CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-05

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The number of simple kG-modules equals the number of p-regular conjugacy classes

Statement

The number of isomorphism classes of simple $k G$ -modules equals the number of $p$ -regular conjugacy classes of $G$ .

Facts & Assumptions

Given: A finite group $G$ over a splitting field $k$ of characteristic $p$ .

[L1]

The irreducible Brauer characters form a basis of the class functions on $G^{0}$ (Irreducible Brauer characters form a basis of the p-regular class functions).

[A1]

A class function on $G^{0}$ is determined by its values on the $p$ -regular conjugacy classes, so the indicator functions of those classes form a basis.

Proof

technique · direct

1.1

By [A1], the vector space of class functions on $G^{0}$ has dimension equal to the number of $p$ -regular conjugacy classes.

A1given

2.1

By [L1], the irreducible Brauer characters form a basis of that same space. The number of basis vectors is therefore the number of irreducible Brauer characters, equivalently the number of simple $k G$ -modules.

L1step 1.1

3.1

Hence the number of simple $k G$ -modules equals the number of $p$ -regular conjugacy classes.

step 2.1∎

Depends on

Irreducible Brauer characters form a basis of the p-regular class functions

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

J. Miquel Martinez, Modular Representation Theory of Finite Groups (standard reference, not scraped)
Tudor Ciurca, Representation Theory (standard reference, not scraped)