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left and right cosets in a double coset
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(Theorem)
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Let $H$ and $K$ be subgroups of a group $G$ . Every double coset $HgK$ , with $g \in G$ , is a union of right or left cosets, since
but these unions need not be disjoint. In particular, from the above equality we cannot say how many right (or left) cosets fit in a double coset.
The following proposition aims to clarify this.
$\,$
Proposition - Let $H$ and $K$ be subgroups of a group $G$ and $g \in G$ . We have that
hold as disjoint unions. In particular, the number of right and left cosets in $HgK$ is respectively given by
- The number of right and left cosets in a double coset does not coincide in general, not even for double cosets of the form $HgH$ .
- 1
- A. Krieg, Hecke algebras, Mem. Amer. Math. Soc., no. 435, vol. 87, 1990.
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Cross-references: number, disjoint unions, proposition, cosets, equality, disjoint, left cosets, union, double coset, group, subgroups
This is version 4 of left and right cosets in a double coset, born on 2008-12-06, modified 2008-12-06.
Object id is 11312, canonical name is LeftAndRightCosetsInADoubleCoset.
Accessed 409 times total.
Classification:
| AMS MSC: | 20A05 (Group theory and generalizations :: Foundations :: Axiomatics and elementary properties) |
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Pending Errata and Addenda
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![$\displaystyle HgK = \bigcup_{[k]\, \in\, (K \cap g^{-1}Hg) \backslash K} Hgk\; = \bigcup_{[h]\, \in\, H / (H \cap gKg^{-1})} hgK$](http://images.planetmath.org:8080/cache/objects/11312/js/img2.png "$\displaystyle HgK = \bigcup_{[k]\, \in\, (K \cap g^{-1}Hg) \backslash K} Hgk\; = \bigcup_{[h]\, \in\, H / (H \cap gKg^{-1})} hgK$")
![$\displaystyle \char93 (H \backslash HgK) = [K: K \cap g^{-1}Hg]$](http://images.planetmath.org:8080/cache/objects/11312/js/img3.png "$\displaystyle \char93 (H \backslash HgK) = [K: K \cap g^{-1}Hg]$"){kind=small}
![$\displaystyle \char93 (HgK/K) =[H: H \cap gKg^{-1}]$](http://images.planetmath.org:8080/cache/objects/11312/js/img4.png "$\displaystyle \char93 (HgK/K) =[H: H \cap gKg^{-1}]$"){kind=small}