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

TL;DR

This paper extends Landau's theorem to pi-blocks of pi-separable groups, establishing bounds on defect groups based on irreducible characters, and generalizes several classical results using the classification of finite simple groups.

Contribution

It introduces a bound on defect group order in pi-blocks, generalizing Brauer's Problem 21 and Landau's theorem for broader classes of groups.

Findings

Bound on defect group order in pi-blocks in terms of irreducible characters

Generalization of Brauer's Problem 21 to pi-separable groups

Extension of Landau's theorem to pi-separable groups

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.