Brauer's three main theorems

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). If the idempotent is non-zero, it 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 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 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 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).

References

• gives a detailed proof of the Brauer's main theorems.
• Walter Feit, The representation theory of finite groups. North-Holland Mathematical Library, 25. North-Holland Publishing Co., Amsterdam-New York, 1982. xiv+502 pp.