The probabilistic zeta function of PSL(2,q), of the Suzuki groups ²B₂(q) and of the Ree groups ²G₂(q)

Abstract

We study the Dirichlet polynomial P G (s) of the groups G = PSL(2, q), 2 B 2 (q), and 2 G 2 (q). For such G we show that if H is a group satisfying P H (s) = P G (s), then H/Frat(H) = G. We also prove that, when q is not a prime number, P G (s) is irreducible in the ring of Dirichlet polynomials. Finally, we prove that the coset poset of G is noncontractible.