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

• algeb
In summary, the conversation discusses the proof that a p group has a subgroup of order p^m for every 0<=m<=n. The participants mention using induction on n and Sylow theorems to prove this, and also discuss the definition of a p-group and its nontrivial centers. It is recommended to study a proof of the appropriate Sylow theorem for this proof.
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.

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.

## 1. What is a P-group?

A P-group is a type of mathematical group that satisfies certain properties, namely that every element in the group has a power that equals the identity element. This means that raising any element in the group to a power will result in the identity element, which is usually denoted as e.

## 2. How do you determine if a group is a P-group?

To determine if a group is a P-group, you need to check if every element in the group has a power that equals the identity element. This can be done by raising each element to different powers and seeing if any of them result in the identity element. If every element satisfies this property, then the group is a P-group.

## 3. What is the significance of P-groups?

P-groups are important in abstract algebra because they have many interesting properties and can be used to study other types of groups. They also have applications in other areas of mathematics, such as in number theory and geometry.

## 4. How are P-groups related to cyclic groups?

Cyclic groups are a special type of P-group where every element can be generated by a single element. This means that every element in the group is a power of this generator. However, not all P-groups are cyclic, as there can be multiple generators in a P-group.

## 5. Are there any real-world examples of P-groups?

Yes, there are many real-world examples of P-groups. Some examples include the integers under addition, the non-zero rational numbers under multiplication, and the symmetries of a regular polygon. P-groups can also be found in chemistry, physics, and computer science.

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