The probabilistic zeta function of finite simple groups

Abstract

Let G be a finite group; there exists a uniquely determined Dirichlet polynomial P(G,s) such that if t is a positive integer, then P(G,t) gives the probability of generating G with t randomly chosen elements. We show that it may be recognized from the knowledge of P(G,s) whether G/Frat(G) iis a simple group

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

“…For instance, If G and H are groups such that P G (s) = P H (s) and G is soluble (or p-soluble, or perfect), then H has the same property [Damian and Lucchini 2003;Detomi and Lucchini 2003b]. If G is simple and P G (s) = P H (s), then H/Frat(H ) is simple [Damian and Lucchini 2007].…”

Recognizing the characteristic of a simple group of Lie type from its Probabilistic Zeta function

“…There are two ways to see that H is a simple group. The first one is by [8,Theorem 7]. The second one is the following: we know that if G is a simple group of Lie type, then the Dirichlet polynomial P G (s) is reducible if and only if G ∼ = PSL 2 (p) with p = 2 e − 1 and e ≡ 3 (mod 4) (see [27]).…”

The Probabilistic Zeta Function of Alternating and Symmetric Groups

“…This new function has again a probabilistic meaning: if P is a Sylow 2-subgroup of G and t ∈ N, then P G, odd (t) gives the conditional probability that t random elements of G generate G together with P given that their product normalizes P (see [8,Proposition 1]). So for example, P G, odd (1) = 0 if G does not have a normal Sylow 2-subgroup.…”

“…An easy computation shows that This is done in [8] for the sporadic simple groups and the simple groups of Lie type; in these cases, there exists a prime divisor p of the order of G with the property that no proper subgroup (or only few subgroups in some particular bad cases) can contain a Sylow 2-subgroup and an element of order p and it is easy to prove that α p (G) = 1. However, it does not seem that it is possible to choose a prime p with similar properties when G is an alternating group, and in [8] the problem has been left open whether α(Alt(n)) = 1.…”

“…However, it does not seem that it is possible to choose a prime p with similar properties when G is an alternating group, and in [8] the problem has been left open whether α(Alt(n)) = 1. We solve this problem in the case of the alternating group, giving a complete description of the subgroups of odd index with non-zero Möbius function.…”