字的有效性与群的幂零性
2009-5
北京邮电大学出版社
李千路
161
We are familiar with abelian and finite groups. Although， of course， these by no means account for all groups， every group is connected with them in some way or other. For instance the free groups of rank are not abelian and not finite，yet，as we have seen，they possess descending normal chains with abelian factors， and they are residually finite. There are a great many papers on group theory devoted to establishing connections such as these with finiteness and commutativity. As a result there exists a welter of conditions generalizing the conditions of finiteness and abelianness， ranging from the sublime to the ridiculous. The most important generalizations of commutativity are solubility and nilpotence. Soluble groups are those that can be constructed from abelian groups by means of a finite number of successive extensions. They are especially well known for their relevance to the problem of solving algebraic equations by radicals ，whence their name. Nilpotent groups form a class smaller than that of soluble groups but larger than that of abelian groups. Their definition is more complicated， but they can be more intimately studied than soluble groups. Thus nilpotent groups occupy an important position in the theory of groups.
幂零群是介于交换群与可解群之间的一类群，在群论中占有十分重要的位置。本书研究群的广义幂零性。幂零群被有限幂指数群的扩张群，是比幂零群范围更广的一类群；同时它们也遗传了许多幂零群的良好性质，因而对这类群的研究具有十分重要的意义。作者在自己研究成果的基础上，总结了多年来在该领域的一些典型成果，从群定律与群结构两方面论述了群的幂零性。本书分两部分。第一部分（2，3，4，5章）研究自由群及字（元素）的性质。第二部分研究群的结构。并着重研究了塌缩群，正定群，Milnor群，多项循环群sB一群等形态群的幂零性。 该书适合作为本科高年级学生的群论教材或参考资料，也可作为数学专业学生的双语课教材。
Chapter 1 PrefaceChapter 2 Fundamental Concept I:Free GroupsChapter 3 Words in the Free Group F2 Chapter 4 General WordsChapter 5 Properties of the Standard Exponents of WordsChapter 6 Fundamental Concept H:Nilpotent GroupsChapter 7 Collapsing GroupsChapter 8 Groups Satisfying Positive WordsChapter 9 Milnor Groups and Groups Satisfying Efficient WordsChapter 10 Polycyclic GroupsChapter 11 Varieties of GroupsChapter 12 Metabelian Subvarieties of GroupsChapter 13 Finitely Generated f-Milnor GroupsChapter 14 Criteria for Almost NllpotenceBibliography
