I apologize for what might be a boring technical question, but in reading about double cosets, I want to understand the following idea which may be of use.
If $H$ and $K$ are subgroups of some group, then why can $HK$ be a disjoint union of cosets $xK$ where $x$ ranges over a set of representatives of $H/(H\cap K)$? In particular, why do the representatives range over $H/(H\cap K)$?
Could someone provide an explanation of how this works exactly? The notes I'm reading say this is more or less obvious, but I don't follow.