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.
Group presentation by generators and relations
Definition
Let be a free group and let be a set of words, called relations. The group with presentation
is the quotient by the normal closure of . The members of are its generators. In this quotient, every relation in becomes the identity, as do all consequences forced by normality.
Depends on
Used by
- Every finite group has a finite presentation from its multiplication table Corollary
- In ⟨ a,b∣ ab, aba⟩, delete-only relator rewriting sends aba either to the empty word or to the stuck word a Counterexample
- In ⟨ a,b∣ ab⟩, the trivial word ba is stuck under free cancellation and delete-only relator rewriting Counterexample
- In ⟨ a,b∣ aba⁻¹b⁻¹⟩, the trivial word a²b²a⁻²b⁻² is stuck under free cancellation and delete-only relator rewriting Counterexample
- Relators and relations; finitely generated, finitely related, and finite presentations Definition
- Tietze transformations: dictionary generators, redundant relators, renaming, and their inverses Definition
- ⟨ a,b∣ a², b², aba⁻¹b⁻¹⟩≅(ℤ/2)×(ℤ/2) Example
- ⟨ a,b∣ aba⁻¹b⁻¹⟩≅(ℤ,+)×(ℤ,+) Example
- ⟨ a∣ aⁿ⟩≅(ℤ/n,+) for every n≥ 1 Example
- Amalgamating infinite cyclic groups by multiplication by m and n gives the presentation with relation xᵐ=yⁿ Example
- Dₙ≅⟨ r,s∣ rⁿ, s², srs⁻¹r⟩ for the dihedral group Dₙ=⟨{ρ,σ}⟩leqSym(ℤ/n), n≥ 3 Example
- Sym({0,1,2})≅⟨ s,t∣ s², t², (st)³⟩ Example
- In ⟨ X∣ R⟩, the words u and v represent the same element if and only if u⁻¹v∈⟨⟨ R⟩⟩ Proposition
- A free product has the union presentation of presentations of its factors Theorem
- A free product with amalgamation has the factor presentations plus the amalgamating relations Theorem
- Every group admits a presentation Theorem
- Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group Theorem
Dependency tree · next 3 levels
Direct dependencies and their dependencies through the next three levels: 22 results over 12 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
- Brittenham, Group Presentations, Class Notes (standard reference, not scraped)