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
$HgK={\displaystyle \bigcup _{[k]\in (K\cap {g}^{1}Hg)\backslash K}}Hgk={\displaystyle \bigcup _{[h]\in H/(H\cap gK{g}^{1})}}hgK$ 
hold as disjoint unions^{}. In particular, the number of right and left cosets in $HgK$ is respectively given by
$\mathrm{\#}(H\backslash HgK)=[K:K\cap {g}^{1}Hg]$  
$\mathrm{\#}(HgK/K)=[H:H\cap gK{g}^{1}]$ 
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 $HgH$.
References
 1 A. Krieg, , Mem. Amer. Math. Soc., no. 435, vol. 87, 1990.
Title  left and right cosets in a double coset 

Canonical name  LeftAndRightCosetsInADoubleCoset 
Date of creation  20130322 18:35:10 
Last modified on  20130322 18:35:10 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  7 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 20A05 