Composition series

A composition series is a finite chain of subgroups in which each subgroup is normal in the next and every successive quotient is simple. Its factor groups are determined up to order and isomorphism by the group.

Normal subgroup: A subgroup invariant under conjugation by every element of its ambient group. Each term must be normal in the next for the successive quotient to be a group.
Jordan–Hölder theorem: A theorem stating that composition factors of a finite-length group are unique up to order and isomorphism. It guarantees that different composition series yield the same factors, counted with multiplicity.
Cyclic group: A group generated by a single element. Every finite cyclic group has prime-order composition factors.
Chief series: A normal series whose factors are minimal normal subgroups of the whole group. Unlike composition factors, chief factors need not be simple groups.
Simple group: A nontrivial group whose only normal subgroups are the identity subgroup and the whole group. A composition series is defined by requiring every successive quotient to be simple.
Refinement of a normal series: The process of inserting intermediate subgroups into a normal series. A normal series becomes a composition series when refined until all factors are simple.
Symmetric group: The group of all permutations of a finite set, under composition. Its composition factors illustrate how a nonabelian simple group can occur.
Derived series: A descending sequence formed by repeatedly taking commutator subgroups. It measures solvability rather than refining a group into simple quotients.
Quotient group: A group formed from the cosets of a normal subgroup in a group. The simple quotients between consecutive terms are the series' factors.
Normal series: A chain of subgroups in which each subgroup is normal in the next. A composition series is a normal series with simple successive quotients and no removable terms.

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