Brauer's three main theorems
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 formula_3 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 formula_4 of the (linear) projection  from formula_5 to formula_6 whose kernel is spanned by the elements of "G" outside formula_3. The image of this map is contained in formula_8, 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. Brauer's first main theorem. Brauer's first main theorem states that if formula_9 is a finite group and formula_10 is a formula_11-subgroup of formula_9, then there is a bijection between the set of (characteristic "p") blocks of formula_9 with defect group formula_10 and blocks of the normalizer formula_15 with defect group "D". This bijection arises because when formula_16, each block of "G" with defect group "D" has  a unique Brauer correspondent block of "H", which also has defect group "D". Brauer's second main theorem. Brauer's second main theorem gives, for an element "t" whose order is a power of a prime "p", a criterion for a (characteristic "p") block of formula_17 to correspond to a given block of formula_9, via "generalized decomposition numbers". These are the coefficients which occur when the restrictions of ordinary characters of formula_9 (from the given block) to elements of the form "tu", where "u" ranges over elements of order prime to "p" in formula_17, are written as linear combinations of  the irreducible Brauer characters of formula_17. The content of the theorem is that it is only necessary to use Brauer characters from blocks of formula_17 which are Brauer correspondents of the chosen block of "G". Brauer's third main theorem. Brauer's third main theorem states that when "Q" is a "p"-subgroup of the finite group "G", and "H" is a subgroup of "G," containing formula_1, and contained in formula_2, then the principal block of "H" is the only Brauer correspondent of the principal block of "G" (where the blocks referred to are calculated in characteristic "p"). Brauer's three