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 free group whose basis contains two distinct elements is not abelian

Example

If a free basis contains distinct elements $x$ and $y$ , then the free group is not abelian.

Facts & Assumptions

Given: A set $X$ with distinct elements $x, y \in X$ , and a free group on $X$ .

[L1]

Every class in $W (X) / \sim$ contains exactly one reduced word (Every class in $W (X) / \sim$ contains exactly one reduced word).

[L2]

The word-quotient group $W (X) / \sim$ , with $x \mapsto [x]$ , is a free group on $X$ (The word-quotient group $W (X) / \sim$ satisfies the universal property of the free group on $X$ ).

[L3]

Free groups on the same set are uniquely isomorphic compatibly with their generators (Free groups on the same set are uniquely isomorphic compatibly with their generators).

Verification

technique · direct

1.1

The words $x y$ and $y x$ are reduced, and they are literally different because $x \neq y$ .

L1given

2.1

Uniqueness in [L1] makes their word classes different, so $[x] [y] \neq [y] [x]$ in the word-quotient free group.

L1step 1.1

3.1

By [L2] and [L3], every free group on $X$ is isomorphic to that model by an isomorphism fixing the generators, so the two chosen basis elements do not commute and the group is not abelian.

L2L3step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

13 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)