Jordan-Holder theorem

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Statement

Suppose $G$ is a group of finite composition length. In other words, $G$ has a composition series of finite length $l$:

$\{e\}=N_{0}\triangleleft N_{1}\triangleleft N_{2}\triangleleft \dots \triangleleft N_{l}=G$

where each $N_{i-1}$ is a proper normal subgroup of $N_{i}$ and $N_{i}/N_{i-1}$ is a simple group. Then, the following are true:

1. Any composition series for $G$ has length $l$.
2. The list of composition factors is the same for any two composition series. In other words, if $N_{i}$ form one composition series and $M_{i}$ form another, then for any simple group $S$, the number of $i$ for which $S$ is isomorphic to $N_{i}/N_{i-1}$ equals the number of $i$ for which $S$ is isomorphic to $M_{i}/M_{i-1}$.

Related facts

• Jordan-Holder theorem for chief series: An analogous result, which states that any two chief series of a group have the same length and that the list of chief factors is the same.
• Finite composition length implies every subnormal series can be refined to a composition series
• Finite chief length implies every normal series can be refined to a chief series
• Jordan-Holder theorem for groups with operators

Some other related facts:

• Finite not implies composition factor-unique: There can exist finite groups for which there are different composition series with the composition factors occurring in different orders.
• Composition factor-unique not implies composition series-unique: Even if all composition series for a group have the same composition factors occurring in the same order, there may be more than one composition series.
• Finite not implies composition factor-permutable: There can exist finite groups for which not all possible orderings of the composition factors can be achieved using composition series.