DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-09-04

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Equivalence of group extensions with fixed kernel and fixed quotient

Definition

Fix groups $N$ and $Q$ . Two extensions of $Q$ by $N$ ,

$1 \to N \overset{i_{1}}{\to} E_{1} \overset{π_{1}}{\to} Q \to 1, 1 \to N \overset{i_{2}}{\to} E_{2} \overset{π_{2}}{\to} Q \to 1,$

are equivalent with fixed kernel and fixed quotient when there is a group isomorphism (Group isomorphisms, automorphisms and the set $Aut (G)$ ) $φ : E_{1} \to E_{2}$ such that

$φ \circ i_{1} = i_{2}, π_{2} \circ φ = π_{1} .$

Thus the middle groups may change, but the identified copies of $N$ and $Q$ are held fixed. In the language of Group extensions, sections, complements, and split extensions, this is the isomorphism relation on short exact sequences that preserves the displayed kernel and quotient identifications.

Depends on

Used by

Morphisms of group extensions Definition
FALSE: isomorphic middle groups force equivalent extensions with fixed kernel and quotient False statement
A group extension induces a well-defined outer action on its kernel Lemma

Dependency tree · two levels

6 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

J. S. Milne, Group Theory (standard reference, not scraped)