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

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.

A retraction of the kernel in a group extension

Definition

For a group extension

$1 \to N \overset{i}{\to} E \overset{π}{\to} Q \to 1,$

a retraction of the kernel is a group homomorphism $r : E \to N$ (Monoid homomorphism and group homomorphism) such that

$r \circ i = {id}_{N} .$

So $r$ restricts to the inverse of the chosen kernel inclusion on the embedded copy of $N$ . This notion is attached to the short exact sequence of Group extensions, sections, complements, and split extensions, not merely to the abstract middle group.

Depends on

Used by

A group extension splits exactly when it has a complement or a compatible semidirect-product model, and a kernel retraction forces a direct product Theorem

Dependency tree · two levels

8 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)