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 nontrivial 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.
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
Since it is a ring homomorphism, for any block B of FG, the Brauer homomorphism sends the identity element of B to an idempotent element (possibly to 0).
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.
, then there is a bijection between the set of (characteristic p) blocks of
with defect group D. This bijection arises because when
, 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 1944, 1959) gives, for an element t whose order is a power of a prime p, a criterion for a (characteristic p) block of
These are the coefficients which occur when the restrictions of ordinary characters of
, are written as linear combinations of the irreducible Brauer characters of
The content of the theorem is that it is only necessary to use Brauer characters from blocks of