Alphabeta MathHow 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.
Characteristic subgroups
Definition
A subgroup is characteristic in , written , if for every automorphism (Group isomorphisms, automorphisms and the set , Subgroup).
Equivalently, every automorphism of restricts to an automorphism of . Characteristicity requires invariance under all automorphisms, not only under inner automorphisms.
Depends on
Used by
- A subgroup of an abelian group need not be characteristic Counterexample
- K is normal in HcharG always implies K is normal in G False statement
- Characteristic subgroups are normal, and characteristicity is transitive Lemma
- If K is characteristic in N and N is normal in G, then K is normal in G Lemma
- Minimal normal subgroups of finite groups are characteristically simple Lemma
- The Frattini subgroup of a finite group is characteristic Lemma
- The socle is characteristic and decomposes as a direct product of minimal normal subgroups Proposition
- Finite characteristically simple groups are direct products of isomorphic simple groups Theorem
- The derived subgroup is characteristic and the abelianization is universal Theorem
Dependency tree · two levels
8 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)