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.
First cohomology via crossed homomorphisms
Definition
Let act on an abelian group . The first cohomology group of with coefficients in is
where is the abelian group of crossed homomorphisms and is the subgroup of principal crossed homomorphisms from With abelian coefficients, crossed homomorphisms form an abelian group and Principal crossed homomorphisms form a subgroup.
Depends on
Used by
- For a finite group, uniquely divisible coefficients have trivial first cohomology Corollary
- Restriction, inflation, and the quotient conjugation action on first cohomology Definition
- The affine group AGL(1,p) has one kernel-conjugacy class of complements to its translation subgroup Example
- First cohomology classifies complements up to kernel conjugacy Theorem
- The inhomogeneous one-cocycle model agrees with crossed homomorphisms in degree one Theorem
Dependency tree · two levels
4 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)