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

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An extension inducing a prescribed abelian kernel action

Definition

Let $M$ be an abelian group and $G$ a group acting on $M$ . An extension

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

induces the prescribed action when for each $g \in G$ and each lift $\tilde{g} \in E$ with $π (\tilde{g}) = g$ , the rule

$g \cdot m : = i^{- 1} (\tilde{g} i (m) {\tilde{g}}^{- 1})$

recovers the given action of $G$ on $M$ .

This is well defined because changing the lift by an element of $i (M)$ does not change the conjugation action on $i (M)$ when $M$ is abelian. Equivalence of such extensions is the fixed-kernel fixed-quotient equivalence of Equivalence of group extensions with fixed kernel and fixed quotient.

Depends on

Equivalence of group extensions with fixed kernel and fixed quotient

Used by

Baer sum of abelian-kernel extensions Definition
Normalized set-theoretic section and factor set Definition
Twisted product extension from a two-cocycle Definition

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)