On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function
نویسنده
چکیده
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 longstanding 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 maxG |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]. We also apply these results to prove the conjecture made in [29] that the symmetric group Sn has n o(1) conjugacy classes of primitive maximal subgroups. 2000 Mathematics Classification Numbers: 20E28, 20G15, 20D06.
منابع مشابه
On the type of conjugacy classes and the set of indices of maximal subgroups
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