字的有效性與群的冪零性

前言

　　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.

內(nèi)容概要

冪零群是介于交換群與可解群之間的一類群，在群論中占有十分重要的位置。本書研究群的廣義冪零性。冪零群被有限冪指數(shù)群的擴(kuò)張群，是比冪零群范圍更廣的一類群；同時它們也遺傳了許多冪零群的良好性質(zhì)，因而對這類群的研究具有十分重要的意義。作者在自己研究成果的基礎(chǔ)上，總結(jié)了多年來在該領(lǐng)域的一些典型成果，從群定律與群結(jié)構(gòu)兩方面論述了群的冪零性。本書分兩部分。第一部分（2，3，4，5章）研究自由群及字（元素）的性質(zhì)。第二部分研究群的結(jié)構(gòu)。并著重研究了塌縮群，正定群，Milnor群，多項(xiàng)循環(huán)群sB一群等形態(tài)群的冪零性。    該書適合作為本科高年級學(xué)生的群論教材或參考資料，也可作為數(shù)學(xué)專業(yè)學(xué)生的雙語課教材。

書籍目錄

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

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