ExampleConstruction: AI-adaptedVerification: AI-generatedprecheck pass

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.

$C_{2}$ free-product $C_{2}$ is the infinite dihedral group, and the product of its generators has infinite order

Example

The free product $C_{2} * C_{2}$ has presentation $⟨ s, t ∣ s^{2} = e, t^{2} = e ⟩ .$ This is the standard infinite dihedral group $D_{\infty}$ , and $s t$ has infinite order.

Facts & Assumptions

Given: The objects and hypotheses in the example.

[L1]

Suppose each $G_{i}$ has a presentation $⟨ X_{i} ∣ R_{i} ⟩$ , with the alphabets replaced by disjoint copies. Then $*_{i} G_{i} ≅ ⟨ ⨆_{i} X_{i} | ⋃_{i} R_{i} ⟩ .$ (A free product has the union presentation of presentations of its factors).

[L2]

Every element of $*_{i \in I} G_{i}$ has a unique reduced syllable expression. The identity is represented by the empty word, and no nonempty reduced word represents the identity. (Normal form theorem for free products).

[L3]

For every $n \in N$ , view $n$ as its canonical nonnegative integer and put $n Z : = {n k : k \in Z}$ . Then the left cosets of $n Z$ in $(Z, +)$ are exactly the congruence classes modulo $n$ , and coset addition is the published addition of congruence classes. Thus $(Z, +) / n Z = (Z / n, +)$ as the same group on the same underlying set. This includes $n = 0$ and $n = 1$ . (For every $n \in N$ , the congruence-class group $(Z / n, +)$ is the quotient group $(Z, +) / n Z$ ).

Verification

technique · direct

1.1givenL1L2L3

The union-presentation theorem gives the displayed presentation from the two cyclic factors.

2.1step 1.1∎

For every $n > 0$ , the word $(s t)^{n}$ is a nonempty reduced word, so normal form makes it nonidentity. Hence $s t$ has infinite order and the group is infinite.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

George D. Torres, Combinatorial Group Theory, §2 (standard reference, not scraped)
B. H. Neumann, Lectures on Topics in the Theory of Infinite Groups, Ch. 9 (standard reference, not scraped)

C2C2​ free-product C2C2​ is the infinite dihedral group, and the product of its generators has infinite order