Statement

Let be a finite group and be a field. If then is a semisimple ring.

Proof

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For a field with characteristic 0, the is not a restriction. That means is semisimple for and , for any group .
(Non-example:) , , then the ring is not semisimple, not in .
Let , be a smooth manifold, and is a finite group that acts on . This makes smooth functions a left -module using the action

then by Maschke’s lemma, where are simple submodules. This is the main idea behind Fourier analysis.

Given the conditions needed, i.e. the characteristic of the

todo - Lecture 24