|
In mathematics, given a group ''G'' and two subgroups ''H'' and ''K'', not necessarily distinct, a double coset (or more precisely an (''H'',''K'') double coset) is a set ''HgK'' for some fixed element ''g'' in ''G''. Equivalently, an (''H'',''K'') double coset in ''G'' is an equivalence class for the equivalence relation defined on ''G'' by :''x'' ~ ''y'' if there are ''h'' in ''H'' and ''k'' in ''K'' with ''hxk'' = ''y''. The basic properties of double cosets follow immediately from the fact that they are equivalence classes; namely, two double cosets ''HxK'' and ''HyK'' are either disjoint or identical and ''G'' is partitioned into its (''H'',''K'') double cosets. Furthermore, each double coset ''HgK'' is a union of ordinary right cosets ''Hy'' of ''H'' in ''G'' and left cosets ''zK'' of ''K'' in ''G''. In particular, the number of right cosets of ''H'' in ''HgK'' is the index (∩ ''g''-1''Hg'' ) and the number of left cosets of ''K'' in ''HgK'' is the index (∩ ''g''-1''Hg'' ).〔 Double cosets are in fact orbits for the group action of ''H''×''K'' on ''G'' with . The set of double cosets is often denoted by : == Algebraic structure == It is possible to define a product operation of double cosets in an associated ring. Given two double cosets and , we decompose each into right cosets and . If we write , then we can define the product of with as the formal sum . An important case is when ''H'' = ''K'' = ''L'', which allows us to define an algebra structure on the associated ring spanned by linear combinations of double cosets. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「double coset」の詳細全文を読む スポンサード リンク
|