Rationality of the probabilistic zeta functions of finitely generated profinite groups

Abstract

We prove that if the probabilistic zeta function P-G(s) of a finitely generated profinite group G is rational and all but finitely many nonabelian composition factors of G are groups of Lie type in a fixed characteristic or sporadic simple groups, then G contains only finitely many maximal subgroups

Finiteness of Profinite Groups with a Rational Probabilistic Zeta Function

“…We are now looking for w(X) := min{m|m ∈ Ω(X)} where X is an almost simple group whose socle S a finite simple group appeared in Theorem 1. In case S is either PSL(2, p) for some prime p ≥ 5 or a sporadic simple group, the values w(X) are computed in [9] and [8] respectively. When S = Alt(p) for a prime p ≥ 5, then Alt(p − 1) is the unique maximal subgroup of S of index p (cf.…”

“…Let T be the set of almost simple groups X such that there are infinitely many indices i ∈ J with X i ∼ = X, and let I = {i ∈ J : X i ∼ = X}. Our assumptions combined with [8,Lemma 4.4], imply that J \ I is finite. In order to prove that J is finite, we need to prove that I is empty.…”

“…However, the conjecture is still open in general and it is very hard to give an affirmative answer for the conjecture when S contains non-comparable families of finite simple groups. In this paper, using the methods developed in [9] and [8], we obtain the following result.…”

“…[6]) and several classes of non-prosolvable groups (cf. [7], [8], [9]). A consequence of those results is that if almost every composition factor of G is isomorphic to a given simple group S, then the conjecture holds.…”

The Möbius function of PSU(3, 2^{2^n})

“…The property (1), which we call (µ, λ)-property, does not hold in general for non-solvable groups; see [1]. Pahlings [23] proved that, if G is solvable, then A lot of work was done by several authors about probabilistic functions for groups; see for instance [19,20,6,10]. In particular, Mann posed in [19] a conjecture, the validity of which would imply that the sum H µ(H, G) [G : H] s over all subgroups H < G of finite index of a positively finitely generated profinite group G is absolutely convergent for s in some right complex half-plane and, for s ∈ N large enough, represents the probability of generating G with s elements.…”