double coset
Let and be subgroups of a group .
An -double coset is a set of the form for some .
Here is defined in the obvious way as
Note that the -double cosets are just the right cosets of ,
and the -double cosets are just the left cosets of .
In general, every -double coset is a union of right cosets of ,
and also a union of left cosets of .
The set of all -double cosets is denoted .
It is straightforward to show that is a partition (http://planetmath.org/Partition) of ,
that is, every element of lies in exactly one -double coset.
In contrast to the situation with ordinary cosets (http://planetmath.org/Coset),
the -double cosets need not all be of the same cardinality.
For example, if is the symmetric group (http://planetmath.org/SymmetricGroup) ,
and and ,
then the two -double cosets
are and .
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 |