On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function
On conjugacy classes of maximal subgroups of finite simple groups, and a related zeta function
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关于有限单群的最大子群的共轭类以及相关的zeta函数
DOI：
10.1215/s0012-7094-04-12834-9
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发表时间：
2005
影响因子：
2.5
通讯作者：
A. Shalev
中科院分区：
文献类型：
--
作者：
M. Liebeck;B. Martin;A. Shalev
证明了有限秩李型群中有界阶极大子群的共轭类个数是有界的。对于特殊群体来说，这解决了一个长期存在的开放问题。在其他工具中，证明使用了几何不变理论的一些方法。利用这个结果，我们给出了固定秩李型群的极大子群的共轭类总数的一个精确界，并得出了相应的zeta函数的性质的结论|G：M|- s，它出现在许多概率应用中。更具体地说，我们能够证明，对于单群G和任何固定的真实的数s > 1，|G| → ∞。这证实了[27]中的一个猜想。我们还应用这些结果证明了文[29]中的猜想：对称群Sn有no（1）个本原极大子群的共轭类. 2000年数学分类号：20 E28、20 G15、20 D 06。
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.