Let be a group and let . Prove that and have the same order.
Recall that the order of a group element is either a positive integer or infinity.
Suppose is infinite and that for some . Then , a contradiction. So if is infinite, must also be infinite. Likewise, if is infinte, then is also infinite.
Suppose now that and are both finite. Then we have , so that . Likewise, . Hence and and have the same order.