RemarkRemark: Literature-sourcedProof: Not suppliedaudited 2026-09-04‡ sources checked 2026-09-04‡ not proved here

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.

‡ Recorded, not proved here. This statement is included so the library can refer to it honestly, with a citation to the literature. It is not proved anywhere in this library: the track that would prove it has not been developed here yet.

Nonabelian extension obstructions live in H^3 and realized classes form an H^2-torsor

Remark

For an abstract kernel $α : Q \to Out (N)$ , the full Eilenberg-Mac Lane theorem identifies a canonical obstruction class in $H^{3} (Q, Z (N))$ . The outer action $α$ is realized by an extension if and only if that obstruction vanishes.

When the obstruction vanishes, the set of equivalence classes of extensions realizing $α$ is not naturally a group, but it is a torsor under $H^{2} (Q, Z (N))$ . This page records that boundary faithfully and defers the actual cohomological theorem to the later cohomology pages.

Depends on

Abstract kernels and the general extension problem

Used by

Nonabelian extension obstruction in H³ Remark

Dependency tree · two levels

2 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

Samuel Eilenberg and Saunders Mac Lane, Cohomology Theory in Abstract Groups. II. Group Extensions with a non-Abelian Kernel (standard reference, not scraped)