# Jordan-Holder theorem

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## Statement

Suppose is a Group of finite composition length (?). In other words, has a Composition series (?) of finite length :

where each is a proper normal subgroup of and is a simple group. Then, the following are true:

- Any composition series for has length .
- The list of composition factors is the same for any two composition series. In other words, if form one composition series and form another, then for any simple group , the number of for which is isomorphic to equals the number of for which is isomorphic to .

## 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.