Jordan-Holder theorem

Statement
Suppose $$G$$ is a fact about::group of finite composition length. In other words, $$G$$ has a fact about::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 $$M_i/M_{i-1}$$.

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.