IMC 2001 · Problem 2

Let $r$, $s$, $t$ be positive integers which are pairwise relatively prime. If $a$ and $b$ are elements of a commutative multiplicative group with unity element $e$, and $a^r = b^s = (ab)^t = e$, prove that $a = b = e$.

Does the same conclusion hold if $a$ and $b$ are elements of an arbitrary non-commutative group?