Titlebook: Linear Representations of Finite Groups; Jean-Pierre Serre Textbook 1977 Springer Science+Business Media New York 1977 Darstellung (Math.

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ExamplesThis is the group of order n consisting of the powers 1, ., ..., . of an element . such that . = 1. It can be realized as the group of rotations through angles 2./. around an axis. It is an abelian group.

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The group algebraLet G be a group of finite order g, and let K be a commutative ring. We denote by K[G] the . G . K; this algebra has a basis indexed by the elements of G, and most of the time we identify this basis with G. Each element . of K[G] can then be uniquely written in the form . and multiplication in K[G] extends that in G.

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Examples of induced representations A . G, .: G → .(V) . G. .:

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A theorem of BrauerLet . be an element of a finite group G. We say that . is a . (or is .) if . has order a power of .; we say that x is a . (or is .) if its order is prime to ..

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Applications of Brauer’s theoremLet B be a subring of C and let G be a finite group.

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Rationality questionsSo far we have only studied representations defined over the field C of complex numbers. In fact, all the proofs of the preceding sections still hold over an . field of characteristic zero, for example, an algebraic closure of .. Now we are going to see what happens for fields which are not algebraically closed.

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The , triangleWe shall define homomorphisms .,., and . which form a commutative triangle:

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