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This book provides an accessible introduction to the state of the art of representation theory of finite groups. Starting from a basic level that is summarized at the start, the book proceeds to cover topics of current research interest, including open problems and conjectures.

The central themes of the book are block theory and module theory of group representations, which are comprehensively surveyed with a full bibliography. The individual chapters cover a range of topics within the subject, from blocks with cyclic defect groups to representations of symmetric groups.

Assuming only modest background knowledge at the level of a first graduate course in algebra, this guidebook, intended for students taking first steps in the field, will also provide a reference for more experienced researchers. Although no proofs are included, end-of-chapter exercises make it suitable for student seminars.

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1 The Basics.- 2 Blocks and Their Characters.- 3 Modules.- 4 The Local-Global Principle.- 5 Blocks with Cyclic Defect Groups.- 6 Blocks with Non-Cyclic Defect Groups.- 7 Clifford Theory.- 8 Representation of Symmetric Groups.- 9 Representations of Groups of Lie Type.- References.- Index.

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David A. Craven is a Royal Society Research Fellow and Senior Birmingham Fellow at the University of Birmingham. He has worked on a variety of areas within representations of groups, from the local-global conjectures to representations of symmetric groups, and groups of Lie type.

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