Prove that a finite group is nilpotent if and only if whenever with , then . [Hint: Use Theorem 3.]
Suppose is a finite nilpotent group. Let with . By Theorem 6.3, is the internal direct product of its Sylow subgroups; thus we may write and , where . Moreover, and ; if some , then divides . so does not divide , and . Similarly, if then .
By relabeling the Sylow subgroups of and collecting factors, we have the internal direct product , with and . Thus .
Suppose is a finite group in which the hypothesis holds. Let be a Sylow -subgroup. If is a Sylow subgroup for some other prime , then . In particular, contains a subgroup of maximal -power order for each dividing ; thus , so that is normal. Since is arbitrary, all Sylow subgroups of are normal. By Theorem 3, is nilpotent.