# Coset

(Difference between revisions)
 Revision as of 10:53, 13 May 2017 (view source)m (Style edits to align with printed edition)← Older edit Revision as of 15:26, 10 October 2017 (view source)m (two languages added)Newer edit → Line 1: Line 1: - Co-ensemble (''Fr''). Restklasse (''Ge''). Classe laterale (''It''). 剰余類 (''Ja''). + مجموعة مشاركة (''Ar''); Co-ensemble (''Fr''); Restklasse (''Ge''); Classe laterale (''It''); 剰余類 (''Ja''); Clase lateral (''Sp''). ==Definition== ==Definition==

## Revision as of 15:26, 10 October 2017

مجموعة مشاركة (Ar); Co-ensemble (Fr); Restklasse (Ge); Classe laterale (It); 剰余類 (Ja); Clase lateral (Sp).

## Definition

If G is a group, H a subgroup of G, and g an element of G, then

gH = { gh : hH } is a left coset of H in G,
Hg = { hg : hH } is a right coset of H in G.

The decomposition of a group into cosets is unique. Left coset and right cosets however in general do not coincide, unless H is a normal subgroup of G.

Any two left cosets are either identical or disjoint: the left cosets form a partition of G, because every element of G belongs to one and only one left coset. In particular the identity is only in one coset, and that coset is H itself; this is also the only coset that is a subgroup. The same holds for right cosets.

All left cosets and all right cosets have the same order (number of elements, or cardinality), equal to the order of H, because H is itself a coset. Furthermore, the number of left cosets is equal to the number of right cosets and is known as the index of H in G, written as [G : H] and given by Lagrange's theorem:

|G|/|H| = [G : H].

Cosets are also sometimes called associate complexes.

## Example

The coset decomposition of the twin lattice point group with respect to the point group of the individual gives the different possible twin laws. Each element in a coset is a possible twin operation.