DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

Group extensions, sections, complements, and split extensions

Definition

A short exact sequence of groups

$1 ⟶ N ⟶^{i} G ⟶^{π} H ⟶ 1$

consists of group homomorphisms (Monoid homomorphism and group homomorphism) with $i$ injective, $π$ surjective, and $im i = \ker π$ , using the kernel and image of The kernel and image of a group homomorphism. It is also called an extension of $H$ by $N$ .

A section is a homomorphism $s : H \to G$ such that $π \circ s = {id}_{H}$ . The extension splits when it has a section. A complement to the kernel is a subgroup (Subgroup) $K \leq G$ such that $G = (\ker π) K$ and $(\ker π) \cap K = {1}$ .

Depends on

Used by

1→ Cₚ→ C_p²→ Cₚ→1 does not split Counterexample
1→⟨ i⟩→ Q₈→ Q₈/⟨ i⟩→1 does not split, with nonabelian middle group Counterexample
A retraction of the kernel in a group extension Definition
Equivalence of group extensions with fixed kernel and fixed quotient Definition
FALSE: a set-theoretic section of an extension is automatically a homomorphism False statement
False: every short exact sequence of groups splits False statement
False: the kernel and quotient determine a group extension up to isomorphism False statement
In a group extension the kernel is normal and the quotient recovers the base Lemma
Splitting lemma for groups: a section, a complement, and a semidirect-product decomposition are equivalent Theorem
The configuration braid short exact sequence 1→ PBₙ→ Bₙᶜᵒⁿᶠ→ Sₙ→ 1 Theorem

Dependency tree · two levels

9 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)