$a$ and $b$ are elements of a group $G$, with the property that $ab=ba$.
Problems:
a) Demonstrate that if $a$ and $b$ have infinite order, then $ab$ may have either finite or infinite order.
b) If $a$ has finite order and $b$ has infinite order, then determine and prove whether $ab$ has infinite or finite order.
c) If $a$ and $b$ have finite order, show with proof whether $ab$ has finite or infinite order.