False statementConstruction: AI-adaptedVerification: AI-adaptedprecheck passaudited 2026-08-28

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False statement: the sum of the irreducible degrees equals $∣ G ∣$

Statement

False claim. For a finite group $G$ , the sum of the degrees of the irreducible representations equals $∣ G ∣$ .

Facts & Assumptions

Given: The group $S_{3}$ .

[L1]

The complex group algebra of $S_{3}$ is $C \times C \times M_{2} (C)$ ( $C [S_{3}] ≅ C \times C \times M_{2} (C)$ ).

[L2]

Over an algebraically closed field of characteristic prime to $∣ G ∣$ , the correct identity is the sum-of-squares formula (If $k$ is algebraically closed and $char k ∤ ∣ G ∣$ , then $\sum_{i} (\dim_{k} V_{i})^{2} = ∣ G ∣$ ).

Refutation

technique · direct

1.1L1givenalgebra

From [L1], the irreducible complex representations of $S_{3}$ have degrees $1$ , $1$ , and $2$ . Their sum is $1 + 1 + 2 = 4$ .

2.1L2step 1.1givenalgebra∎

But $∣ S_{3} ∣ = 6$ , so the unsquared sum is not the group order. Indeed [L2] gives the correct identity $1^{2} + 1^{2} + 2^{2} = 6.$ Therefore the claim is false.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

6 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

Pavel Etingof et al., Introduction to Representation Theory, Theorem 3.1(ii) (standard reference, not scraped)

False statement: the sum of the irreducible degrees equals ∣G∣∣G∣