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Brauer's main theorems are three theorems in representation theory of finite groups linking the blocks of a finite group (in characteristic ''p'') with those of its ''p''-local subgroups, that is to say, the normalizers of its non-trivial ''p''-subgroups. The second and third main theorems allow refinements of orthogonality relations for ordinary characters which may be applied in finite group theory. These do not presently admit a proof purely in terms of ordinary characters. All three main theorems are stated in terms of the Brauer correspondence. ==Brauer correspondence== There are many ways to extend the definition which follows, but this is close to the early treatments by Brauer. Let ''G'' be a finite group, ''p'' be a prime, ''F'' be a ''field'' of characteristic ''p''. Let ''H'' be a subgroup of ''G'' which contains : for some ''p''-subgroup ''Q'' of ''G,'' and is contained in the normalizer :, where is the centralizer of ''Q'' in ''G''. The Brauer homomorphism (with respect to ''H'') is a linear map from the center of the group algebra of ''G'' over ''F'' to the corresponding algebra for ''H''. Specifically, it is the restriction to of the (linear) projection from to whose kernel is spanned by the elements of ''G'' outside . The image of this map is contained in , and it transpires that the map is also a ring homomorphism. Since it is a ring homomorphism, for any block ''B'' of ''FG'', the Brauer homomorphism sends the identity element of ''B'' either to ''0'' or to an idempotent element. In the latter case, the idempotent may be decomposed as a sum of (mutually orthogonal) primitive idempotents of ''Z(FH).'' Each of these primitive idempotents is the multiplicative identity of some block of ''FH.'' The block ''b'' of ''FH'' is said to be a Brauer correspondent of ''B'' if its identity element occurs in this decomposition of the image of the identity of ''B'' under the Brauer homomorphism. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Brauer's three main theorems」の詳細全文を読む スポンサード リンク
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