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

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The p-regular classes of S3

Example

For $S_{3}$ , the $2$ -regular conjugacy classes are the identity class and the class of $3$ -cycles, while the $3$ -regular conjugacy classes are the identity class and the class of transpositions.

Facts & Assumptions

Given: The symmetric group $S_{3}$ .

[F1]

An element is $p$ -regular exactly when $p$ does not divide its order (p-regular and p-singular elements).

[L1]

Brauer characters are constant on $p$ -regular conjugacy classes (Brauer characters are class functions on p-regular elements).

Verification

technique · direct

1.1

In $S_{3}$ , the conjugacy classes are represented by $1$ , a transposition $(12)$ of order $2$ , and a $3$ -cycle $(123)$ of order $3$ .

givenalgebra

2.1

By [F1], the $2$ -regular elements are those of order $1$ or $3$ , so they form the classes of $1$ and $(123)$ . Likewise the $3$ -regular elements are those of order $1$ or $2$ , so they form the classes of $1$ and $(12)$ .

F1step 1.1

3.1

Hence a Brauer character of $S_{3}$ in characteristic $2$ or $3$ is determined by the two values on the corresponding classes, in agreement with [L1].

L1step 2.1∎

Depends on

Used by

The decomposition matrix of S3 in characteristic two Example

Dependency tree · two levels

4 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)