False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-29

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
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FALSE: a complex character of a finite group is always a group homomorphism

Statement

The statement "a complex character of a finite group is always a group homomorphism" is false: the standard character $χ_{2}$ of $S_{3}$ is not a homomorphism $S_{3} \to C^{\times}$ .

Facts & Assumptions

Given: The group $S_{3}$ and its standard character $χ_{2}$ .

[F1]

The character table of $S_{3}$ gives $χ_{2} ((12)) = 0$ on the class of the transposition $(12)$ (The character table of $S_{3}$ ).

[A1]

A group homomorphism $φ : S_{3} \to C^{\times}$ takes values in the multiplicative group $C^{\times}$ , in which $0$ is not an element.

Refutation

technique · construct

1.1

By [F1], $χ_{2} ((12)) = 0$ .

F1given

2.1

If $χ_{2}$ were a homomorphism $S_{3} \to C^{\times}$ , then by [A1] its value at $(12)$ would be an element of $C^{\times}$ , in particular nonzero, contradicting step 1.1.

A1step 1.1

3.1

Hence $χ_{2}$ is not a group homomorphism, so the claimed statement is refuted.

step 2.1discharge-construct: counterexample∎

Depends on

The character table of $S_3$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

4 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 3 (standard reference, not scraped)