I'm getting a little confused about what exactly Sylow's first theorem says. On Wikipedia, it says that Sylow's First Theorem says "For any prime factor p with multiplicity n of the order of a finite group G, there exists a Sylow p-subgroup of G, of order p^n." Then in the section of the proof it is restated as "A finite group G whose order |G| is divisible by a prime power p^k has a subgroup of order p^k." To me, these seem like they are saying two different things. The first seems like its only guaranteeing subgroups of prime power where the prime power is maximal (p^k where |G|=p^km where p does not divide m), while the second seems to be saying there are subgroups of order p^1, p^2, p^3, ... p^(k-1), p^k. Am I misunderstanding something?