有限群的基本理论与应用

The Basic Theory and Applications of Finite Groups

摘要: 有限群是群论中一个重要且广泛研究的分支，对于数学、物理学、化学以及密码学等领域都具有重要意义。本论文旨在介绍有限群的定义、性质及分类，Abel群的定义，有限p群的定义和结构定理，即Sylow 定理。并探讨有限群的相关理论在不同领域中的应用，包括几何学(对称性与立体几何、曲面理论、张量积分与变换、状态空间搜索)、数论、物理学(对称性与粒子物理学、能带理论与固体物理学、场论和粒子物理)和密码学等领域。

Abstract: Finite group is an important and widely studied branch of group theory, which is of great signifi-cance for fields such as mathematics, physics, chemistry, and cryptography. This paper aims to in-troduce the definition, properties, and classification of finite group, the definition of Abel group, the definition and structure theorem of finite p-groups, namely the Sylow theorem. And explore the application of related theories of finite groups in different fields, including geometry (symmetry and solid geometry, surface theory, tensor integration and transformation, state space search), number theory Fields such as physics (symmetry and particle physics, band theory and solid state physics, field theory and particle physics) and cryptography.