Finite Group Theory

There are many textbooks on group theory. This one is aimed at graduate students who know the basics (however, there is an appendix that covers the introductory topics) and who seek solid knowledge of classical techniques. The book is written with obvious care and the exposition is clear and covers many topics that are not very accessible or not very well explained elsewhere. For example, chapter 1 is essentially about Sylow theorems and consequences but it also covers the Chermak-Delgado measure. Chapter 2 is on subnormality (the Wielandt zipper lemma and the theorems of Baer, Zenkov and Lucchini). Chapter 3 presents the standard material on split extensions (e.g. Hall subgroups and Glaubermann's lemma). Chapter 4, called Commutators, contains, amongst other things, various consequences of the Hall-Witt identity, the theorem of Mann and Thompson's PxQ lemma.

Chapter 5 introduces the technique of transfer, digresses into infinite groups by proving theorems of Schur and Dietzmann, continues with the Burnside theorem on normal p-complements, introduces focal subgroups and finishes with the Frobenius theorem on normal p-complements. The nilpotency of Frobenius kernel is proved in chapter 6. In fact, a part of the proof relies upon the properties of the Thompson subgroup, to which chapter 7 is devoted. This chapter also presents a proof of Burnside's theorem that is independent of linear representation theory (the proof follows arguments of Goldschmidt, Matsuyama and Bender). The ensuing chapter contains standard material on permutation groups, followed by a discussion of the orbital graph and subdegrees, including the proofs of the theorems of Weiss and Manning. The penultimate chapter starts with a definition of the generalised Fitting subgroup and of components and continues with a discussion of their basic properties. It also proves Wielandt's results on automorphism towers, Schenkman’s theorem, Thompson's result on corefree maximal subgroups (in connection to the Sims conjecture) and some facts about strong conjugacy. The final chapter returns to the transfer (theorems of Yoshida, Huppert's result on metacyclic groups and connections with the group ring). Each chapter has several subsections (A, B, C, D and sometimes E, F and G) and each subsection finishes with a list of problems, many of which are far from routine.

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Published: 
2008
ISBN: 
978-0-8218-4344-4
Price: 
USD 59
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