Abstract

In the factorial ring of Dirichlet polynomials we explore the connections between how the Dirichlet polynomial P_G(s) associated with a finite group G factorizes and the structure of G. If P_G(s) is irreducible, then G∕FratG is simple. We investigate whether the converse is true, studying the factorization in the case of some simple groups. For any prime p ≥ 5 we show that if P_G(s) = P_Alt(p)(s), then G∕FratG≅Alt(p) and P_Alt(p)(s) is irreducible. Moreover, if P_G(s) = P_PSL(2,p)(s), then G∕FratG is simple, but P_PSL(2,p)(s) is reducible whenever p = 2^t − 1 and t = 3 mod 4.

In the factorial ring of Dirichlet polynomials we explore the connections between how the Dirichlet polynomial P_G(s) associated with a finite group G factorizes and the structure of G. If P_G(s) is irreducible, then G∕FratG is simple. We investigate whether the converse is true, studying the factorization in the case of some simple groups. For any prime p ≥ 5 we show that if P_G(s) = P_Alt(p)(s), then G∕FratG≅Alt(p) and P_Alt(p)(s) is irreducible. Moreover, if P_G(s) = P_PSL(2,p)(s), then G∕FratG is simple, but P_PSL(2,p)(s) is reducible whenever p = 2^t − 1 and t = 3 mod 4.

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 $t=3mod4$.

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 $t=3mod4$.

In the factorial ring of Dirichlet polynomials we explore the connections between how the Dirichlet polynomial P_G(s) associated with a finite group G factorizes and the structure of G. If P_G(s) is irreducible, then G ∕ FratG is simple. We investigate whether the converse is true, studying the factorization in the case of some simple groups. For any prime p ≥ 5 we show that if P_G(s) = P_Alt(p)(s), then G ∕ FratG≅Alt(p) and P_Alt(p)(s) is irreducible. Moreover, if P_G(s) = P_PSL(2,p)(s), then G ∕ FratG is simple, but P_PSL(2,p)(s) is reducible whenever p = 2^t − 1 and t = 3 mod 4.

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