False statementConstruction: Literature-sourcedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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.

FALSE: if $∣ G ∣ > 1$ , then $k [G]$ is a field

Statement

False claim. If $G$ is a nontrivial finite group and $k$ is a field, then $k [G]$ is a field.

Facts & Assumptions

Given: A nontrivial finite group $G$ and a field $k$ .

[L1]

The group algebra $k [G]$ has nonzero zero divisors (Any nontrivial finite group algebra has zero divisors coming from a nonidentity cyclic subgroup).

[L2]

In a field every nonzero element is invertible (Field).

Refutation

technique · direct

1.1

By [L1], choose nonzero elements $x, y \in k [G]$ with $x y = 0$ .

L1givenchoose

2.1

If $k [G]$ were a field, then [L2] would make $x$ invertible, and multiplying $x y = 0$ by $x^{- 1}$ would give $y = 0$ , contradicting step 1.1. So the claim is false.

step 1.1L2algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Peter Webb, A Course in Finite Group Representation Theory, Chapter 1 Section 1.1 (standard reference, not scraped)

FALSE: if ∣G∣>1∣G∣>1, then k[G]k[G] is a field