Solvability of Generalized Monomial Groups

TL;DR

This paper proves that a generalized class of groups, extending monomial groups through properties of Artin L-series, are also solvable, showing limitations in deriving the Artin conjecture from solvable groups.

Contribution

It introduces a broader class of groups related to Artin L-series and proves their solvability using the classification of finite simple groups.

Findings

Generalized monomial groups are solvable.

Properties of Artin L-series do not imply solvability of non-solvable groups.

Limits on deriving the Artin conjecture from solvable groups.

Abstract

The solvability of monomial groups is a well-known result in character theory. Certain properties of Artin L-series suggest a generalization of these groups, namely to such groups where every irreducible character has some multiple which is induced from a character phi of U with solvable factor group U/ker(phi). Using the classification of finite simple groups, we prove that these groups are also solvable. This means in particular that the mentioned properties do not enable one to deduce a proof of the famous Artin conjecture for any non-solvable group from a possible proof for solvable groups.