# 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

${\displaystyle QC_{G}(Q)}$

for some p-subgroup Q of G, and is contained in the normalizer

${\displaystyle N_{G}(Q)}$,

where ${\displaystyle C_{G}(Q)}$ 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 ${\displaystyle Z(FG)}$ of the (linear) projection from ${\displaystyle FG}$ to ${\displaystyle FC_{G}(Q)}$ whose kernel is spanned by the elements of G outside ${\displaystyle C_{G}(Q)}$. The image of this map is contained in ${\displaystyle Z(FH)}$, 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 (Brauer 1944, 1956, 1970) states that if ${\displaystyle G}$ is a finite group a ${\displaystyle D}$ is a ${\displaystyle p}$-subgroup of ${\displaystyle G}$, then there is a bijection between the set of (characteristic p) blocks of ${\displaystyle G}$ with defect group ${\displaystyle D}$ and blocks of the normalizer ${\displaystyle N_{G}(D)}$ with defect group D. This bijection arises because when ${\displaystyle H=N_{G}(D)}$, 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 (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 ${\displaystyle C_{G}(t)}$ to correspond to a given block of ${\displaystyle G}$, via generalized decomposition numbers. These are the coefficients which occur when the restrictions of ordinary characters of ${\displaystyle G}$ (from the given block) to elements of the form tu, where u ranges over elements of order prime to p in ${\displaystyle C_{G}(t)}$, are written as linear combinations of the irreducible Brauer characters of ${\displaystyle C_{G}(t)}$. The content of the theorem is that it is only necessary to use Brauer characters from blocks of ${\displaystyle C_{G}(t)}$ which are Brauer correspondents of the chosen block of G.

## 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 ${\displaystyle QC_{G}(Q)}$, and contained in ${\displaystyle N_{G}(Q)}$, 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).