# Tagged: finite set

## 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
$AB=\{ab \mid a\in A, b\in B\}.$

## Problem 145

Let $G$ be a finite group of order $n$ and let $m$ be an integer that is relatively prime to $n=|G|$. Show that for any $a\in G$, there exists a unique element $b\in G$ such that
$b^m=a.$