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The regular module of Cp in characteristic p is indecomposable with a unique simple quotient

Example

Let $C_{p} = ⟨ g ⟩$ and let $k$ have characteristic $p$ . Then the regular $k C_{p}$ -module is indecomposable, and its unique simple quotient is the trivial module $k$ .

Facts & Assumptions

Given: The cyclic group $C_{p}$ and a field $k$ of characteristic $p$ .

[L1]

Over every field of characteristic $p$ , the group algebra of a finite $p$ -group 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).

Verification

technique · direct

1.1

By [L1], the algebra $A : = k C_{p}$ is local. A nontrivial decomposition of the left regular module would give a nontrivial idempotent projection in ${End}_{A} (_{A} A) ≅ A^{o p}$ , but a local algebra has no nontrivial idempotents. Hence the regular module is indecomposable.

L1givenalgebra

2.1

The augmentation ideal is maximal because its quotient is $k$ , and it is the unique maximal left ideal because $A$ is local. Every simple quotient of the regular module has a maximal left ideal as its kernel, so it is the augmentation quotient $k$ , with the trivial $C_{p}$ -action. Thus the regular module has the asserted unique simple quotient.

L1step 1.1algebra∎

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

Nothing in the library uses this result yet.

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)