Alphabeta MathDefinitionDefinition: Literature-sourcedProof: Not applicableaudited 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.
Simple groups
Definition
A group is simple if and its only normal subgroups are and , where normality is as in Normal subgroup: invariance under conjugation.
Depends on
Used by
- No group of order p²q for distinct primes is simple Corollary
- No group of order pq for distinct primes is simple Corollary
- Almost simple finite groups Definition
- Composition series, composition factors, and composition length Definition
- Quasisimple groups, components, and the layer Definition
- Cyclic simple groups of prime order Example
- CFSG says every finite group is simple False statement
- FALSE: Aₙ is simple for every n≥4 False statement
- FALSE: the O'Nan-Scott theorem is the classification of finite simple groups False statement
- Aₙ is simple for every n≥5 Theorem
- Every group of order 105 has normal Sylow 5- and 7-subgroups and is not simple Theorem
- Every group of order 30 has normal Sylow 3- and 5-subgroups and is not simple Theorem
- Finite characteristically simple groups are direct products of isomorphic simple groups Theorem
Dependency tree · two levels
3 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
- T. Judson, Abstract Algebra: Theory and Applications, Simplicity of $A_n$ (standard reference, not scraped)