Editing Coset (section)

=== 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}}.