# zbMATH — the first resource for mathematics

Reduction theorems for the Brauer conjecture on the number of characters in a p-block. (Russian) Zbl 0721.20006
The paper deals with Brauer’s conjecture on the number of irreducible complex characters in a block of finite groups. It is well-known that there are sporadic results in this direction only. Let G be a finite group, p be a prime, B be a p-block of G. Denote by D the defect group of B. Let $$k(B)$$ be the number of the irreducible complex characters in B and $$l(B)$$ be the number of the irreducible Brauer characters in B. Also, $$q(B)$$ be the minimum of the Hermitian form $$Q(e,e)$$ with the matrix $$| D| C_ B$$ for non-zero integral vectors e, where $$C_ B$$ is the Cartan matrix of B. As follows from Brauer’s result [see W. Feit, The representation theory of finite groups (North-Holland, 1982; Zbl 0493.20007), theorem V.9.17], if $$q(B)\geq l(B)$$, the Brauer conjecture is valid.
In the paper we investigate behaviour of q(B) under Fong’s reduction. The main result is: Theorem 1. Let G be a finite p-solvable group, B be a p- block of G. There exist an elementary Abelian p-group P and a $$p'$$-group A acting on P and a p-block b of AP such that $$l(b)\geq l(B)$$ and $$q(B)\geq q(b)$$. If q(b)$$\geq l(b)$$, then $$q(B)\geq l(B)$$. In particular, if $$q(b)\geq l(b)$$, then Brauer’s conjecture is valid for B. Theorem 1 may be used in the following situation. Theorem 2. Let L be a finite group, B be a p- block of L, D be a defect group of B, $$\pi\in Z(G)$$. Suppose that $$C_ G(\pi)$$ is p-solvable. Denote a block of $$C_ G(\pi)$$ by $$\tilde B$$ such that $$(\tilde B)^ G=B$$ and the group and its block by AP and b respectively as in theorem 1, applied to $$G=C_ G(\pi)$$ and to the p- block $$\tilde B.$$ If $$q(b)\geq l(b)$$, Brauer’s conjecture is valid for B. Corollary. Let G be a finite group, B be a p-block of G, D be a defect group of B. Suppose that for some $$\pi\in Z(D)$$, $$C_ G(\pi)$$ is p- solvable and has an Abelian Hall $$p'$$-subgroup. Then Brauer’s conjecture is valid for B.

##### MSC:
 20C20 Modular representations and characters 20D20 Sylow subgroups, Sylow properties, $$\pi$$-groups, $$\pi$$-structure 20D60 Arithmetic and combinatorial problems involving abstract finite groups 20D10 Finite solvable groups, theory of formations, Schunck classes, Fitting classes, $$\pi$$-length, ranks
Full Text: