The Classification of Finite Simple Groups

研究者和他的同事继续他们的项目，写一个完整的，有效的和实质上自包含的证明有限单群的分类。在完成了这个项目的四卷之后，他们正在准备接下来的四卷，构成证明的核心，一般情况和特殊奇数情况。这些例子涵盖了奇型有限单群的识别，一类包括几乎所有的交替群和限定在奇阶有限域上的李型群，以及26个零星群中的5个。在过去的两个世纪里，被称为“群”的数学对象一直是整个数学领域中最基本、研究最多的对象之一。群是一个抽象的数学对象，人们可以通过它来表述和研究对称，无论是化学分子、晶体、原子、密码、网络还是抽象的数学结构。每一个这样的物体都有一个被称为“对称群”的群，它捕捉了物体的对称特性，并使某些基于对称的有用计算成为可能。例如，对称群的识别和利用在许多复杂的计数方案、许多编码和解码算法、x射线晶体学和量子物理学中都是决定性的。最基本的群是所谓的“简单”群；例如，所有有限群（与具有有限数量对称的对象相对应的群）都是由有限的“简单”群构成的。事实上，许多在重要应用中出现的有限群是简单的或近似简单的。由于这些原因，确定所有有限单群的问题从理论和应用的角度都是至关重要的。对这一问题的深入研究始于20世纪50年代初，通过研究者和他的同事以及其他研究人员的工作，该研究已接近成果。

The investigator and his colleague continue their project to write a complete, efficient and substantially self-contained proof of the classification of the finite simple groups. Having completed four volumes of this project, they are preparing the next four volumes, constituting the heart of the proof, the generic case and the special odd case. These cases cover the identification of the finite simple groups of odd type, a class including almost all of the alternating groups and the groups of Lie type defined over finite fields of odd order, as well as five of the twenty-six sporadic groups.For the past two centuries the mathematical objects called ``groups'' have been among the most fundamental and most studied in the entire landscape of mathematics. A group is an abstract mathematical object by which one may formulate and study symmetry, whether in chemical molecules, crystals, atoms, codes, networks, or abstract mathematical structures. Each such object possesses a group called its ``symmetry group'' which captures the symmetry properties of the object and which makes possible certain kinds of useful calculations based on symmetry. For example, the recognition and exploitation of symmetry groups is decisive in many complex counting schemes, in many algorithms for coding and decoding, in X-ray crystallography and in quantum physics. The most fundamental groups are the so-called ``simple'' groups; for example, all the finite groups (those corresponding to objects possessing a finite number of symmetries) are built out of finite ``simple'' groups. Indeed many finite groups which arise in important applications are simple or nearly simple. For these reasons, the problem of determining all the finite simple groups is of paramount importance from both the theoretical and applied perspectives. The intensive investigation of this problem began in the early 1950's, and is nearing fruition through the work of the investigator and his colleague, as well as other researchers.