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.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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FALSE: distinct irreducible complex characters of a finite group have distinct degrees

Statement

The statement "distinct irreducible complex characters of a finite group have distinct degrees" is false: the group $Q_{8}$ has four distinct irreducible characters of degree $1$ .

Facts & Assumptions

Given: The group $Q_{8}$ and its character table.

[F1]

The character table of $Q_{8}$ has the four distinct irreducible characters $χ_{1}$ , $χ_{2}$ , $χ_{3}$ , $χ_{4}$ , whose rows all begin with the value $1$ (The character table of $Q_{8}$ ).

[A1]

The degree of an irreducible character is its value at the identity, the first column of the table.

Refutation

technique · construct

1.1

By [F1], the four characters $χ_{1}, χ_{2}, χ_{3}, χ_{4}$ of $Q_{8}$ are distinct irreducible characters.

F1given

2.1

By [F1] and [A1], each of these four has degree $1$ , since each row begins with the value $1$ at the identity.

F1A1step 1.1

3.1

Hence four distinct irreducible characters share the degree $1$ , so the claimed statement is refuted.

step 2.1discharge-construct: counterexample∎

Depends on

The character table of $Q_8$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

7 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)