Condition for Composition Series

Theorem

Let $G$ be a finite group.

A normal series $\mathcal H$ for $G$ is a composition series for $G$ iff:

Every factor group of $\mathcal H$ is a simple group.

Proof

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

Let:

$(1): \quad \left\{{e}\right\} = G_0 \triangleleft G_1 \triangleleft \cdots \triangleleft G_{n-1} \triangleleft G_n = G$

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:

$\left\{{e}\right\} \triangleleft G' \triangleleft G_{k+1} / G_k$

It follows that:

$G_k \triangleleft G'' \triangleleft G_{k+1}$

where:

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

and:

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

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:

$G_k \triangleleft G'' \triangleleft G_{k+1}$

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

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

• Allan Clark: Elements of Abstract Algebra (1971)... (previous)... (next): $\S 74$