Alphabeta Math

Free Groups and Presentations: Examples and Counterexamples

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✓ 12 results · all verified · 0 also independently AI-judged
Every result on this page is machine-checked by a proof checker and read in full and owner-audited; the judge is an additional, independent cross-model AI review of the proofs. The 12 not AI-judged were verified by owner audit (typically over a confirmed judge false positive), not failures.
1 · Prerequisites
Abstract Algebra
Combinatorics
Number Theory
Real Analysis
Set Theory
2 · Logical flowchart

3 · Items

Example

Reducing the word xx⁻¹yy⁻¹x leaves the reduced word x

Example

The free group on the empty set is the trivial group

Example

The free group on one generator is isomorphic to (ℤ,+)

Example

A free group whose basis contains two distinct elements is not abelian

Example

⟨ a∣ aⁿ⟩≅(ℤ/n,+) for every n≥ 1

Example

Dₙ≅⟨ r,s∣ rⁿ, s², srs⁻¹r⟩ for the dihedral group Dₙ=⟨{ρ,σ}⟩leqSym(ℤ/n), n≥ 3

Example

⟨ a,b∣ a², b², aba⁻¹b⁻¹⟩≅(ℤ/2)×(ℤ/2)

Example

⟨ a,b∣ aba⁻¹b⁻¹⟩≅(ℤ,+)×(ℤ,+)

Example

Sym({0,1,2})≅⟨ s,t∣ s², t², (st)³⟩

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

In ⟨ a,b∣ ab, aba⟩, delete-only relator rewriting sends aba either to the empty word or to the stuck word a

Sources