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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?

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The following is probably nonsense but maybe it will provoke a good reposnse:

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 $[A:x^{-1}Bx\cap A]$.

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  • $\begingroup$ 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"? $\endgroup$
    – Jack M
    Nov 7 '13 at 21:49
  • $\begingroup$ Just that I wrote it very quickly and didn't have time to check for any possible silly errors $\endgroup$
    – aPaulT
    Nov 8 '13 at 8:03

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