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A course in group theory. (English) Zbl 0843.20001

Oxford: Oxford Univ. Press. xii, 279 p., £35.00/hbk (1996).
Aimed at beginners, this text can be used to taylor a year-long course for undergraduates. The focus is on finite groups and the techniques needed to obtain a complete list of the groups of order less than 32. The arguments are clear and full proofs are given. There could have been more exercises included in each chapter, but since there are a number of problem books on the market this is not a serious inconvenience.
The author has contributed to establishing a new breed of text books for beginners, texts which contain a description of the infinite family of simple groups \(\text{PSL}(2,n)\). He even makes the next logical step forward and includes a discussion of the first Mathieu group \(M_{11}\) and the related Mathieu groups, a nice opportunity to illustrate occurrences of multiple transitive groups. In this way, the door is opened for an overview of the classification of the finite simple groups, a feature unheard of in textbooks at this level some 10-15 years ago.
The whole text is actually built up around the idea of classification theorems. The inherent limitations (lack of space and time) of such an approach put aside, such glimpses of a distant horizon can do a lot towards stimulating the students to find more about the subject for themselves.

MSC:

20-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to group theory
20D05 Finite simple groups and their classification
94B05 Linear codes (general theory)
20Dxx Abstract finite groups
20G40 Linear algebraic groups over finite fields
20D20 Sylow subgroups, Sylow properties, \(\pi\)-groups, \(\pi\)-structure
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