TITLE: Why do we stop at double cosets and do not consider triple, ... n-ple cosets?
QUESTION [6 upvotes]: The obvious answer seems to be that a group has only two sides, and once you're done taking quotients on the left and on the right, you can't quotient by another subgroup from another side.
But if $G$ is a group with subgroups $H$ and $K$, why can we not consider a quotient like $G/H/\bar K$, where $\bar K$ is the image of $K$ in $G/H$ and the whole should be understood as the set of cosets $xHK$?

REPLY [4 votes]: That's because you cannot build an equivalence relation on a double quotient on one side. Note that you can't get transitivity:
$$\begin{aligned} &a = xhk \\ &b = xh'k' \\ &b = yh''k'' \\ &c = y'''k'''\end{aligned}$$ so that $a \sim b$ and $b \sim c$, but a bit of manipulation will show that you can get $a\sim c$ when you have at most one coset on each side.