PERMUTATION GROUPS (London Mathematical Society Student Texts 45)
PERMUTATION GROUPS (London Mathematical Society Student Texts 45)
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DOI：
10.1112/s0024609300217359
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发表时间：
2000-11
影响因子：
0.9
通讯作者：
G. Jones
中科院分区：
文献类型：
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作者：
G. Jones
群论的早期发展在很大程度上是置换群的发展，特别是伽罗瓦群，置换多项式的根。到19世纪末的世纪，然而，公理化的方法，群论开始接管，尽管发现约旦，伯恩赛德，弗罗贝纽斯，舒尔和Wielandt置换群，在未来60-70年看到群论越来越关注抽象群体：马歇尔霍尔的书[3]在20章中只用了一章来讨论置换群，而申克曼[4]几乎完全忽略了它们，即使在讨论弗罗贝纽斯群时也是如此。然而，近几十年来，这一主题又死灰复燃。在有限单群的分类过程中，构造或消除一个受限单群的一种有效技术是将其表示为置换群;分类器（或多或少）实现了他们的目标，通过将有限置换群理论的大部分内容从公理研究转变为列表研究来回报赞美。密切的联系已与代数组合学，通过这样的概念，如轨道图，相干配置和协会计划，而无限置换群理论也发现了富有成效的联系与模型理论和拓扑结构。计算的发展已经把有限群理论和代数组合学变成了实验科学，GAP、MAGMA、GRAPE和COCO等系统允许研究置换群和组合结构，这些结构以前无法达到的大小和复杂性。尽管所有这些令人兴奋的发展，学生的置换群，直到最近举行了回来，缺乏一个全面的和最新的教科书：Wielandt的经典有限置换群[6]简明地总结了直到1960年的位置，卡梅隆，Passman，Tamaschke，Tsuzuku和Wielandt的书籍阐明了这个主题的特定领域，卡梅隆[1]和其他人的调查文章给出了一个广泛的概述，但直到1996年狄克逊和莫蒂默[2]和Suprunenko [5]的书出版，才有了全面的治疗方法，考虑到上面列出的发展。卡梅隆的最新著作是基于一个为期一周的课程，在欧拉研究所离散数学及其应用，与五个七章（一般理论，表示理论，奥南斯科特定理，oligomorphic群，和missieea）每个对应一天的工作;一章连贯的配置和一章的表格后增加。正如作者所承认的，“主题的选择有点特殊;这不是对主题的完整处理”，这本书最好在访问狄克逊和莫蒂默的情况下阅读，卡梅隆将证明一些更详细结果的任务委托给他们。某些主题，如弗罗贝纽斯群，是明智地避免，因为彻底的治疗可以在其他地方找到。尽管如此，卡梅隆的精湛的简洁风格使他能够涵盖大量的地面，往往只是简单地勾勒出一个证明或说明它与一个精心挑选的例子，并鼓励读者获得经验，解决一个
The early development of group theory was, to a large extent, that of permutation groups—specifically Galois groups, permuting the roots of polynomials. By the end of the nineteenth century, however, the axiomatic approach to group theory was beginning to take over, and despite the discoveries of Jordan, Burnside, Frobenius, Schur and Wielandt on permutation groups, the next 60–70 years saw group theory increasingly concerned with abstract groups: Marshall Hall’s book [3] devotes just one chapter out of twenty to permutation groups, and Schenkman [4] almost completely ignores them, even when discussing Frobenius groups. Recent decades, however, have seen a rebirth of the subject. During the classification of finite simple groups, an effective technique for constructing or eliminating a conjectured simple group was to represent it as a permutation group; the classifiers, having (more or less) achieved their goal, have repaid the compliment by turning much of finite permutation group theory from a study of axioms into a study of lists. Close links have been developed with algebraic combinatorics, through such concepts as orbital graphs, coherent configurations and association schemes, while infinite permutation group theory has also found fruitful links with model theory and topology. Developments in computing have turned finite group theory and algebraic combinatorics into experimental sciences, with systems such as GAP, MAGMA, GRAPE and COCO allowing the investigation of permutation groups and combinatorial structures of previously inaccessible size and complexity. Despite all these exciting developments, students of permutation groups have until recently been held back by the lack of a comprehensive and up-to-date textbook: Wielandt’s classic Finite permutation groups [6] concisely summarises the position up to about 1960, books by Cameron, Passman, Tamaschke, Tsuzuku and Wielandt illuminate particular areas of the subject, and survey articles by Cameron [1] and others give a broad overview, but it was not until the publication in 1996 of the books by Dixon and Mortimer [2] and by Suprunenko [5] that comprehensive treatments were available, taking account of the developments listed above. Cameron’s latest book is based on a one-week course given at the Euler Institute for Discrete Mathematics and its Applications, with five of the seven chapters (on general theory, representation theory, the O’Nan–Scott Theorem, oligomorphic groups, and miscellanea) each corresponding to a day’s work; a chapter on coherent configurations and a chapter of tables were added later. As the author admits, ‘ the choice of topics is a bit idiosyncratic ; this is not a complete treatment of the subject ’, and the book is best read with access to Dixon and Mortimer, to whom Cameron delegates the task of proving some of the more detailed results. Certain topics, such as Frobenius groups, are sensibly avoided since thorough treatments can be found elsewhere. Nevertheless, Cameron’s masterly expository style allows him to cover an enormous amount of ground, often simply sketching a proof or illustrating it with a well-chosen example, and encouraging the reader to gain experience by tackling a