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In mathematics, a composition series of a group G is a chain of subgroups of G satisfying
- <math>1\triangleleft H_1\triangleleft \cdots \triangleleft H_n\triangleleft G,<math>
where <math>\triangleleft<math> stands for normal subgroup, such that the quotient group of each link in the chain is a simple group.
For a finite group G, such composition series always exist. In fact, the isomorphism classes of simple groups are unique, up to permutation (more precisely, as a multiset). This result is the Jordan-Hölder theorem, named after Camille Jordan and Otto Hölder.
See also normal series.
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