Condition for Composition Series

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Theorem

Let $G$ be a finite group.


Then:

a normal series $\HH$ for $G$ is a composition series for $G$

if and only if:

every factor group of $\HH$ is a simple group.


Proof

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

Let:

$(1): \quad \set e = G_0 \lhd G_1 \lhd \cdots \lhd G_{n - 1} \lhd G_n = G$

be a normal series for $G$.


Necessary Condition

Suppose there exists $k$ such that $G_{k + 1} / G_k$ is not a simple group.

Then there exists a normal subgroup $G'$ such that:

$\set e \lhd G' \lhd G_{k + 1} / G_k$

It follows that:

$G_k \lhd G \lhd G_{k + 1}$

where:

$G$ is a normal subgroup of $G_{k + 1}$

and:

$G' = G / G_{k + 1}$


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Thus $(1)$ has a proper refinement and so is not a composition series.

By the Rule of Transposition it follows that if normal series is a composition series, every factor group of that normal series is a simple group.

$\Box$


Sufficient Condition

Suppose $(1)$ is not a composition series for $G$.

Then a proper refinement of $(1)$ can be constructed by inserting a group $G$ into the series somewhere, for example:

$G_k \lhd G \lhd G_{k + 1}$

It follows that $G / G_k$ is a normal subgroup of $G_{k + 1} / G$.


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Thus, by definition, $G_{k + 1} / G_k$ is not a simple group.

Thus it has been shown that if a normal series is not a composition series, then it contains at least one factor group which is not a simple group

By the Rule of Transposition it follows that if every factor group of a normal series is a simple group, then that normal series is a composition series.

$\Box$


Hence the result.

$\blacksquare$


Sources

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