On finite subgroups of the classical groups
On finite subgroups of the classical groups
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DOI：
10.1515/jgth-2015-0012
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发表时间：
2015-07
影响因子：
0.5
通讯作者：
M. J. Collins
中科院分区：
文献类型：
--
作者：
M. J. Collins
摘要 1878 年，Jordan 证明 GL(n,ℂ)${\operatorname{GL}(n,\mathbb {C})}$ 的有限子群必须拥有一个阿贝尔正规子群，其索引仅受 n 函数的限制。在之前的论文中，作者获得了最优边界；特别是，当 n≥71${n\ge 71}$ 时，通过对称群 Sn+1 获得通用界限 (n+1)!${(n+1)!}$ 。本文对于复辛群和正交群的有限子群得到了类似的界。在 Sp(2n,ℂ)${\operatorname{Sp}(2n,\mathbb {C})}$ 的情况下，最佳界限是 (60) n ·n!${(60)^{n}\cdot n!}$ ，通过自然作用的花环积 SL 2 (5)wrS n ${\operatorname{SL}_{2}(5)\operatorname{wr}S_{n}}$ 实现n 个二维空间的直和；对于正交群 O(n,ℂ)${\mathrm {O}(n,\mathbb {C})}$ ，只要 n≥25${n\ge 25}$ 即可实现 (n+1)!${(n+1)!}$ 的通用线性群界。
Abstract In 1878, Jordan showed that a finite subgroup of GL(n,ℂ)${\operatorname{GL}(n,\mathbb {C})}$ must possess an abelian normal subgroup whose index is bounded by a function of n alone. In previous papers, the author obtained optimal bounds; in particular, a generic bound (n+1)!${(n+1)!}$ was obtained when n≥71${n\ge 71}$ , achieved by the symmetric group Sn+1. In this paper, analogous bounds are obtained for the finite subgroups of the complex symplectic and orthogonal groups. In the case of Sp(2n,ℂ)${\operatorname{Sp}(2n,\mathbb {C})}$ the optimal bound is (60) n ·n!${(60)^{n}\cdot n!}$ , achieved by the wreath product SL 2 (5)wrS n ${\operatorname{SL}_{2}(5)\operatorname{wr}S_{n}}$ acting naturally on the direct sum of n 2-dimensional spaces; for the orthogonal groups O(n,ℂ)${\mathrm {O}(n,\mathbb {C})}$ , the generic linear group bound of (n+1)!${(n+1)!}$ is achieved as soon as n≥25${n\ge 25}$ .