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

Massimiliano Patassini

Vol. 240 (2009), No. 1, 185–200

DOI: 10.2140/pjm.2009.240.185

Abstract

We study the Dirichlet polynomial P_G(s) of the groups G = PSL(2,q), ² B₂(q), and ²G₂(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.

Keywords

probabilistic zeta function, simple Lie groups, Suzuki groups, Ree groups, simple linear groups, coset poset

Mathematical Subject Classification 2000

Primary: 20D30

Secondary: 20P05, 11M41, 20D06, 20D60, 20E28

Milestones

Received: 26 February 2008

Revised: 31 July 2008

Accepted: 14 November 2008

Published: 2 March 2009