Jordan–Hölder theorem

Jordan–Hölder theorem
Name	Jordan–Hölder theorem
Field	Group theory, Module theory
Introduced	1889
Discoverer	Camille Jordan, Otto Hölder

Jordan–Hölder theorem The Jordan–Hölder theorem is a foundational result in Group theory and module theory that asserts uniqueness (up to order and isomorphism) of composition factors for finite composition series. It connects work of Camille Jordan, Otto Hölder, Émile Mathieu, William Burnside and informs structures studied by Évariste Galois, Arthur Cayley, Felix Klein and Richard Dedekind.

Statement

The theorem states that if a finite group or a finite-length module over a ring admits two composition series, then the series have the same length and the same multiset of simple factors up to isomorphism and permutation. Classical formulations appear in writings of Camille Jordan and Otto Hölder and are used in contexts considered by Hermann Weyl, Emmy Noether, David Hilbert, Issai Schur and Emil Artin.

Proofs

Original proofs derive from inductive methods and the Schreier refinement theorem associated with Otto Schreier and later simplifications use Zassenhaus' butterfly lemma credited to Hans Zassenhaus. Modern expositions employ arguments popularized by Jean-Pierre Serre, Bertram Kostant, Nathan Jacobson and Paul Halmos or categorical approaches influenced by Saunders Mac Lane and Samuel Eilenberg. Proofs often combine chain refinement from Otto Schreier with isomorphism theorems used by Richard Dedekind and techniques echoed in papers of Issai Schur and Emmy Noether.

Composition series and composition factors

A composition series is a maximal subnormal chain whose consecutive quotients are simple objects; classical examples include composition series for finite groups studied by Camille Jordan, William Burnside and Émile Mathieu, and for modules appearing in work of Emil Artin and Emmy Noether. Composition factors are the simple quotients, as in analyses by Issai Schur, Richard Dedekind and Évariste Galois; their behaviour under extensions is central to research by Claude Chevalley, Claude Shannon (in analogy), John von Neumann and Hermann Weyl.

Uniqueness and consequences

The uniqueness statement yields invariants such as the length of a composition series and the multiset of composition factors, which play roles in classification efforts by William Burnside, Huppert, Bertram Huppert, Michael Aschbacher and Daniel Gorenstein. Consequences include refinements of solvability criteria used by Évariste Galois and structure theorems exploited by Feit, John Thompson, Walter Feit and Richard Thompson; they inform the Jordan–Hölder property in categories studied by Grothendieck and Alexander Grothendieck-inspired algebraists and figure in representation-theoretic work of George Lusztig and Israel Gelfand.

Examples and applications

Classic examples include composition series of symmetric groups S_n and alternating groups A_n examined by Évariste Galois and Arthur Cayley, of dihedral groups treated by Camille Jordan and William Burnside, and of modules over principal ideal domains studied by Emil Artin and David Hilbert. Applications extend to representation theory in the work of Frobenius, Issai Schur, Emil Artin, Hermann Weyl and Richard Brauer, to classification results used by Daniel Gorenstein and Michael Aschbacher, and to computational group theory developed by C. C. Sims, John Cannon and Ákos Seress.

Generalizations and related results

Generalizations include the Schreier refinement theorem of Otto Schreier, the Zassenhaus butterfly lemma of Hans Zassenhaus, and categorical formulations in abelian and exact categories influenced by Emmy Noether, Saunders Mac Lane and Alexander Grothendieck. Related results appear in the Jordan–Hölder property for derived categories studied by Maxim Kontsevich and Amnon Neeman, and in lattice-theoretic contexts linked to work of Birkhoff and Garrett Birkhoff; parallels arise in the Krull–Schmidt theorem used by Otto Krull and Wolfgang Schmidt and in the Krull–Schmidt–Azumaya theory extended by Goro Azumaya.

Category:Theorems in group theory