论划分环上特殊线性群的对角线子群

IF 0.3 Q4 MATHEMATICS

Acta Mathematica Vietnamica Pub Date : 2024-08-02 DOI:10.1007/s40306-024-00544-6

Bui Xuan Hai

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引用次数: 0

摘要

让 K 是一个以 Z(K) 为中心的分环，n 是一个正整数。让 \(\textrm{SL}(n,K)\) 是 K 上 n 度的特殊线性群，而 \(\textrm{SD}(n,K)\) 是由 Dieudonne 行列式为 \(\overline{1}) 的所有对角矩阵组成的子群。我们证明了在\(n\ge 3\) 的情况下，Z(K)是一个无限域；或者在\(n\ge 5\) 的情况下，Z(K)是一个至少包含七个元素的有限域，那么\(\textrm{SD}(n,K)\)在\(\textrm{SL}(n,K)\)中是弱正则的，但不是正则的。

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On the Diagonal Subgroup of the Special Linear Group Over a Division Ring

Let K be a division ring with center Z(K), and n a positive integer. Let \(\textrm{SL}(n,K)\) be the special linear group of degree n over K and \(\textrm{SD}(n,K)\) its subgroup consisting of all diagonal matrices whose Dieudonne’s determinant is \(\overline{1}\). We prove that \(\textrm{SD}(n,K)\) is weakly pronormal, but not pronormal in \(\textrm{SL}(n,K)\) provided either Z(K) is an infinite field in case \(n\ge 3\) or Z(K) is a finite field containing at least seven elements in case \(n\ge 5\).

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来源期刊

Acta Mathematica Vietnamica MATHEMATICS-

CiteScore

0.90

自引率

0.00%

发文量

期刊介绍： Acta Mathematica Vietnamica is a peer-reviewed mathematical journal. The journal publishes original papers of high quality in all branches of Mathematics with strong focus on Algebraic Geometry and Commutative Algebra, Algebraic Topology, Complex Analysis, Dynamical Systems, Optimization and Partial Differential Equations.