ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-11

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.

$⟨ a ∣ a^{n} ⟩ ≅ (Z / n, +)$ for every $n \geq 1$

Example

For every natural number $n \geq 1$ ,

$⟨ a ∣ a^{n} ⟩ ≅ (Z / n, +),$

with the generator $a$ corresponding to the residue class $[1]_{n}$ . At $n = 1$ both groups are trivial.

Facts & Assumptions

Given: A natural number $n \geq 1$ and the presentation $P = ⟨ a ∣ a^{n} ⟩$ .

[L1]

For integers $k$ and positive $n$ , there are integers $q, r$ with $k = q n + r$ and $0 \leq r < n$ (Division with remainder in $Z$ : for $a \in Z$ and $b > 0$ there are unique $q, r \in Z$ with $a = q b + r$ and $0 \leq r < b$ ).

[L2]

Every class in $Z / n$ contains exactly one integer $r$ with $0 \leq r < n$ (For $n \geq 1$ , every class in $Z / n$ has one representative $r$ with $0 \leq r < n$ , so $∣ Z / n ∣ = n$ ; while $Z / 0$ is in bijection with $Z$ ).

[L3]

A map of generators that sends every relator to the identity extends uniquely to a homomorphism from the presented group (Von Dyck's theorem: maps of generators that satisfy the relators extend uniquely from a presented group).

[L4]

Group powers satisfy $g^{m + n} = g^{m} g^{n}$ and $(g^{m})^{q} = g^{m q}$ for integer exponents (Exponent laws in a group: $g^{m + n} = g^{m} g^{n}$ and $(g^{m})^{n} = g^{m n}$ for all $m, n \in Z$ , and $(g h)^{n} = g^{n} h^{n}$ when $g$ and $h$ commute).

Verification

technique · constructive

1.1

In the additive group $Z / n$ , the $n$ -fold multiple of $[1]_{n}$ is $[n]_{n} = [0]_{n}$ , so [L3] constructs a homomorphism $π : P \to Z / n$ with $π (a) = [1]_{n}$ .

L3construct

1.2

Every word on one generator represents $a^{k}$ for some $k \in Z$ ; write $k = q n + r$ by [L1]. Since $a^{n} = e$ in $P$ , [L4] gives $a^{k} = (a^{n})^{q} a^{r} = a^{r}$ with $0 \leq r < n$ .

L1L4given

2.1

The image of $a^{r}$ is $[r]_{n}$ , and [L2] makes these images distinct and exhaustive for $0 \leq r < n$ ; combined with step 1.2, this proves that $π$ is injective and surjective.

L2step 1.1step 1.2

3.1

Hence $π$ is the claimed isomorphism; when $n = 1$ , the sole normal form is $a^{0} = e$ and the sole residue is $[0]_{1}$ .

step 2.1discharge-construct∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

42 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

John McKernan, Presentations and Groups of Small Order, Lecture 12 (standard reference, not scraped)
J. Aspnes, Group Theory (standard reference, not scraped)

⟨a∣an⟩≅(Z/n,+)⟨a∣an⟩≅(Z/n,+) for every n≥1n≥1