TheoremStatement: Literature-sourcedProof: Literature-sourcedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

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The Jordan–Hölder theorem for groups

Statement

If a group $G$ has two composition series, then the series have the same length and their composition factors agree up to isomorphism and permutation.

Facts & Assumptions

Given: Two composition series of the same group $G$ .

[F1]

A composition series is a strictly descending subnormal series with simple factors (Composition series, composition factors, and composition length).

[L1]

Any two finite subnormal series of a group have equivalent refinements (The Schreier refinement theorem).

[L2]

For $N ⊴ H$ , the maps $K \mapsto K / N$ and inverse image under $H \to H / N$ give inverse inclusion-preserving bijections between subgroups above $N$ and subgroups of $H / N$ , and they preserve normality (Correspondence theorem: subgroups of $G / N$ correspond to subgroups of $G$ containing $N$ , with normality preserved).

Proof

technique · direct

1.1givenL1

By [L1], the two composition series have equivalent refinements.

1.2F1L2

A subnormal refinement cannot insert a term strictly between adjacent terms $H ▹ N$ : the first inserted term $K$ would satisfy $N < K < H$ and $K ⊴ H$ , so [L2] would make $K / N$ a nontrivial proper normal subgroup of the simple factor $H / N$ .

2.1step 1.2

Therefore each refinement differs from its original composition series only by repeated adjacent terms, and deleting those repetitions recovers the original series.

3.1step 1.1step 2.1∎

Equivalence of the refinements now pairs the original nontrivial factors; hence the two series have the same number of factors, and a permutation matches their factors up to isomorphism.

Depends on

Used by

Dependency tree · two levels

12 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Group Theory, Chapter 6 (standard reference, not scraped)
K. Conrad, Subgroup Series I (standard reference, not scraped)
K. Igusa, Notes on Jordan-Hölder, section 5 (standard reference, not scraped)