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Solution of Brauer’s k(B)-conjecture for π-blocks of π-separable groups

  • Benjamin Sambale ORCID logo EMAIL logo
Published/Copyright: January 13, 2018

Abstract

Answering a question of Pálfy and Pyber, we first prove the following extension of the k(GV)-problem: Let G be a finite group and let A be a coprime automorphism group of G. Then the number of conjugacy classes of the semidirect product GA is at most |G|. As a consequence, we verify Brauer’s k(B)-conjecture for π-blocks of π-separable groups which was proposed by Y. Liu. This generalizes the corresponding result for blocks of p-solvable groups. We also discuss equality in Brauer’s Conjecture. On the other hand, we construct a counterexample to a version of Olsson’s Conjecture for π-blocks which was also introduced by Liu.

Keywords:
MSC 2010: 20C15; 20D20

Communicated by Freydoon Shahidi


Award Identifier / Grant number: SA 2864/1-1

Award Identifier / Grant number: SA 2864/3-1

Funding statement: This work is supported by the German Research Foundation by the projects SA 2864/1-1 and SA 2864/3-1.

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Received: 2017-07-11
Published Online: 2018-01-13
Published in Print: 2018-07-01

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