无穷单三角形矩阵群中循环群的自由积

Q3 Mathematics Matematychni Studii Pub Date : 2023-09-22 DOI:10.30970/ms.60.1.28-33
A. Oliynyk
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引用次数: 0

摘要

研究了结合酉环上的无限单三角形矩阵群。这些群自然作用于底层环上的无限维自由模。如果下环是有限的,它们就是无限的。利用它们与有限自动机所定义的群的联系,研究了用带无限单棱矩阵构造群的自由积的忠实表示问题。对于任意素数p,给出了特征为p的酉结合环上的带无限酉三角形矩阵有限集生成循环p群自由积的充分条件。这些条件是基于下环上有限维自由模作用的某些性质。结果表明,这些条件都是满足的。对于有限个循环p群的任意自由积,给出了特征p的唯一结合环上的无限单三角形矩阵集生成给定自由积的构造性实例。这些无限矩阵是由有限维的幂零约旦块构成的。关于所提出的例子和其他类型的忠实表示的性质的几个开放问题被制定。
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Free products of cyclic groups in groups of infinite unitriangular matrices
Groups of infinite unitriangular matrices over associative unitary rings are considered. These groups naturally act on infinite dimensional free modules over underlying rings. They are profinite in case underlying rings are finite. Inspired by their connection with groups defined by finite automata the problem to construct faithful representations of free products of groups by banded infinite unitriangular matrices is considered.For arbitrary prime p a sufficient conditions on a finite set of banded infinite unitriangular matrices over unitary associative rings of characteristic p under which they generate the free product of cyclic p-groups is given. The conditions are based on certain properties of the actions on finite dimensional free modules over underlying rings.It is shown that these conditions are satisfied. For arbitrary free product of finite number of cyclic p-groups constructive examples of the sets of infinite unitriangular matrices over unitar associative rings of characteristic p that generate given free product are presented. These infinite matrices are constructed from finite dimensional ones that are nilpotent Jordan blocks.A few open questions concerning properties of presented examples and other types of faithful representations are formulated.
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来源期刊
Matematychni Studii
Matematychni Studii Mathematics-Mathematics (all)
CiteScore
1.00
自引率
0.00%
发文量
38
期刊介绍: Journal is devoted to research in all fields of mathematics.
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