Editing Coset (section)

== Double cosets ==
{{main|Double coset}}
Given two subgroups, {{math|''H''}} and {{math|''K''}} (which need not be distinct) of a group {{math|''G''}}, the '''double cosets''' of {{math|''H''}} and {{math|''K''}} in {{math|''G''}} are the sets of the form {{math|1=''HgK'' = {{mset|''hgk'' : ''h'' an element of ''H'', ''k'' an element of ''K''}}}}. These are the left cosets of {{math|''K''}} and right cosets of {{math|''H''}} when {{math|1=''H'' = 1}} and {{math|1=''K'' = 1}} respectively.<ref>{{harvnb|Scott|1987|loc= p. 19}}</ref>

Two double cosets {{math|''HxK''}} and {{math|''HyK''}} are either disjoint or identical.<ref name=Hall>{{harvnb|Hall|1959|loc=pp. 14–15}}</ref> The set of all double cosets for fixed {{mvar|H}} and {{mvar|K}} form a partition of {{mvar|G}}.

A double coset {{math|''HxK''}} contains the complete right cosets of {{mvar|H}} (in {{mvar|G}}) of the form {{math|''Hxk''}}, with {{mvar|k}} an element of {{mvar|K}} and the complete left cosets of {{mvar|K}} (in {{mvar|G}}) of the form {{math|''hxK''}}, with {{mvar|h}} in {{mvar|H}}.<ref name=Hall />

=== Notation ===
Let {{math|''G''}} be a group with subgroups {{math|''H''}} and {{math|''K''}}. Several authors working with these sets have developed a specialized notation for their work, where<ref>{{citation|first=Gary M.|last=Seitz|chapter=Double Cosets in Algebraic Groups|editor1-first=R.W.|editor1-last=Carter|editor2-first=J.|editor2-last=Saxl|title=Algebraic Groups and their Representation| year=1998|pages=241–257|publisher=Springer|doi=10.1007/978-94-011-5308-9_13|isbn=978-0-7923-5292-1}}</ref><ref>{{citation | first=W. Ethan|last=Duckworth|title=Infiniteness of double coset collections in algebraic groups|journal=Journal of Algebra|volume=273|issue=2|pages=718–733|year=2004|publisher=Elsevier|doi=10.1016/j.jalgebra.2003.08.011|arxiv=math/0305256| s2cid=17839580}}</ref>
* {{math|''G''{{hsp}}/{{hsp}}''H''}} denotes the set of left cosets {{math|{{mset|''gH'' : ''g'' in ''G''}}}} of {{math|''H''}} in {{math|''G''}}.
* {{math|''H''{{hsp}}\{{hsp}}''G''}} denotes the set of right cosets {{math|{{mset|''Hg'' : ''g'' in ''G''}}}} of {{math|''H''}} in {{math|''G''}}.
* {{math|''K''{{hsp}}\{{hsp}}''G''{{hsp}}/{{hsp}}''H''}} denotes the set of double cosets {{math|{{mset|''KgH'' : ''g'' in ''G''}}}} of {{math|''H''}} and {{math|''K''}} in {{math|''G''}}, sometimes referred to as ''double coset space''.
* {{math|''G''{{hsp}}//{{hsp}}''H''}} denotes the double coset space {{math|''H''{{hsp}}\{{hsp}}''G''{{hsp}}/{{hsp}}''H''}} of the subgroup {{mvar|H}} in {{mvar|G}}.