Alphabeta MathHow 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.
The quotient group and coset product
Definition
Let be a group and let be a normal subgroup (Normal subgroup: invariance under conjugation). The quotient group, or factor group, has the left cosets
as its elements (Left and right cosets and of a subgroup, The coset set and the index of a subgroup), with product
Independence of the chosen representatives is proved in Coset multiplication is well defined if and only if is normal ↗, and the group axioms are proved in For , the cosets form a group with identity and inverse ↗.