Abstract

In the factorial ring of Dirichlet polynomials we explore the connections between how the Dirichlet polynomial $P_G\left(s\right)$ associated with a finite group $G$ factorizes and the structure of $G$. If $P_G\left(s\right)$ is irreducible, then $G∕\text{ Frat}G$ is simple. We investigate whether the converse is true, studying the factorization in the case of some simple groups. For any prime $p\ge 5$ we show that if $P_G\left(s\right)=P_{\text{ Alt}\left(p\right)}\left(s\right)$, then $G∕\text{ Frat}G\cong \text{ Alt}\left(p\right)$ and $P_{\text{ Alt}\left(p\right)}\left(s\right)$ is irreducible. Moreover, if $P_G\left(s\right)=P_{\text{ PSL}\left(2,p\right)}\left(s\right)$, then $G∕\text{ Frat}G$ is simple, but $P_{\text{ PSL}\left(2,p\right)}\left(s\right)$ is reducible whenever $p=2^t-1$ and $t=3mod4$.

Milestones