Alternating sums over pi-subgroups

TL;DR

This paper extends Dade's conjecture to sets of primes for pi-separable groups, connecting local subgroup structures with character counts, and relates to a pi-version of Alperin's weight conjecture.

Contribution

It proves a pi-version of Dade's conjecture for pi-separable groups, generalizing the p-solvable case and linking to a pi-version of Alperin's weight conjecture.

Findings

Proves a pi-version of Dade's conjecture for pi-separable groups

Extends known results from p-solvable to pi-separable groups

Establishes a connection with a pi-version of Alperin's weight conjecture

Abstract

Dade's conjecture predicts that if p is a prime, then the number of irreducible characters of a finite group of a given p-defect is determined by local subgroups. In this paper we replace $p$ by a set of primes pi and prove a pi-version of Dade's conjecture for pi-separable groups. This extends the (known) p-solvable case of the original conjecture and relates to a pi-version of Alperin's weight conjecture previously established by the authors.