CorollaryStatement: Literature-sourcedProof: AI-generatedprecheck passjudge pass (gpt-5.6-terra)audited 2026-09-04

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A finite p-group has only the trivial simple module over a field of characteristic p

Statement

Let $P$ be a finite $p$ -group and $k$ a splitting field of characteristic $p$ . Then the only simple $k P$ -module is the trivial module $k$ .

Facts & Assumptions

Given: A finite $p$ -group $P$ and a splitting field $k$ of characteristic $p$ .

[L1]

The group algebra $k P$ is local (For a finite group and a field of characteristic p, the group algebra is local exactly when the group is a p-group).

Proof

technique · direct

1.1L1given

By [L1], the algebra $k P$ has a unique simple module up to isomorphism.

2.1step 1.1givenalgebra∎

The augmentation map $k P \to k$ makes the ground field $k$ into a simple $k P$ -module on which every element of $P$ acts trivially. Since step 1.1 says there is only one simple module, it must be this trivial module.

Depends on

For a finite group and a field of characteristic p, the group algebra is local exactly when the group is a p-group

Used by

Brauer characters of a p-group Example
Defect zero and p group boundaries Example

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 (23 Feb 2016 draft) (standard reference, not scraped)