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

$HgK={\displaystyle \bigcup _{k\in K}}Hgk={\displaystyle \bigcup _{h\in H}}hgK,$

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

$HgK={\displaystyle \bigcup _{[k]\in (K\cap g^{-1}Hg)\backslash \backslash K}}Hgk={\displaystyle \bigcup _{[h]\in H/(H\cap gK{g}^{-1})}}hgK$

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

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

$\mathrm{\#}(HgK/K)=[H:H\cap gK{g}^{-1}]$

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 $HgH$.