Alphabeta Math
False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passaudited 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: every representation is faithful

Statement

False claim. Every representation of a group is faithful.

Facts & Assumptions

Given: A nontrivial group G and its trivial representation over a field k.

[L1]

In the trivial representation, every group element acts as the identity map (The trivial representation, the regular representation, and permutation representations from finite G-sets).

[L2]

A representation is faithful when only the identity group element acts as the identity map (Intertwiners, the spaces Hom⁡G(V,W) and End⁡G(V), equivalent representations, and faithful representations).

Refutation

technique · direct
1.1

By [L1], every element of the nontrivial group G acts as the identity in the trivial representation.

L1given
2.1

Since some element of G is not the identity, [L2] shows that this representation is not faithful. Therefore the stated claim is false.

step 1.1L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

9 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