chosen as representative for the next coset.

Examples

  • (i) Let [{\cal G} = {3m}] be the symmetry group of an equilateral triangle and [{\cal H} = \langle {3}^+ \rangle] its subgroup containing the rotations. Then for every reflection [m \in {\cal G}] the elements [\ispecialfonts{\sfi e}, m] form a system of coset representatives of [{\cal G}] relative to [{\cal H}] and the coset decomposition is [{\cal G} = \{ {1}, {3}^+, {3}^{-} \} \,\cup] [ \{ {m}_{10}, {m}_{01}, {m}_{11} \}].

  • (ii) For any integer n, the set [n {\bb Z}: = \{ n a \mid a \in {\bb Z} \}] of multiples of n forms an infinite subgroup of index n in [\bb Z]. A system of coset representatives of [\bb Z] relative to [n \bb Z] is formed by the numbers [0, 1, 2, \ldots, n-1]. The coset with representative 0 is [\{ \ldots, -n, 0, n, 2n, \ldots \}], the coset with representative 1 is [\{ \ldots, -n+1, 1, n+1, 2n+1, \ldots \}] and an integer a belongs to the coset with representative k if and only if a gives remainder k upon division by n.








































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