We describe a new approach for classifying conjugacy classes of elementary abelian subgroups in simple algebraic groups over an algebraically closed field, and understanding the normaliser and centraliser structure of these. For toral subgroups, we give an effective classification algorithm. For non-toral elementary abelian subgroups, we focus on algebraic groups of exceptional type with a view to future applications, and in this case we provide tables explicitly describing the subgroups and their local structure. We then describe how to transfer results to the corresponding finite groups of Lie type using the Lang-Steinberg Theorem; this will be used in forthcoming work to complete the classification of elementary abelian p p -subgroups for torsion primes p p in finite groups of exceptional Lie type. Such classification results are important for determining the maximal p p -local subgroups and p p -radical subgroups, both of which play a crucial role in modular representation theory.

我们描述了一种在代数闭域上将初等阿贝尔子群的共轭类分类为简单代数群的新方法，并理解这些子群的归一化器和中心化结构。对于 toral 亚群，我们给出了一种有效的分类算法。对于非 toral 初等阿贝尔子群，我们专注于异常类型的代数群，以期将来的应用，在这种情况下，我们提供明确描述子群及其局部结构的表格。然后，我们描述了如何使用 Lang-Steinberg 定理将结果转移到相应的 Lie 类型有限群;这将用于即将到来的工作，以完成异常 Lie 类型的有限群中扭转引物 p p 的基本阿贝尔 p 子群的分类。这样的分类结果对于确定最大 p p -局部子群和 p p -根子群很重要，这两者都在模表示理论中起着至关重要的作用。