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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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