A review of the development and advanced research of group theory
群论的发展与前沿研究综述

摘要

群论作为数学领域的一个重要分支，其历史脉络、发展轨迹与前沿探索一直是数学界关注的焦点。自群论诞生以来，它经历了从基础理论的构建到现代数学中的广泛应用，再到前沿领域的深入探索的历程。在这个过程中，群论不仅为数学研究提供了强大的工具和方法，还为其他科学领域的发展提供了重要的支撑。在群论的发展历程中，众多杰出的数学家为其做出了卓越的贡献。在现代数学中，群论的应用已经渗透到各个分支领域。例如，在代数几何中，群论被用来研究几何对象的对称性和变换性质，为几何学的发展提供了新的视角和方法。在拓扑学中，群论被用来研究空间的连续变换和拓扑结构，为拓扑学的研究提供了有力的工具。此外，在数论、量子力学、密码学等领域，群论也发挥着重要的作用。这些应用不仅展示了群论的广泛适用性，也进一步推动了群论的发展。随着科学技术的不断进步和数学研究的深入发展，群论的应用领域将会更加广泛，其理论体系也将不断完善和创新。

Abstract

Group theory, as an important branch of mathematics, has always been the focus of the mathematical community for its historical context, development trajectory and frontier exploration. Since the birth of group theory, it has experienced a course from the construction of basic theory to the wide application of modern mathematics, and then to the in-depth exploration of frontier fields. In this process, group theory not only provides powerful tools and methods for mathematical research, but also provides important support for the development of other scientific fields. In the development of group theory, many outstanding mathematicians have made outstanding contributions to it. In modern mathematics, the application of group theory has penetrated into various branches. For example, in algebraic geometry, group theory is used to study the symmetries and transformation properties of geometric objects, providing new perspectives and methods for the development of geometry. In topology, group theory is used to study the continuous transformation and topological structure of space, which provides a powerful tool for the study of topology. In addition, in number theory, quantum mechanics, cryptography and other fields, group theory also plays an important role. These applications not only demonstrate the wide applicability of group theory, but also further promote the development of group theory. With the continuous progress of science and technology and the in-depth development of mathematical research, the application field of group theory will be more extensive, and its theoretical system will continue to improve and innovate.