论文标题
将单位性问题与理性组代数的结构联系起来
Connecting monomiality questions with the structure of rational group algebras
论文作者
论文摘要
最近，在两个不同方向上对单一组进行了大量积极的研究。尽管小组理论家对他们的正常亚组和霍尔亚组的研究感兴趣，但由于应用多样化，组环理论家的兴趣在于其理性群体代数的结构。本文的目的是将两个线程绑定在一起。重新访问达德（Dade）著名的嵌入定理，该定理指出，有限的可解决方案可以嵌入一些单一群体中，因此在这里证明，嵌入确实是在某个广义的强烈单元组中完成的。所谓的广义强烈单族在作者的最新作品中出现了，同时了解理性群体代数的代数结构。通过证明所有已回答的单一群体的类别都是概括性的单一群体。这项研究还提出了一些有趣的问题，比Dornhoff和Isaacs在调查中所要求的问题要弱。
In recent times, there has been a lot of active research on monomial groups in two different directions. While group theorists are interested in the study of their normal subgroups and Hall subgroups, the interest of group ring theorists lie in the structure of their rational group algebras due to varied applications. The purpose of this paper is to bind the two threads together. Revisiting Dade's celebrated embedding theorem which states that a finite solvable group can be embedded inside some monomial group, it is proved here that the embedding is indeed done inside some generalized strongly monomial group. The so called generalized strongly monomial groups arose in a recent work of authors while understanding the algebraic structure of rational group algebras. Still unresolved monomiality questions have been correlated by proving that all the classes of monomial groups where they have been answered are generalized strongly monomial. The study also raises some intriguing questions weaker than those asked by Dornhoff and Isaacs in their investigations.
