TITLE: If cosets $xH \ne Hy$ then, their intersection is empty?
QUESTION [3 upvotes]: I am revising basic Group Theory and was hoping to check if my understanding about this is correct.
http://www.proofwiki.org/wiki/Left_and_Right_Coset_Spaces_are_Equivalent says that the left and right coset spaces are equivalent.
Does this mean if cosets $xH \ne Hy$ then, their intersection is empty?

REPLY [0 votes]: Just to add to what's already been said, you might be interested to hear that a lot of research has been done by people who asked questions similar to yours.  A big deal is often made in advanced group theory about $xH \cap Hx$ (usually written in the form $H \cap x^{-1}Hx)$, which is called the "twist" of $H$ by $x$.  Furthermore, "double cosets" are also a thing in the big leagues, which are written $HxK$ and work kind of like "shared" cosets between two groups $H$ and $K$.
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