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 $p\geq 5$ we show that if $P_G(s) = P_{Alt(p)}(s)$, then $G/FratG \cong Alt(p)$ 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 $t = 3 mod 4$.

SOME PROPERTIES ON THE PROBABILISTIC ZETA FUNCTION OF FINITE SIMPLE GROUPS / DAMIAN E; LUCCHINI A; MORINI F.. - In: PACIFIC JOURNAL OF MATHEMATICS. - ISSN 0030-8730. - 215(2004), pp. 3-14.

SOME PROPERTIES ON THE PROBABILISTIC ZETA FUNCTION OF FINITE SIMPLE GROUPS

MORINI, Fiorenza
2004

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 $p\geq 5$ we show that if $P_G(s) = P_{Alt(p)}(s)$, then $G/FratG \cong Alt(p)$ 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 $t = 3 mod 4$.
SOME PROPERTIES ON THE PROBABILISTIC ZETA FUNCTION OF FINITE SIMPLE GROUPS / DAMIAN E; LUCCHINI A; MORINI F.. - In: PACIFIC JOURNAL OF MATHEMATICS. - ISSN 0030-8730. - 215(2004), pp. 3-14.
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/11381/1894781
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