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.

Authors