# Jordan-Holder theorem

This article describes a fact or result that is not basic but it still well-established and standard. The fact may involve terms that are themselves non-basic
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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 $l$:

$\{ e \} = N_0 < N_1 < N_2 < \dots < N_l = G$

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 $M_i/M_{i-1}$.

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