## The Existence of an Element in an Abelian Group of Order the Least Common Multiple of Two Elements

## Problem 497

Let $G$ be an abelian group.

Let $a$ and $b$ be elements in $G$ of order $m$ and $n$, respectively.

Prove that there exists an element $c$ in $G$ such that the order of $c$ is the least common multiple of $m$ and $n$.

Also determine whether the statement is true if $G$ is a non-abelian group.

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