论文标题
在多项式环上的基质组中柱的稳定器
The stabilizer of a column in a matrix group over a polynomial ring
论文作者
论文摘要
提出了一种原始的非标准方法，用于描述$ n \ times n $矩阵在多项式环或多项式环或laurent $ n $变量的多项式环上的结构。稳定器被描述为使用$（N-1）\ times（n-1）$矩阵类型的$（n-1）\ times（n-1）$矩阵类型的子组的扩展，这是$ n-1 $变量的相应环。在本文中，我们考虑$ n \ leq 3. $ n = 2 $的$ n = 2 $，稳定器定义为单参数子组，并且通过直接计算进行证明。情况$ n = 3 $是不平凡的；上述方法应用于它。为获得的结果提供了推论。特别是，我们证明，对于问题中的稳定剂，它不是由它的有限子集以及给定柱的所谓驯服稳定器生成的。我们将在即将发表的论文中研究$ n \ geq 4 $的案例。请注意，组的自动形态组的许多关键亚组被定义为矩阵组中的列稳定器。例如，这描述了自动形态的亚组（$ m_r $），它们是相同的Modulo，是免费的Metabelian $ m_r $ m_r $ r $ r $的通勤者。这种方法证明了对许多知名群体存在的群体和基质组的群体和基质组的理论的平行性。这使我们可以在基质组上使用结果来描述自称组。在这项工作中，使用了Suslin，Cohn以及Bachmuth和Mochizuki的经典定理。
An original non-standard approach to describing the structure of a column stabilizer in a group of $n \times n$ matrices over a polynomial ring or a Laurent polynomial ring of $n$ variables is presented. The stabilizer is described as an extension of a subgroup of a rather simple structure using the $(n-1) \times (n-1)$ matrix group of congruence type over the corresponding ring of $n-1$ variables. In this paper, we consider cases where $n \leq 3.$ For $n = 2$, the stabilizer is defined as a one-parameter subgroup, and the proof is carried out by direct calculation. The case $n = 3$ is nontrivial; the approach mentioned above is applied to it. Corollaries are given to the results obtained. In particular, we prove that for the stabilizer in the question, it is not generated by its a finite subset together with the so-called tame stabilizer of the given column. We are going to study the cases when $n \geq 4$ in a forthcoming paper. Note that a number of key subgroups of the groups of automorphisms of groups are defined as column stabilizers in matrix groups. For example, this describes the subgroup IAut($M_r$) of automorphisms that are identical modulo a commutant of a free metabelian group $M_r$ of rank $r$. This approach demonstrates the parallelism of theories of groups of automorphisms of groups and matrix groups that exists for a number of well-known groups. This allows us to use the results on matrix groups to describe automorphism groups. In this work, the classical theorems of Suslin, Cohn, as well as Bachmuth and Mochizuki are used.
