# Coset

### From Online Dictionary of Crystallography

Co-ensemble (*Fr*). Restklasse (*Ge*). Classe laterale (*It*). 剰余類 (*Ja*).

## Definition

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

*gH*= {*gh*:*h*∈*H*} is a**left coset of**in*H**G*,*Hg*= {*hg*:*h*∈*H*} is a**right coset of**in*H**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:

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*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.