Understanding Sylow's First Theorem: A Guide to Prime Power Subgroups

[doubleaxel195]

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?

 

[Robert1986]

The book has a slightly stronger statement as Sylow's Theorem than what the wikipedia article says. However, if P is a p-group of order p^k then there is a subgroup of order p^j for 0 <= j <= k. To prove this, you can use induction and the fact that p-groups have non-trivial centers and the fact that abelian groups (you can only use abelian properties for the center) have subgroups of order p for every prime that divides the order of the group (there is probably an exercise or this is a theorem where this fact is proven).

 

[jdinatale]

Sylow's firs theorem is basically saying that 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.