# Relations between left and right cosets

I have two subgroups $A$ and $B$ of the same finite group. Given a left coset of A, I would like to know how many right cosets of $B$ it intersects.

If I wanted intersections between left (or right) cosets of both $A$ and $B$, I think I could get some answers using this theorem. But in my case this doesn't work.

I'd probably be very lucky if there were some known formula for what I want, but are there at least general properties on the interactions between left and right cosets, either of different subgroup or of the same subgroup, that might be useful to me?

You have a coset $xA$ and you want to know how many cosets of the form $By$ it intersects. Well, $xA$ intersects $By$ exactly when $y$ is in the double coset $BxA$. The double coset is a disjoint union of right cosets of $B$ so we just need to know how many right cosets of $B$ constitute $BxA$, ie the value $|BxA|/|B|$. From here that is $[A:x^{-1}Bx\cap A]$.
• Wait, doesn't this provide an essentially complete answer to the question? You have a formula for the number of intersections, which depends only on $x$, $A$ and $B$. Why do you say that that's "probably nonsense"? Commented Nov 7, 2013 at 21:49