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.
Contents
- Brauer correspondence
- Brauers first main theorem
- Brauers second main theorem
- Brauers third main theorem
- References
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
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 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 (Brauer 1944, 1956, 1970) states that if
Brauer's second main theorem
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
Brauer's third main theorem
Brauer's third main theorem (Brauer 1964, theorem3) states that when Q is a p-subgroup of the finite group G, and H is a subgroup of G, containing