Finite Group has Composition Series

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Theorem

Let $G$ be a finite group.


Then $G$ has a composition series.


Proof

Let $G$ be a finite group whose identity is $e$.

Either $G$ has a proper non-trivial normal subgroup or it does not.

If not, then:

$\set e \lhd G$

is the composition series for $G$.

Otherwise, $G$ has one or more proper non-trivial normal subgroup.

Of these, one or more will have a maximum order.

Select one of these and call it $G_1$.

Again, either $G_1$ has a proper non-trivial normal subgroup or it does not.

If not, then:

$\set e \lhd G_1 \lhd G$

is a composition series for $G$.

By the Jordan-Hölder Theorem, there can be no other composition series which is longer. As $G_1$ is a proper subgroup of $G$:

$\order {G_1} < \order G$

where $\order G$ denotes the order of $G$.

Again, if $G_1$ has one or more proper non-trivial normal subgroup, one or more will have a maximum order.

Select one of these and call it $G_2$.

Thus we form a normal series:

$\set e \lhd G_2 \lhd G_1 \lhd G$


The process can be repeated, and at each stage a normal subgroup is added to the series of a smaller order than the previous one.

This process can not continue infinitely.

Eventually a $G_n$ will be encountered which has no proper non-trivial normal subgroup.

Thus a composition series:

$\set e \lhd G_n \cdots \lhd G_2 \lhd G_1 \lhd G$

will be the result.

$\blacksquare$


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