Introduces the richness of group theory to advanced undergraduate and graduate students, concentrating on the finite aspects. Provides a wealth of exercises and problems to support self-study. Additional online resources on more challenging and more specialised topics can be used as extension material for courses, or for further independent study.

This book offers a systematic introduction to recent achievements and development in research on the structure of finite non-simple groups, the theory of classes of groups and their applications. In particular, the related systematic theories are considered and some new approaches and research methods are described – e.g., the F-hypercenter of groups, X-permutable subgroups, subgroup functors, generalized supplementary subgroups, quasi-F-group, and F-cohypercenter for Fitting classes. At the end of each chapter, we provide relevant supplementary information and introduce readers to selected open problems.

This book covers the latest achievements of the Theory of Classes of Finite Groups. It introduces some unpublished and fundamental advances in this Theory and provides a new insight into some classic facts in this area. By gathering the research of many authors scattered in hundreds of papers the book contributes to the understanding of the structure of finite groups by adapting and extending the successful techniques of the Theory of Finite Soluble Groups.

The theory of groups, especially of finite groups, is one of the most delightful areas of mathematics. Its proofs often have elegance and crystalline beauty. This textbook is intended for the reader who has been exposed to about three years of serious mathematics. The notion of a group appears widely in mathematics and even further afield in physics and chemistry, and the fundamental idea should be known to all mathematicians. In this textbook a purely algebraic approach is taken and the choice of material is based upon the notion of conjugacy. The aim is not only to cover basic material, but also to present group theory as a living, vibrant and growing discipline, by including references and discussion of some work up to the present day.

The theory of groups, especially of finite groups, is one of the most delightful areas of mathematics, its proofs often having great elegance and beauty. This textbook is intended for the reader who has been exposed to about three years of serious mathematics. The notion of a group appears widely in mathematics and even further afield in physics and chemistry, and the fundamental idea should be known to all mathematicians. In this textbook a purely algebraic approach is taken and the choice of material is based upon the notion of conjugacy. The aim is not only to cover basic material, but also to present group theory as a living, vibrant and growing discipline, by including references and discussion of some work up to the present day. Request Inspection Copy

"An excellent up-to-date introduction to the theory of groups. It is general yet comprehensive, covering various branches of group theory. The 15 chapters contain the following main topics: free groups and presentations, free products, decompositions, Abelian groups, finite permutation groups, representations of groups, finite and infinite soluble groups, group extensions, generalizations of nilpotent and soluble groups, finiteness properties." —-ACTA SCIENTIARUM MATHEMATICARUM

This book is concerned with the generalizations of Sylow theorems and the related topics of formations and the fitting of classes to locally finite groups. It also contains details of Sunkov's and Belyaev'ss results on locally finite groups with min-p for all primes p. This is the first time many of these topics have appeared in book form. The body of work here is fairly complete.

Part 1 (ISBN 978-3-7568-0801-4) of the Trilogy is based on the BoD-Book "Characterising locally finite groups satisfying the strong Sylow Theorem for the prime p - Revised edition" (see ISBN 978-3-7562-3416-5). The First edition of Part 1 (see ISBN 978-3-7543-6087-3) removes the highlights in light green of the Revised edition, adds 14 pages to the AGTA paper and 10 pages to the Revised edition. It includes Reference [11] resp. [10] as Appendix 1 resp. Appendix 2 and calls to mind Professor Otto H. Kegel's contribution to the conference Ischia Group Theory 2016. The Second edition introduces a uniform page numbering, adds page numbers to the appendices, improves 19 pages, adds Pages 109 to 112 and a Table of Contents. Part 2 (ISBN 978-3-7543-3642-8) of the Trilogy is based on the author's research paper "About the Strong Sylow Theorem for the Prime p in Simple Locally Finite Groups". We first give an overview of simple locally finite groups and reduce their Sylow theory for the prime p to a conjecture of Prof. Otto H. Kegel about the rank-unbounded ones of the 19 known families of finite simple groups. Part 2 introduces a new scheme to describe the 19 families, the family T of types, defines the rank of each type, and emphasises the rôle of Kegel covers. This part presents a unified picture of known results and is the reason why our title starts with "About". We then apply new ideas to prove the conjecture for the alternating groups (see Page ii). Thereupon we remember Kegel covers and *-sequences. Finally we suggest a plan how to prove the conjecture step-by-step which leads to further conjectures thereby unifying Sylow theory in locally finite simple groups with Sylow theory in locally finite and p-soluble groups. In Part 3 (ISBN 978-3-7578-6001-1) of the Trilogy we continue the program begun in [10] to optimise along the way 1) its Theorem about the first type "An" of infinite families of finite simple groups step-by-step to further types by proving it for the second type "A = PSLn". We start with proving the Conjecture 2 of [10] about the General Linear Groups by using new ideas (see Page ii), and then break down this insight to the Special Linear and the PSL Groups. We close with suggestions for future research regarding the remaining rank-unbounded types (the "Classical Groups") and the way 2), the (locally) finite and p-soluble groups, and Augustin-Louis Cauchy's and Évariste Galois' contributions to Sylow theory in finite groups.

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