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

1.1L1given

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

2.1L1step 1.1

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

3.1L2L3step 2.1∎

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.

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)