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easy group theory

Source: IMC 2001 day 1 problem 2

October 29, 2005
number theoryrelatively primesuperior algebrasuperior algebra unsolved

Problem Statement

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