## Lectures on Block TheoryBlock theory is a part of the theory of modular representation of finite groups and deals with the algebraic structure of blocks. In this volume Burkhard Klshammer starts with the classical structure theory of finite dimensional algebras, and leads up to Puigs main result on the structure of the so called nilpotent blocks, which he discusses in the final chapter. All the proofs in the text are given clearly and in full detail, and suggestions for further reading are also included. For researchers and graduate students interested in group theory or representation theory, this book will form an excellent self contained introduction to the theory of blocks. |

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### Contents

Foundations | 1 |

Idempotents | 5 |

Simple and Semisimple Algebras | 12 |

Points and Maximal Ideals | 18 |

Miscellaneous Results on Algebras | 25 |

Modules | 29 |

Groups Acting on Algebras | 35 |

Pointed Groups | 41 |

Group Algebras | 63 |

Blocks of Group Algebras | 68 |

Nilpotent Blocks | 77 |

The Source Algebra of a Nilpotent Block | 81 |

Puigs Theorem | 91 |

94 | |

97 | |

List of Symbols | 103 |

### Common terms and phrases

A-submodules abstract algebras algebra over F algebraically closed field called classes of G conjugacy classes defect groups defect pointed subgroup denote direct embedding embedding of interior epimorphism FCG(P FG with nilpotent finite group follows easily form a basis FP-module Frobenius formula G on FG group algebra Hence homomorphism of algebras homomorphism of interior iFGi implies interior G-algebra isomorphic to Mat(n isomorphism of algebras isomorphism of interior K C H kernel Krull-Schmidt theorem Let F let H let P7 local algebra Mat(n,F maximal ideal maximal local pointed modules Moreover nilpotent blocks nilpotent ideal non-zero idempotent orthogonal primitive idempotents p-group p-subgroup p'-section pairwise orthogonal primitive particular point of G pointed group positive integer prime characteristic projective left A-module Proof proper subgroup RCG(Q result is proved satisfying simple algebra subgroup H subgroup of Hp suffices to show symmetric algebra theory unique point unitary subalgebra

### Popular passages

Page 95 - On the structure of block ideals in group algebras of finite groups, Comm. Algebra 8 (1980), 1867-1872 18.

Page 95 - T. Okuyama and Y. Tsushima, Local properties of p-block algebras of finite groups, Osaka J. Math. 20 (1983), 33-41.

Page 94 - Group representation theory Part A," Marcel Dekker, New York 1971 11. L. DORNHOFF, "Group representation theory Part B," Marcel Dekker, New York 1972 12.

Page 94 - Methods of representation theory Vol. I," Wiley-Interscience, New York 1981 9. CW CURTIS and I. REINER, "Methods of representation theory Vol.

Page 94 - The Representation Theory of Finite Groups, North-Holland, Amsterdam, 1982. 13. W. Feit, M. Hall, and }. G. Thompson, "Finite groups in which the centrali2er of any non-identity element is nilpotent,

Page 95 - B. KULSHAMMER, Crossed products and blocks with normal defect groups, Comm. Algebra 13 (1985), 147-168.