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 < 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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