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The augmentation ideal and Loewy series of kCp can be written explicitly

Example

For $C_{p} = ⟨ g ⟩$ over a field $k$ of characteristic $p$ , write $x = g - 1$ . Then

$k C_{p} ≅ k [x] / (x^{p}),$

the augmentation ideal is $(x)$ , and the Loewy series of the regular module is

$k C_{p} \supset (x) \supset (x^{2}) \supset \dots \supset (x^{p - 1}) \supset (x^{p}) = 0.$

Facts & Assumptions

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

[F1]

The head and Loewy series are the quotient by the radical and its iterated powers (The radical, socle, head, and Loewy series of a finite-dimensional module).

[L1]

Over every field of characteristic $p$ , the group algebra $k C_{p}$ 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).

Verification

technique · direct

1.1

In characteristic $p$ , one has $g^{p} - 1 = (g - 1)^{p} = x^{p}$ , so the relation $g^{p} = 1$ becomes $x^{p} = 0$ . Every element of $k C_{p}$ is a polynomial in $g$ , hence in $x = g - 1$ , and the basis $1, g, \dots, g^{p - 1}$ becomes the basis $1, x, \dots, x^{p - 1}$ . Therefore $k C_{p} ≅ k [x] / (x^{p})$ .

givenalgebra

2.1

Under that identification, the augmentation map kills $x$ and sends $1$ to $1$ , so its kernel is $(x)$ . Since [L1] makes the ring local, $(x)$ is its radical. Therefore [F1] gives the displayed Loewy series by successive powers of $(x)$ .

F1L1step 1.1algebra∎

Depends on

Used by

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Dependency tree · two levels

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Sources

Peter Webb, A Course in Finite Group Representation Theory (23 Feb 2016 draft) (standard reference, not scraped)