Landau's Theorem for pi-blocks of pi-separable groups

DOI10.32037/AGTA-2019-013zbMATH Open1456.20007arXiv1810.05827OpenAlexW4393142405MaRDI QIDQ5219194

Publication date: 6 March 2020

Abstract: Slattery has generalized Brauer's theory of p-blocks of finite groups to pi-blocks of pi-separable groups where pi is a set of primes. In this setting we show that the order of a defect group of a pi-block B is bounded in terms of the number of irreducible characters in B. This is a variant of Brauer's Problem 21 and generalizes K"ulshammer's corresponding theorem for p-blocks of p-solvable groups. At the same time, our result generalizes Landau's classical theorem on the number of conjugacy classes of an arbitrary finite group. The proof relies on the classification of finite simple groups.

Full work available at URL: https://arxiv.org/abs/1810.05827

zbMATH Keywords

\(\pi\)-blocks number of characters Brauer's problem 21

Mathematics Subject Classification ID

Ordinary representations and characters (20C15) Modular representations and characters (20C20)

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