Every finite group has a finite composition series

This article gives the statement and possibly, proof, of an implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., finite group) must also satisfy the second group property (i.e., group of finite composition length)
View all group property implications | View all group property non-implications
Get more facts about finite group|Get more facts about group of finite composition length

Statement

Every finite group $G$ has a composition series of finite length, that is, a sequence

$\{e\}\triangleleft H_{1}\triangleleft H_{2}\triangleleft \dots \triangleleft H_{k}\triangleleft G$

where $K\triangleleft H$ denotes that $K$ is a proper normal subgroup of $H$, and no $K$ exists such that $H_{i}\triangleleft K\triangleleft H_{i+1}$ for $i=1,2,\dots ,k-1$.

That is, every finite group is a group of finite composition length.

Proof

Induction on $|G|$.

When $|G|=1$, the trivial composition series $\{e\}$ is a composition series.

Otherwise, pick a proper normal subgroup $H\triangleleft G$ with $|H|$ maximal.

Then by induction, $H$ has a composition series $\{e\}\triangleleft H_{1}\triangleleft H_{2}\triangleleft \dots \triangleleft H_{k}\triangleleft H$ of finite length.

Then $\{e\}\triangleleft H_{1}\triangleleft H_{2}\triangleleft \dots \triangleleft H_{k}\triangleleft H\triangleleft G$ is a composition series for $G$ of finite length.