## If Two Subsets $A, B$ of a Finite Group $G$ are Large Enough, then $G=AB$

## Problem 493

Let $G$ be a finite group and let $A, B$ be subsets of $G$ satisfying

\[|A|+|B| > |G|.\]
Here $|X|$ denotes the cardinality (the number of elements) of the set $X$.

Then prove that $G=AB$, where

\[AB=\{ab \mid a\in A, b\in B\}.\]