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Patassini, Massimiliano On the irreducibility of the Dirichlet polynomial of a simple group of Lie type. [Preprint]

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Abstract (english)

Given a finite group $G$ and a normal subgroup $N$ of $G$, the Dirichlet polynomial of $G$ given $G/N$ is $$P_{G,N}(s)=\sum_{\tiny \begin{array}{c} H\leq G\\ NH=G
\end{array}\normalsize} \frac{\mu_G(H)}{|G:H|^s}.$$
In this paper, we assume that $G$ is a primitive monolithic group with non-abelian socle $\soc(G)\cong S^n$ for some simple group $S$ of Lie type. Under some assumptions on the Lie rank of $S$, we prove that $P_{G,\soc(G)}(s)$ is irreducible in the ring of finite Dirichlet series. Moreover, we show that the Dirichlet polynomial $P_S(s)=P_{S,S}(s)$ of a simple group $S$ of Lie type is reducible if and only if $S$ is isomorphic to $A_1(p)$ where $p$ is a Mersenne prime such that $\log_2(p+1)\equiv 3\pmod{4}$.


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EPrint type:Preprint
Codice ID:3271
Depositato il:13 Jan 2011 09:13
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