AI 中文总结
研究有限可解一致半有理群可实现的素图，通过对亚幂零群素图分类及引入MP*群，给出其素图完整分类，还得到该群在不同子类中可实现素图情况并对N-素图分类，解决了一致半有理群的素图问题。
AI 中文摘要
本文研究了有限可解一致半有理群可实现的素图（也称为Gruenberg-Kegel图）。主要通过对亚幂零群可实现的素图进行分类来实现。还引入了一类新的群，称为MP*群，它扩展了具有Magnus性质的群类，是一致半有理群的一个自然子类。对于MP*群，给出了它们可实现的素图的完整分类。在此过程中，假设一致半有理群处于可解群的不同重要子类中，即2-弗罗贝尼乌斯群、亚循环群、亚交换群、幂零-交换群、交换-循环群和循环-交换群，得到了它们可实现素图的情况。还对所有这些群类的所谓N-素图进行了分类。也解决了一致半有理群的素图问题。
英文摘要
In this paper, we study the prime graph, also known as Gruenberg-Kegel graph, realizable by finite solvable uniformly semi-rational groups. This is primarily achieved by classifying the prime graphs realizable by metanilpotent groups. We also introduce a new class of groups, called MP* groups, which extends the class of groups with Magnus property and forms a natural subclass of uniformly semi-rational groups. For MP* groups, we give a complete classification of the prime graphs realized by them. In the process, we obtain the realizability of prime graphs by uniformly semi-rational groups, assuming them to be in varied substantial subclasses of solvable groups, namely, 2-Frobenius, metacyclic, metabelian, nilpotent-by-abelian, abelian-by-cyclic and cyclic-by-abelian. We also classify the so called N-prime graphs for all these classes of groups. The prime graph question for uniformly semi-rational groups has also been addressed.
