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

Abstract

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'' ${\zeta }_G\left(s\right)=\sum _{\mathrm{M\; max}G}{∣ G:M∣ }^{-s}$, which appears in many probabilistic applications. More specifically, we are able to show that for simple groups $G$ and for any fixed real number $s>1$, ${\zeta }_G\left(s\right)\to 0$ as $∣ G∣ \to \infty$. 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 $n^{o\left(1\right)}$ conjugacy classes of primitive maximal subgroups.