阶至多2000的有限单纯可约群的分类
A classification of finite simply reducible groups of order at most 2000
- School of Mathematics, Shandong University（山东大学数学学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过计算机辅助分类，给出了阶至多2000的所有非平凡有限单纯可约群（共7889个）的同构分类，并特别构造了阶1024的803个群，推导了计数公式与界。
AI中文摘要：
我们给出了阶至多2000的所有非平凡有限单纯可约群在同构意义下的计算机辅助分类。这样的群共有7889个，且均为偶数阶。我们提供了代表元以及每个阶的群数表格。阶1024在计算机代数系统GAP的SmallGroups库中没有作为群对象目录提供，因此需要单独构造。结合后代生成与对$\n\mathbb F_2$上二次映射的分类，我们恰好得到803个该阶的群，其中包括221个幂零类为2的群。我们还推导了若干无限阶族上的精确计数公式以及来自结构子类的显式界。
英文摘要：
We give a computer-assisted classification, up to isomorphism, of all nontrivial finite simply reducible groups of order at most 2000. There are 7889 such groups, all of even order. We provide representatives and a table of their numbers at each order. Order 1024, which is not available as a catalogue of group objects in the SmallGroups library of the computer algebra system GAP, requires a separate construction. Combining descendant generation with a classification of quadratic maps over $\mathbb F_2$, we obtain exactly 803 groups of this order, including 221 of nilpotency class two. We also derive exact counting formulas on several infinite families of orders and explicit bounds from structural subclasses.
