AI 中文总结
该博士论文开发归约方法，将非可解群的扩张计算归约为可解子群中的计算，结合(Zφ)-扩张与补方法，完成了阶小于23040的有限非可解群的同构分类，还提供了相关扩张构造与识别算法。
AI 中文摘要
本博士论文研究群扩张的分类，其主要成果是对阶小于23040的有限非可解群进行同构意义下的分类。为使该分类具备计算有效性，我们开发了归约方法，将涉及有限非可解群的扩张计算替换为在合适的更小可解子群中的计算。这种从非可解计算到可解计算的归约，大幅降低了计算时间，使得在2002年可用的标准个人计算机上，能够对约840万个阶小于23040的非可解群进行分类。给定一个非可解群E由其完全核P（即完全剩余）扩展得到，扩展的商群为可解群H=E/P。所得非可解群E的同构问题被归约为涉及小幂零子群U<P与商群H的扩张之间的同构计算。我们引入并使用(Zφ)-扩张及补方法，以在一系列有限情形中获得明确分类。这些方法被应用于阶小于23040的有限非可解群的分类，并提供了用于构造和识别同构意义下扩张的算法。
英文摘要
This PhD thesis studies the classification of group extensions, whose main result is the classification, up to isomorphism, of finite nonsolvable groups of order less than 23,040. To make this classification computationally effective, we develop reduction methods that replace extension computations involving a finite nonsolvable group by computations in suitable smaller solvable subgroups. This reduction from nonsolvable to solvable computations led to a major decrease in computing time and made it possible, on a standard personal computer available in 2002, to classify around 8.4 million nonsolvable groups of order less than 23,040. A nonsolvable group E is constructed from a given perfect kernel P, its perfect residual, extended by a given solvable quotient H=E/P. The isomorphism problem for the resulting nonsolvable groups E is reduced to isomorphism computations between extensions involving a small nilpotent subgroup U<P and the quotient H. We introduce and use (Zϕ)-extensions and supplement methods to obtain explicit classifications in a range of finite cases. These methods are applied to the classification of finite nonsolvable groups of order less than 23,040 and provide algorithms for constructing and identifying extensions up to isomorphism.
CommentsDoctoral thesis, Université Libre de Bruxelles, 2003. 110 pages. This arXiv version was prepared in 2026 from the original thesis sources, with minor typographical corrections and updated numerical tables; the mathematical framework and main results are those of the 2003 thesis
