Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2018 / Paper 2 / 8 / b

The two products are conjugate:Conjugation by $a^{-1}$ is a group automorphism, so part (a) provesDirectly, cancellation gives $(ba{)}^r=a^{-1}(ab{)}^{r}a$ for every positive integer $r$. Thus a power of one product is the identity exactly when the same power of the other is the identity. This proves that conjugation preserves element order, even without the finiteness assumption.