On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function

We prove that the number of conjugacy classes of maximal subgroups of bounded order in a finite group of Lie type of bounded rank is bounded. For exceptional groups this solves a long-standing open problem. The proof uses, among other tools, some methods from geometric invariant theory. Using this result, we provide a sharp bound for the total number of conjugacy classes of maximal subgroups of Lie-type groups of fixed rank, drawing conclusions regarding the behaviour of the corresponding "zeta function" ζG(S) = ∑_{M max G} |G : M|^-s, which appears in many probabilistic applications. More specifically, we are able to show that for simple groups G and for any fixed real number s > 1, ζG(s) → 0 as |G| → ∞. This confirms a conjecture made in [27, page 84]. We also apply these results to prove the conjecture made in [28, Conjecture 1, page 343], that the symmetric group S_n has n°⁽¹⁾ conjugacy classes of primitive maximal subgroups.