# Double coset

### From Online Dictionary of Crystallography

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Double coset (*Fr*). Doppelte Nebenklasse (*Ge*). Doppio coset (*It*). 両側剰余類 (*Ja*). Clase lateral doble (*Sp*).

Let *G* be a group, and *H* and *K* be two subgroups of *G*. One says that the two elements *g*_{1} ∈ *G* and *g*_{2} ∈ *G* belong to the same **double coset** of *G* relative to *H* and *K* if there exist elements *h _{i}* ∈

*H*and

*k*∈

_{j}*K*such that

*g*_{2} = *h _{i}g*

_{1}

*k*.

_{j}The complex *Hg*_{1}*K* is called a **double coset**.

The partition of *G* into double cosets relative to *H* and *K* is a classification, *i.e.* each *g _{i}* ∈

*G*belongs to exactly one dobule coset. It is also a generalization of the coset decomposition, because the double coset

*Hg*

_{1}

*K*contains complete left cosets of

*K*and complete right cosets of

*H*.