Is there a version of Jordan-Hölder theorem for infinite composition series?

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In summary, the Jordan-Hölder theorem states that if a group has a composition series, then there is a permutation that satisfies the relation M_{\pi (i)} / M_{\pi (i-1)} .
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Hello friends.

I am working trough "Abstract Algebra" by Dummit & Foote.
I recently got to section 3.4, on composition series and "the Hölder program".

The Jordan-Hölder theorem states:

Let G be a finite, non-trivial group. Then:

1) G has a composition series.

2) If

[tex] \{ 1 \} = N_0 \leq N_1 \leq ... \leq N_r = G[/tex]

and

[tex] \{ 1 \} = M_0 \leq M_1 \leq ... \leq M_s = G [/tex]

are two composition series of G, then:

2a) r = s
2b) There is some permutation [tex]\pi[/tex] of {1, 2, ..., r} such that:

[tex]
M_{\pi (i)} / M_{\pi (i) - 1} \cong N_{i} / N_{i-1}
[/tex]

for [tex] 0 \leq i \leq r[/tex].My question is on 2b). For [tex] M_{\pi (i)} / M_{\pi (i) - 1} [/tex] to make sense, we must of course have

[tex] M_{\pi (0)} \leq M_{\pi (1)} \leq ... \leq M_{\pi (s)} [/tex]

But how can any permutation satisfy this relation, i.e. not break the subgroup ordering?

Let me rephrase my question with numbers instead of subgroups.
It is clearly impossible to permute the sequence:

[tex]
1 \leq 3 \leq 4 \leq 7 \leq 9 \leq 11
[/tex]

without breaking the ordering.
The only example I can think of where we may permute is something like:

[tex]
1 \leq 3 \leq 4 \leq 7 \leq 7 \leq 7
[/tex]

Where we are allowed to permute only the last three numbers without breaking the ordering.
But the analog of this last example for subgroups would be pretty pointless, since the assertion of 2b would of course hold true before applying such a permutation.Can somebody please explain what I'm missing?

Edit: Hmm... Is there perhaps one permutation [tex]\pi_i[/tex], say, for each quotient [tex]N_{i} / N_{i-1} [/tex], instead of just a single one for every quotient?
 
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  • #2
Never mind. I found my own error.

I was thinking

[tex]
M_{\pi (i)} / M_{\pi (i-1)}
[/tex]

instead of

[tex]
M_{\pi (i)} / M_{\pi (i) - 1}
[/tex]

which is what the theorem says.

Sorry about that.
Cool theorem, though. :smile:
 
  • #3
Hello Folks,

I would like to know if there is a version of Jordan-Hölder theorem for infinite composition series (in the case of Groups or Modules). If yes, please, give me a reference to its statement and its proof.

Assume that some module is presented as a direct sum of infinite number of simple submodules. Is this presentation unique up to a permutation? What can you say about multiplicities? Choice axiom and Zorn lemma are admissible for me.
 

1. What is the Jordan-Hölder theorem?

The Jordan-Hölder theorem is a fundamental result in abstract algebra that states that any two composition series of a group or a module have the same length and the same composition factors, up to isomorphism.

2. Who discovered the Jordan-Hölder theorem?

The theorem was independently discovered by mathematicians Camille Jordan and Otto Hölder in the late 19th century.

3. What is the significance of the Jordan-Hölder theorem?

The Jordan-Hölder theorem is an important tool for understanding the structure of groups and modules, and it has many applications in algebra and other areas of mathematics.

4. Is the Jordan-Hölder theorem only applicable to finite groups and modules?

No, the Jordan-Hölder theorem also holds for infinite groups and modules, as long as they have a well-defined composition series.

5. Are there any variations or extensions of the Jordan-Hölder theorem?

Yes, there are several variations and extensions of the Jordan-Hölder theorem, including the Krull-Remak-Schmidt theorem and the Fitting lemma. These results provide further insight into the structure of groups and modules.

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