Let be a group and let with normal in . Prove that if and are relatively prime then .
First we prove a lemma.
Lemma: Let be a group, , and an element of finite order . If is the least positive integer such that , then . Proof: If does not divide , we have for some by the division algorithm. Now , and . Thus , which contradicts the minimality of . Thus .
Now to the main result.
Suppose , and let be the least positive integer such that . ( exists since is finite.) By a previous exercise, as an element of , , so that divides . Moreover, we have divides by Lagrange, so that (by the lemma) divides and thus divides . Because and are relatively prime, then, . But then , so , and we have . So . By a previous exercise .