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.

Twisted product extension from a two-cocycle

Definition

Given a normalized two-cocycle $f \in Z^{2} (G, M)$ in the sense of Normalized two-cocycle and two-coboundary, define a multiplication on $M \times G$ by

(m, g) (n, h) : = (m + g \cdot n + f (g, h), g h) .

The cocycle identity gives associativity, normalization gives the identity $(0, 1)$ , and the resulting group is denoted $M \times_{f} G$ and called the twisted product defined by $f$ .

The maps

$i (m) = (m, 1), π (m, g) = g$

then give an extension

$1 \to M \overset{i}{\to} M \times_{f} G \overset{π}{\to} G \to 1$

inducing the original $G$ -action on $M$ .

Depends on

Used by

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

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