double coset

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

HxK=\{hxk\mid h\in H\hbox{ 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 $H=\langle(1,2)\rangle$ and $K=\langle(1,3)\rangle$ , 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