DefinitionDefinition: Literature-sourcedProof: Not applicableSession-authored (Fable 5 assisted)judge pass (gpt-5.6-terra)audited 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.

The class sum $\hat{C}$ of a conjugacy class $C$

Definition

Let $G$ be a finite group, let $k$ be a field, and let $C \subseteq G$ be a conjugacy class (The conjugacy class ${Cl}_{G} (x)$ and centralizer $C_{G} (x)$ of an element). Its class sum in $k [G]$ is

\hat{C} : = \sum_{g \in C} [g] .

Because $C$ is finite, this is a well-defined element of the group algebra $k [G]$ (The group ring $R [G]$ of finitely supported formal $R$ -linear combinations of group elements).

Depends on

Used by

For a finite group, the class sums form a basis of Z(k[G]) Theorem

Dependency tree · two levels

5 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 3.4.2 (standard reference, not scraped)

The class sum C^C of a conjugacy class CC