TITLE: If $G \setminus H$ be a finite set, then $G$ is finite?
QUESTION [5 upvotes]: Let $G$ be a group and $H$ be a proper subgroup of it. I should prove that if $G \setminus H$ be a finite set, then $G$ is finite group. Any suggestion?

REPLY [3 votes]: for any  $a \in G-H$, $aH \cap H$ = $\emptyset$ ...so $H$ is finite so $G$ is finite