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

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Homework Help Overview

The discussion revolves around the properties of p-groups, specifically focusing on whether every p-group has a subgroup of order p^m for every integer m between 0 and n, where n is the order of the group.

Discussion Character

  • Exploratory, Conceptual clarification, Problem interpretation

Approaches and Questions Raised

  • Participants explore the definition of p-groups and consider using induction as a method to approach the problem. There is also mention of the Sylow theorems and their relevance to the proof.

Discussion Status

Participants are actively discussing various definitions and properties of p-groups, with some suggesting induction and others considering the implications of the Sylow theorems. There is recognition of the complexity of the proof, indicating a productive exploration of ideas.

Contextual Notes

Some participants question the assumptions regarding the use of Sylow theorems and the definitions of p-groups, highlighting the need for clarity on these points as they relate to the problem.

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

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