Showing that every finite group has a composition series

In summary: But this doesn't happen, because there is a maximal nontrivial proper normal subgroup which is not itself maximal.
  • #1
Mr Davis 97
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Homework Statement


Prove that for any finite group ##G## there exists a sequence of nested subgroups of ##G##, ##\{e\}=N_0\leq N_1\leq \cdots \leq N_n=G## such that for each integer ##i## with ##1\leq i\leq n## we have ##N_{i-1}\trianglelefteq N_i## and the quotient group ##N_i/N_{i-1}## is simple.

Homework Equations

The Attempt at a Solution


Here is a proof I found online:

We prove the result by induction. Suppose that every group of order less than ##|G|## has a composition series.

Now if ##G## is a simple group then ##\{ e_G \} \triangleleft G## is a composition series. If ##G## is not a simple group then there exists a nontrivial proper normal subgroup of ##G##. Since ##G## is a finite group, a maximal nontrovial proper normal subgroup exists. Denote this subgroup by ##H##. Since ##H## is a proper subgroup of ##G## we have that ##|H|<|G|##. By the induction hypothesis, ##H## has a composition series: $$\{e_G\}=H_0\le H_1\le\dots\le H_k=H.$$ But then ##H \trianglelefteq G## by the maximality of ##H##. So: $$\{e_G\}=H_0\le H_1\le\dots\le H_k=H \le G.$$ The above chain of subgroup is a composition series of ##G##. QEDHere are my questions: How do we know for sure that ##G/H## is simple? Is this implied by the maximality of ##H##? Why?

Also, I am confused by the statement **but then ##H \trianglelefteq G## by the maximality of ##H##.** Why does the maximality of ##H## matter in this case? Don't we already know that ##H \trianglelefteq G##?
 
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  • #2
Mr Davis 97 said:
Here are my questions: How do we know for sure that ##G/H## is simple? Is this implied by the maximality of ##H##? Why?
If ##G/H## is not simple, then there is a normal subgroup ##G/H \trianglerighteq K/H## where ##K\trianglelefteq G## is a non-trivial normal subgroup which contains ##H##.
Also, I am confused by the statement **but then ##H \trianglelefteq G## by the maximality of ##H##.** Why does the maximality of ##H## matter in this case? Don't we already know that ##H \trianglelefteq G##?
You are right. ##H## is chosen to be normal. Maximality is only needed for ##G/H## simple.
 
  • #3
fresh_42 said:
If ##G/H## is not simple, then there is a normal subgroup ##G/H \trianglerighteq K/H## where ##K\trianglelefteq G## is a non-trivial normal subgroup which contains ##H##.

You are right. ##H## is chosen to be normal. Maximality is only needed for ##G/H## simple.
Could you explain this statement: Since ##G## is a finite group, a maximal nontrovial proper normal subgroup exists.

Why must a maximal nontrivial proper normal subgroup exist if ##G## is finite?
 
  • #4
Mr Davis 97 said:
Could you explain this statement: Since ##G## is a finite group, a maximal nontrovial proper normal subgroup exists.

Why must a maximal nontrivial proper normal subgroup exist if ##G## is finite?
We also have that ##G## is not simple, which is the real important property here. The case ##G## simple is trivial, so we assume that ##G## has a non-trivial normal subgroup. Among those we choose a maximal. The non-simplicity of ##G## gives us the subgroup we need, and the finiteness guarantees, that there is a maximal one, for otherwise we could have an infinite sequence ##1 \triangleleft H_1\triangleleft H_2 \triangleleft \ldots G\,.##
 
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Related to Showing that every finite group has a composition series

1. What is a composition series?

A composition series is a sequence of normal subgroups that are distinct from each other and whose quotient groups are simple. In simpler terms, it is a way to break down a group into smaller and simpler parts.

2. Why is it important to show that every finite group has a composition series?

It is important to show that every finite group has a composition series because it helps us understand the structure of a group and its subgroups. It also allows us to prove important theorems about groups, such as the Jordan-Hölder theorem.

3. How do you prove that every finite group has a composition series?

There are a few different ways to prove this statement, but one common approach is to use induction on the order of the group. This involves showing that any group of order n has a composition series, and then using this to prove that any group of order n+1 also has a composition series.

4. What is the significance of simple quotient groups in a composition series?

Simple quotient groups are important in a composition series because they cannot be broken down any further. This means that they are the "building blocks" of the group, and understanding their structure helps us understand the structure of the entire group.

5. Can all groups be broken down into a composition series?

No, not all groups can be broken down into a composition series. For example, infinite groups and some special types of groups, such as the alternating group An, do not have composition series. However, every finite group can be broken down into a composition series.

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