TheoremStatement: Literature-sourcedProof: 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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The regular character is $∣ G ∣$ at $1$ and $0$ away from $1$

Statement

Let $G$ be a finite group and let $χ_{r e g}$ be the character of the regular representation $C [G]$ over $C$ . Then

$χ_{r e g} (g) = {\begin{cases} ∣ G ∣, & g = 1, \\ 0, & g \neq 1. \end{cases}$

Facts & Assumptions

Given: A finite group $G$ , an element $g \in G$ , and the regular representation $C [G]$ .

[F1]

The regular representation acts on the basis $([h])_{h \in G}$ by $g \cdot [h] = [g h]$ (The trivial representation, the regular representation, and permutation representations from finite $G$ -sets).

[F2]

The character of a permutation representation counts fixed points (The character of a permutation representation counts fixed points).

[A1]

In a group, $g x = x$ holds if and only if $g = 1$ , by cancellation.

Proof

technique · direct

1.1

By [F1], the regular action is the permutation representation of the left $G$ -set $G$ with basis indexed by the group elements. By [F2], $χ_{r e g} (g)$ is the number of $h \in G$ with $g \cdot h = h$ .

F1F2given

2.1

By [A1], $g h = h$ has a solution in $G$ only when $g = 1$ , in which case every $h$ is fixed. Hence the count of step 1.1 is $∣ G ∣$ for $g = 1$ and $0$ for $g \neq 1$ .

A1step 1.1algebra∎

Depends on

Used by

The regular character gives a second proof of the sum-of-squares formula Corollary

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, Lemma 3.3.6 (standard reference, not scraped)

The regular character is ∣G∣∣G∣ at 11 and 00 away from 11