# 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 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.

## 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
    $QC_{G}(Q)$
for some p-subgroup Q of G, and is contained in the normalizer
    $N_{G}(Q)$,
where $C_{G}(Q)$ 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 $Z(FG)$ of the (linear) projection from $FG$ to $FC_{G}(Q)$ whose kernel is spanned by the elements of G outside $C_{G}(Q)$. The image of this map is contained in $Z(FH)$, 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 to an idempotent element (possibly to 0). 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 $G$ is a finite group and $D$ is a $p$-subgroup of $G$, then there is a bijection between the set of (characteristic p) blocks of $G$ with defect group $D$ and blocks of the normalizer $N_{G}(D)$ with defect group D. This bijection arises because when $H=N_{G}(D)$, 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 $C_{G}(t)$ to correspond to a given block of $G$, via generalized decomposition numbers. These are the coefficients which occur when the restrictions of ordinary characters of $G$ (from the given block) to elements of the form tu, where u ranges over elements of order prime to p in $C_{G}(t)$, are written as linear combinations of the irreducible Brauer characters of $C_{G}(t)$. The content of the theorem is that it is only necessary to use Brauer characters from blocks of $C_{G}(t)$ 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 $QC_{G}(Q)$ and contained in $N_{G}(Q)$, 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).