Brauer's theorem on induced characters
Brauer's theorem on induced characters, often known as Brauer's induction theorem, and named after Richard Brauer, is a basic result in the branch of mathematics known as character theory, which is, in turn, part of the representation theory of a finite group. Let G be a finite group and let Char(G) denote the subring of the ring of complex-valued class functions of G consisting of integer combinations of irreducible characters. Char(G) is known as the character ring of G, and its elements are known as virtual characters (alternatively, as generalized characters, or sometimes difference characters). It is a ring by virtue of the fact that the product of characters of G is again a character of G. Its multiplication is given by the elementwise product of class functions.
Brauer's induction theorem shows that the character ring can be generated (as an abelian group) by induced characters of the form , where H ranges over subgroups of G and λ ranges over linear characters (having degree 1) of H.
In fact, Brauer showed that the subgroups H could be chosen from a very restricted collection, now called Brauer elementary subgroups. These are direct products of cyclic groups and groups whose order is a power of a prime.
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