Does Every p-Group Have a Subgroup of Every Order Between 0 and n?

[algeb]

Homework Statement

If G is a p group, show that it has a subgroup of order p^m for every 0<=m<=n.

The Attempt at a Solution

The only thing I know about p-groups is that they have nontrivial centers.

 

[morphism]

I take it that the order of G is p^n? And I take it you're not supposed to use Sylow?

How about induction on n then?

 

[algeb]

yes, p-group is by definition a group of order p^n. And it's OK to use Sylow theorems, but how?

 

[Mystic998]

Well, it has another definition where every element has order p^k for some k. Anyway, I think induction is pretty much the only way to go. As I recall, this proof is rather tricky so don't get discouraged.

 

[morphism]

This is really a restatement of one of the Sylow theorems for p-groups. So it's a good idea to try to study a proof of the appropriate Sylow theorem.