Landau's Theorem for pi-blocks of 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.