# Finite Group has Composition Series

## 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.

$\blacksquare$

## Sources

- 1971: Allan Clark:
*Elements of Abstract Algebra*... (previous) ... (next): Chapter $2$: Normal and Composition Series: $\S 73 \alpha$ - 1978: John S. Rose:
*A Course on Group Theory*... (previous) ... (next): $1$: Introduction to Finite Group Theory: $1.8$