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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 of  , that is, every element of  lies in exactly one  -double coset.
In contrast to the situation with ordinary cosets, the  -double cosets need not all be of the same cardinality. For example, if  is the symmetric group  , and
 and
 , then the two  -double cosets are
 and
 .
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