On the irreducibility of the dirichlet polynomial of a simple group of lie type

Abstract

In this paper, we assume that G is a primitive monolithic group with nonabelian socle soc(G) ∼ = S n for some simple group S of Lie type. Under some assumptions on the Lie rank of S, we prove that P G,soc(G) (s) is irreducible in the ring of finite Dirichlet series. Moreover, we show that the Dirichlet polynomial P S (s) = P S,S (s) of a simple group S of Lie type is reducible if and only if S is isomorphic to A 1 (p), where p is a Mersenne prime such that log 2 (p + 1) ≡ 3 (mod 4).

Profinite groups in which the probabilistic zeta function has no negative coefficients

“…In the case of alternating groups, we may use the following result: if G = A n , n ≥ 9 and |A n : K| ≤ n(n − 1), then K is either a point-stabilizer, or a 2-set stabilizer, or the intersection of two point-stabilizers (see for example [11,Theorem 5.2A]). This implies When G is a simple group of Lie type defined over a field of characteristic p, one can consider the inverse of the series P (p) G (s) = (n,p)=1 a n (G)/n s , which can be easily described since it depends only on the parabolic subgroups of G (see [26,Theorem 17]). For example, it is not difficult to see that if G = P SL(2, q) is an untwisted group of Lie type, then (P (p) G (s)) −1 has at least a negative coefficient.…”

Rationality of the probabilistic zeta functions of finitely generated profinite groups

“…We have that: Theorem 3.2. [14,Theorem 17] Let S be a simple group of Lie type defined over a field K = F q of characteristic p and X an almost simple group with socle S. Then…”

Recognizing the Non-Frattini Abelian Chief Factors of a Finite Group from Its Probabilistic Zeta Function

“…Assume that S PSL k q for k ≥ 3 and X does not contain non-trivial graph automorphisms. By [20,Theorem 2], if k q = 3 2 , then the polynomial P L N s is irreducible, so P L N s P p L N s = 1. If S PSL 3 2 then p = 7.…”