IMC 2001 Problem 2

Let $r,s,t$ positive integers which are relatively prime and $a,b \in G$ , $G$ a commutative multiplicative group with unit element $e$ , and $a^r=b^s=(ab)^t=e$ . (a) Prove that $a=b=e$ . (b) Does the same hold for a non-commutative group $G$ ?