False statementConstruction: AI-adaptedVerification: AI-adaptedSession-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 statement: a group with $r$ conjugacy classes has an irreducible representation of degree $r$

Statement

False claim. If a finite group has $r$ conjugacy classes, then it has an irreducible representation of degree $r$ .

Facts & Assumptions

Given: The quaternion group $Q_{8}$ .

[L1]

The group $Q_{8}$ has five conjugacy classes, but its irreducible complex degrees are $1, 1, 1, 1, 2$ ( $C [Q_{8}]$ and $C [Dih (C_{4})]$ both decompose as $C^{4} \times M_{2} (C)$ ).

Refutation

technique · direct

1.1

By [L1], the relevant value is $r = 5$ , while the irreducible degrees of $Q_{8}$ are only $1, 1, 1, 1, 2$ .

L1given

2.1

None of those degrees equals $5$ , so the claim fails even for $Q_{8}$ . Therefore a group with $r$ conjugacy classes need not have an irreducible representation of degree $r$ .

step 1.1∎

Depends on

$\mathbb C[Q_8]$ and $\mathbb C[\operatorname{Dih}(C_4)]$ both decompose as $\mathbb C^4\times M_2(\mathbb C)$

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

8 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 Section 3.4 (standard reference, not scraped)

False statement: a group with rr conjugacy classes has an irreducible representation of degree rr