Composition Series and simple groups

• lmn123
In summary, the conversation discusses finding a group with a composition series of length 1 and a subgroup of length m for any value of m greater than or equal to 2. The equations discussed include the fact that simple groups have length 1, and the length of a group can be determined by its order and normal subgroups. The attempt at a solution involves considering a cyclic group of order p as G, and finding a subgroup H with order p^m. However, this does not work as the subgroup would have a higher order than G. The conversation ends with a question about whether G can be put inside a simple group and what simple groups are known.
lmn123

Homework Statement

For each m >= 2, find a group with a composition series of length 1 with a subgroup of length m.

Homework Equations

Simple groups iff length 1.

If G is abelian of order p1^k1...pr^kr, then length G = k1 + ... + kr

If G has a composition series and K is normal subgroup of G, then Length G = Length K + Length G/K

The Attempt at a Solution

I don't really see how this is even possible considering the fact that simple groups have length 1. So this means that G must be a simple group. I do not understand how to get a group of length m from a simple group.

I think I'm getting somewhere with this:

What I was thinking was have G = the cyclic group of order p. Then G would have length 1

Then I thought to Let H be a subgroup of G such that |G|=p^m, so if G=<a>, st |a|=p, then H=<a^p^(m+1)>

But this doesn't really work because the subgroup would have higher order than G right? Am I missing something, like modular arithmetic?

Any light you guys could shed would be appreciated
Thanks

Fix m, and let G be an abelian group of length m. Can you put G inside a simple group? (What simple groups do you know?)

1. What is a composition series?

A composition series is a sequence of normal subgroups in a group, where each subgroup is a normal subgroup of the next subgroup. This series can be used to understand the structure and properties of a group.

2. How is a composition series related to simple groups?

A composition series can help determine whether a group is simple or not. If a group has no nontrivial normal subgroups, it is considered a simple group. Therefore, a composition series that consists of only the identity and the whole group indicates that the group is simple.

3. What is the difference between a composition series and a chief series?

A composition series includes all normal subgroups of a group, while a chief series only includes chief factors (quotient groups where the normal subgroup is maximal). A chief series can be viewed as a refinement of a composition series.

4. How are composition series and simple groups useful in group theory?

Composition series and simple groups provide a way to classify and understand the various types of groups in mathematics. They help identify the basic building blocks of groups and their properties, which can then be used to study more complex groups.

5. Can every group be decomposed into a composition series?

Yes, every finite group can be decomposed into a composition series. This is known as the Jordan-Hölder theorem and is an important result in group theory. However, not all infinite groups have a composition series.

• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
7
Views
1K
• Calculus and Beyond Homework Help
Replies
3
Views
2K
• Calculus and Beyond Homework Help
Replies
9
Views
1K
• Calculus and Beyond Homework Help
Replies
6
Views
978
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus and Beyond Homework Help
Replies
2
Views
872
• Calculus and Beyond Homework Help
Replies
1
Views
857
• Calculus and Beyond Homework Help
Replies
1
Views
2K
• Calculus and Beyond Homework Help
Replies
5
Views
2K