DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-13

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Composition series, composition factors, and composition length

Definition

A composition series of a group $G$ is a subnormal series $G = G_{0} ▹ G_{1} ▹ \dots ▹ G_{n} = 1$ whose inclusions are strict and whose factors $G_{i} / G_{i + 1}$ are simple groups (Simple groups). The factors are the composition factors, and $n$ is the composition length of this series.

The trivial group has the length-zero composition series consisting only of $1$ . A nontrivial group has a composition series exactly when it has a finite subnormal series with simple factors.

Depends on

Used by

The order of a finite group is the product of the orders of its composition factors Corollary
A composition series and the derived series of S₃ Example
Composition and derived series of S₄ Example
The composition factors determine a finite group up to isomorphism False statement
Every finite group has a composition series Theorem
The Jordan–Hölder theorem for groups Theorem

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 8 results over 4 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click 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)