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.
$\blacksquare$
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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: $1.8$