Alphabeta Math
DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)audited 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]=ghg−1h−1[g,h]=ghg^{-1}h^{-1} and the commutator subgroup [G,G][G,G]

Definition

Let GG be a group. For g,h∈Gg,h\in G, their commutator is

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

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

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

[G,G]:=⟨{[g,h]:g,h∈G}⟩.[G,G]:=\langle\{[g,h]:g,h\in G\}\rangle.

The generated subgroup notation is that of The subgroup ⟨S⟩\langle S \rangle generated by a subset, the cyclic subgroup ⟨g⟩\langle g \rangle, and cyclic groups.

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 11 results over 10 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources