Editing Composition series (section)

==For groups==
If a group ''G'' has a [[normal subgroup]] ''N'', then the factor group ''G''/''N'' may be formed, and some aspects of the study of the structure of ''G'' may be broken down by studying the "smaller" groups ''G/N'' and ''N''. If ''G'' has no normal subgroup that is different from ''G'' and from the trivial group, then ''G'' is a [[simple group]]. Otherwise, the question naturally arises as to whether ''G'' can be reduced to simple "pieces", and if so, whether there are any unique features of the way this can be done.

More formally, a '''composition series''' of a [[group (mathematics)|group]] ''G'' is a [[subnormal series]] of finite length
:<math>1 = H_0\triangleleft H_1\triangleleft \cdots \triangleleft H_n = G,</math>
with strict inclusions, such that each ''H''<sub>''i''</sub> is a [[maximal subgroup|maximal]] proper normal subgroup of ''H''<sub>''i''+1</sub>. Equivalently, a composition series is a subnormal series such that each factor group ''H''<sub>''i''+1</sub> / ''H''<sub>''i''</sub> is [[simple group|simple]]. The factor groups are called '''composition factors'''.

A subnormal series is a composition series [[if and only if]] it is of maximal length. That is, there are no additional subgroups which can be "inserted" into a composition series. The length ''n'' of the  series is called the '''composition length'''.

If a composition series exists for a group ''G'', then any subnormal series of ''G'' can be ''refined'' to a composition series, informally, by inserting subgroups into the series up to maximality. Every [[finite group]] has a composition series, but not every [[infinite group]] has one. For example, <math>\mathbb{Z}</math> has no composition series.

===Uniqueness: Jordan–Hölder theorem===
A group may have more than one composition series. However, the '''Jordan–Hölder theorem''' (named after [[Camille Jordan]] and [[Otto Hölder]]) states that any two composition series of a given group are equivalent. That is, they have the same composition length and the same composition factors, [[up to]] [[permutation]] and [[isomorphism]]. This theorem can be proved using the [[Schreier refinement theorem]].  The Jordan–Hölder theorem is also true for [[Transfinite induction|transfinite]] ''ascending'' composition series, but not transfinite ''descending'' composition series {{Harv|Birkhoff|1934}}. 
{{harvtxt|Baumslag|2006}} gives a short proof of the Jordan–Hölder theorem by intersecting the terms in one subnormal series with those in the other series.

====Example====
For a [[cyclic group]] of order ''n'', composition series correspond to ordered prime factorizations of ''n'',  and in fact yields a proof of the [[fundamental theorem of arithmetic]].

For example, the cyclic group <math>C_{12}</math> has <math>C_1\triangleleft C_2\triangleleft C_6 \triangleleft C_{12}, \ \, C_1\triangleleft C_2\triangleleft C_4\triangleleft C_{12}, </math> and <math>C_1\triangleleft C_3\triangleleft C_6 \triangleleft C_{12}</math> as three different composition series. The sequences of composition factors obtained in the respective cases are <math>C_2,C_3,C_2, \ \, C_2,C_2,C_3, </math> and <math>C_3,C_2,C_2.</math>