double coset

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

HxK={hxkhH and kK}.

Note that the (H,{1})-double cosets are just the right cosetsMathworldPlanetmath 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\G/K. It is straightforward to show that H\G/K is a partitionMathworldPlanetmath ( of G, that is, every element of G lies in exactly one (H,K)-double coset.

In contrast to the situation with ordinary cosets (, the (H,K)-double cosets need not all be of the same cardinality. For example, if G is the symmetric groupMathworldPlanetmathPlanetmath ( S3, and H=(1,2) and K=(1,3), 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