False statementConstruction: Literature-sourcedVerification: AI-generatedprecheck passaudited 2026-08-28

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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.

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FALSE: every degree-one representation is trivial

Statement

False claim. Every degree-one representation is trivial.

Facts & Assumptions

Given: The cyclic group $C_{2} = ⟨ g ⟩$ .

[L1]

Over $C$ , a cyclic group of order $n$ has exactly the $n$ characters $g \mapsto λ$ with $λ^{n} = 1$ (Over $C$ , a cyclic group of order $n$ has exactly $n$ irreducible representations up to equivalence, represented by the characters $g \mapsto λ$ with $λ^{n} = 1$ ).

Refutation

technique · direct

1.1L1given

Applying [L1] with $n = 2$ gives a degree-one representation of $C_{2}$ with $g \mapsto - 1$ .

2.1step 1.1∎

This representation is not trivial because the generator acts by $- 1 \neq 1$ . Therefore the stated claim is false.

Depends on

Over $\mathbb{C}$, a cyclic group of order $n$ has exactly $n$ irreducible representations up to equivalence, represented by the characters $g\mapsto \lambda$ with $\lambda^n=1$

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

Peter Webb, A Course in Finite Group Representation Theory, Example 1.1.2 (standard reference, not scraped)