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 be a subgroup of . Then for the set is called the left coset of with representative . Analogously, the right coset with representative is defined as The coset is called the trivial coset of .

Remarks

 (i) Since two elements and in the same coset can only be the same if , the elements of are in one-to-one correspondence with the elements of . In particular, for a finite subgroup the number of elements in each coset of equals the order of the subgroup . (ii) Every element contained in may serve as representative for this coset, i.e. for every . In particular, if an element is contained in the intersection of two cosets, one has and . 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 is contained in the coset , the cosets of partition the elements of into sets of the same cardinality as (which is of the order of in the case where this is finite).

Definition. If the number of different cosets of a subgroup is finite, this number is called the index of in , denoted by or . Otherwise, is said to have infinite index in .

In the case of a finite group, the partitioning of the elements of into the cosets of shows that both the order of and the index of in divide the order of . This is summarized in the following famous result.

#### Lagrange's theorem

For a finite group and a subgroup of one has 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 .

Whether or not two cosets of a subgroup are equal depends on whether the quotient of their representatives is contained in : for left cosets one has if and only if and for right cosets if and only if .

Definition. If is a subgroup of and are such that for , and every is contained in some left coset , then is called a system of left coset representatives of relative to . It is customary to choose so that the coset is the subgroup itself. The decomposition is called the coset decomposition of into left cosets relative to .

Analogously, is called a system of right coset representatives if for and every is contained in some right coset . Again, one usually chooses and the decomposition is called the coset decomposition of into right cosets relative to .

To obtain the coset decomposition one starts by choosing as the first coset (with representative ). Next, an element with is selected as representative for the second coset . For the third coset, an element with and is required. If at a certain stage the cosets have been defined but do not yet exhaust , an element not contained in the union is chosen as representative for the next coset.

#### Examples

 (i) Let be the symmetry group of an equilateral triangle and its subgroup containing the rotations. Then for every reflection the elements form a system of coset representatives of relative to and the coset decomposition is . (ii) For any integer n, the set of multiples of n forms an infinite subgroup of index n in . A system of coset representatives of relative to is formed by the numbers . The coset with representative 0 is , the coset with representative 1 is and an integer a belongs to the coset with representative k if and only if a gives remainder k upon division by n.