DefinitionDefinition: Literature-sourcedProof: Not applicableaudited 2026-08-02

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.

Commutators $[g, h] = g h g^{- 1} h^{- 1}$ and the commutator subgroup $[G, G]$

Let $G$ be a group. For $g, h \in G$ , their commutator is

$[g, h] : = g h g^{- 1} h^{- 1} .$

This convention is fixed throughout; some sources use its inverse. By the inverse laws of In a group $e^{- 1} = e$ , $(g^{- 1})^{- 1} = g$ and $(g h)^{- 1} = h^{- 1} g^{- 1}$ , the order of the last product being essential, one has $[g, h]^{- 1} = h g h^{- 1} g^{- 1} = [h, g]$ .

The commutator subgroup, or derived subgroup, is the subgroup generated by all commutators:

$[G, G] : = ⟨ {[g, h] : g, h \in G} ⟩ .$

Depends on