DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-05

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.

Normalized set-theoretic section and factor set

Definition

For an extension

$1 \to M \overset{i}{\to} E \overset{π}{\to} G \to 1$

inducing the given action on the abelian kernel $M$ , a normalized set-theoretic section is a map $s : G \to E$ such that

$π \circ s = {id}_{G}, s (1) = 1.$

Its factor set is the function $f_{s} : G \times G \to M$ determined by

$i (f_{s} (g, h)) = s (g) s (h) s (g h)^{- 1} .$

Because $π (s (g) s (h) s (g h)^{- 1}) = 1$ , the right-hand side lies in $i (M)$ , so $f_{s} (g, h)$ is well defined.

Depends on

An extension inducing a prescribed abelian kernel action

Used by

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

Clara Loh, Group Cohomology, SS 2019 (standard reference, not scraped)
Caroline Lassueur, Cohomology of Groups, SS 2021 (standard reference, not scraped)