Weak almost monomial groups and Artin's conjecture

Abstract

We introduce a new class of finite groups, called weak almost monomial, which generalize two different notions of "almost monomial" groups, and we prove it is closed under taking factor groups and direct products. Let $K/\mathbb Q$ be a finite Galois extension with a weak almost monomial Galois group $G$ and $s_0\in \mathbb C\setminus \{1\}$. We prove that Artin conjecture's is true at $s_0$ if and only if the monoid of holomorphic Artin $L$-functions at $s_0$ is factorial. Also, we show that if $s_0$ is a simple zero for some Artin $L$-function associated to an irreducible character of $G$ and it is not a zero for any other $L$-function associated to an irreducible character, then Artin conjecture's is true at $s_0$.