# double coset

Let $H$ and $K$ be subgroups^{} of a group $G$.
An *$\mathrm{(}H\mathrm{,}K\mathrm{)}$-double coset* is a set of the form $HxK$ for some $x\in G$.
Here $HxK$ is defined in the obvious way as

$$HxK=\{hxk\mid h\in H\text{and}k\in K\}.$$ |

Note that the $(H,\{1\})$-double cosets are just the right cosets^{} of $H$,
and the $(\{1\},K)$-double cosets are just the left cosets of $K$.
In general, every $(H,K)$-double coset is a union of right cosets of $H$,
and also a union of left cosets of $K$.

The set of all $(H,K)$-double cosets is denoted $H\backslash G/K$.
It is straightforward to show that $H\backslash G/K$ is a partition^{} (http://planetmath.org/Partition) of $G$,
that is, every element of $G$ lies in exactly one $(H,K)$-double coset.

In contrast to the situation with ordinary cosets (http://planetmath.org/Coset),
the $(H,K)$-double cosets need not all be of the same cardinality.
For example, if $G$ is the symmetric group^{} (http://planetmath.org/SymmetricGroup) ${S}_{3}$,
and $H=\u27e8(1,2)\u27e9$ and $K=\u27e8(1,3)\u27e9$,
then the two $(H,K)$-double cosets
are $\{e,(1,2),(1,3),(1,3,2)\}$ and $\{(2,3),(1,2,3)\}$.

Title | double coset |
---|---|

Canonical name | DoubleCoset |

Date of creation | 2013-03-22 16:17:28 |

Last modified on | 2013-03-22 16:17:28 |

Owner | yark (2760) |

Last modified by | yark (2760) |

Numerical id | 8 |

Author | yark (2760) |

Entry type | Definition |

Classification | msc 20A05 |