Jordan–Hölder theorem

The Jordan–Hölder theorem states that any two finite composition series of a group have isomorphic simple factor groups, counted with multiplicity, though their order may differ.

Composition series: A finite subnormal series whose successive quotient groups are simple. Its factor groups are the objects compared by the theorem.
Schreier refinement theorem: A theorem stating that two subnormal series of a group have equivalent refinements. Refining both composition series gives the route to matching their factors.
Composition factor: A simple quotient appearing between consecutive terms of a composition series. The theorem says these factors are independent of the chosen series, up to order.
Schur–Zassenhaus theorem: A theorem describing complements to normal Hall subgroups in finite groups. It addresses splitting of group extensions, not uniqueness of composition factors.
Simple group: A nontrivial group whose only normal subgroups are the identity subgroup and itself. Every factor in a composition series must be simple.
Refinement (group theory): A subnormal series obtained by inserting additional subgroups into an existing series. Composition series can be compared by refining their subgroup chains.
Finite group: A group with finitely many elements. For finite groups, composition series exist and the theorem applies.
Krull–Schmidt theorem: A theorem asserting uniqueness of decompositions into indecomposable objects under suitable conditions. It concerns direct-sum decompositions rather than subgroup-series factors.
Normal subgroup: A subgroup invariant under conjugation by every element of its group. Composition series require each subgroup to be normal in the next.
Zassenhaus lemma: A group-theoretic lemma that identifies isomorphic sections arising from two subgroup chains. It supplies the isomorphisms used to establish equivalent refinements.

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