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.
Finitely generated groups
Definition
Let be a group. A subset generates if in the sense of The subgroup generated by a subset, the cyclic subgroup , and cyclic groups.
The group is finitely generated if some finite subset generates . Such an is a finite generating set of .
Depends on
Used by
- A group with the word metric of any generating set is a (1,1)-quasi-geodesic space Corollary
- Quasi-isometry invariants and geometric properties of finitely generated groups Definition
- The quasi-isometry type of a finitely generated group Definition
- Word length of a group element with respect to a generating set Definition
- A coarse Lipschitz map between word metric spaces of finitely generated groups is Lipschitz Proposition
- A subgroup of finite index in a finitely generated group is finitely generated, and its inclusion is a quasi-isometry Proposition
- Balls of a word metric are finite if and only if the generating set is finite Proposition
- The quotient map by a finite normal subgroup is a quasi-isometry of word metric spaces Proposition
- The identity map between the word metrics of two finite generating sets is a bilipschitz equivalence Theorem
Dependency tree · two levels
17 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
- C. Drutu and M. Kapovich, Geometric Group Theory, Section 7.9 (standard reference, not scraped)
- C. Loh, Geometric Group Theory: An Introduction (2015 course version), Section 5.2 (standard reference, not scraped)