# Coset

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 H**in*G* - Hg = { hg : h ∈ H } is a
**right coset of H**in*G*.

The

Any two left cosets are either identical or disjoint: the left 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.