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.
Crossed homomorphism for a G-group
Definition
Let be a group acting on a group by automorphisms, written . A map is a crossed homomorphism when
for all .
When is abelian we write its law additively, and the same condition becomes
This is the degree-one cocycle identity for a group object with Left group actions, transitive actions, and faithful actions.
Depends on
Used by
- First group cohomology via inhomogeneous one-cocycles Definition
- First nonabelian cohomology as a pointed set Definition
- Principal crossed homomorphism for abelian coefficients Definition
- Crossed homomorphisms from a cyclic group are determined by the value on a generator Example
- FALSE: every crossed homomorphism is an ordinary homomorphism False statement
- A graph subgroup is a complement exactly for a crossed homomorphism Lemma
- With abelian coefficients, crossed homomorphisms form an abelian group Lemma
Dependency tree · two levels
2 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
- David A. Craven, Finite Group Theory (standard reference, not scraped)