精品文档---下载后可任意编辑Grobner-Shirshov 基理论在群上的若干应用的开题报告【摘要】Grobner-Shirshov 基理论是现代代数的重要分支之一，其主要讨论对象为非交换自由群的基本子代数的结构。本文旨在通过探讨 Grobner-Shirshov 基理论在群上的若干应用，深化理解该理论的概念、方法和应用。文中将从以下三个方面进行讨论和讨论：一、Grobner-Shirshov 基理论的概念和基本思想；二、Grobner-Shirshov 基理论在群上的应用，包括基本子代数的结构、消解算法和微小自由消解等；三、Grobner-Shirshov 基理论在一些具体群类别（如无限周期群和有限群）上的应用实例。估计本文将对 Grobner-Shirshov 基理论的理论和实践讨论有一定的贡献，同时对于相关领域的学者和讨论者也将提供一定的参考和借鉴作用。【关键词】Grobner-Shirshov 基理论；非交换自由群；基本子代数；消解算法；无限周期群；有限群。【Abstract】Grobner-Shirshov basis theory is an important branch of modern algebra, which mainly studies the structure of basic subalgebras of non-commutative free groups. The purpose of this paper is to explore several applications of Grobner-Shirshov basis theory on groups, and to deepen the understanding of the concepts, methods and applications of this theory. The paper will study and discuss from the following three aspects: 1. The concept and basic ideas of Grobner-Shirshov basis theory; 2. The applications of Grobner-Shirshov basis theory on groups, including the structure of basic subalgebras, resolution algorithms, and minimal free resolutions; 3. Examples of the application of Grobner-Shirshov basis theory on some specific categories of groups (such as infinite periodic groups and finite groups). It is expected that this paper will make some contributions to the theoretical and practical research of Grobner-Shirshov basis theory, and also provide some references and inspiration for scholars and researchers in related fields.【Keywords】Grobner-Shirshov basis theory; non-commutative free group; basic subalgebra; resolution algorithm; infinite periodic group; finite group.
