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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
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