# left and right cosets in a double coset

Let $H$ and $K$ be subgroups of a group $G$. Every double coset $HgK$, with $g\in G$, is a union of right (http://planetmath.org/Coset) or left cosets, since

 $\displaystyle HgK=\bigcup_{k\in K}Hgk\;=\bigcup_{h\in H}hgK,$

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.

$\,$

- Let $H$ and $K$ be subgroups of a group $G$ and $g\in G$. We have that

 $\displaystyle HgK=\bigcup_{[k]\,\in\,(K\cap g^{-1}Hg)\backslash K}Hgk\;=% \bigcup_{[h]\,\in\,H/(H\cap gKg^{-1})}hgK$

hold as disjoint unions. In particular, the number of right and left cosets in $HgK$ is respectively given by

 $\displaystyle\#(H\backslash HgK)=[K:K\cap g^{-1}Hg]$ $\displaystyle\#(HgK/K)=[H:H\cap gKg^{-1}]$

## 0.1 Remarks

• The number of right and left cosets in a double coset does not coincide in general, not for double cosets of the form $HgH$.

## References

• 1 A. Krieg, , Mem. Amer. Math. Soc., no. 435, vol. 87, 1990.
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