Let be a finite group and a prime. Suppose that for some , we have for all with . Prove that for all . Deduce that in this case, the number of nonidentity elements of -power order in is precisely .
Let and be distinct Sylow -subgroups. Now by Sylow’s Theorem, is conjugate to in ; say . Now if we also have , then , a contradiction. Note then that , where is some Sylow -subgroup of distinct from . Thus , and we have .
Now every nonidentity element of -power order is in some Sylow -subgroup of , an no such element is in more than one. The number of nonidentity elements in a Sylow -subgroup is and the number of Sylow -subgroups is , and the final conclusion follows.