Difference between revisions of "Double coset"
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Let G be a group, and H and K be two [[subgroup]]s of G. One says that the two elements g<sub>1</sub> ∈ G and g<sub>2</sub> ∈ G belong to the same '''double coset''' of G relative to H and K if there exist elements h<sub>i</sub> ∈ H and k<sub>j</sub> ∈ K such that: | Let G be a group, and H and K be two [[subgroup]]s of G. One says that the two elements g<sub>1</sub> ∈ G and g<sub>2</sub> ∈ G belong to the same '''double coset''' of G relative to H and K if there exist elements h<sub>i</sub> ∈ H and k<sub>j</sub> ∈ K such that: | ||
Revision as of 12:09, 6 February 2012
Let G be a group, and H and K be two subgroups of G. One says that the two elements g1 ∈ G and g2 ∈ G belong to the same double coset of G relative to H and K if there exist elements hi ∈ H and kj ∈ K such that:
g2 = hig1kj
The complex Hg1K is called a double coset
The partition of G into double cosets relative to H and K is a classification, i.e. each gi ∈ G belongs to exactly one dobule coset. It is also a generalization of the coset decomposition, because the double coset Hg1K contains complete left cosets of K and complete right cosets of H.