= Solution
The two products are conjugate:
$$
a^{-1}(ab)a=ba.
$$
<Conjugation> by $a^{-1}$ is a <group automorphism>, so part (a) proves
$$
\boxed{\operatorname{ord}(ab)=\operatorname{ord}(ba)}.
$$
Directly, cancellation gives $(ba)^r=a^{-1}(ab)^ra$ 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.
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