## Representation Theory of Finite Groups and Associative Algebras |

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### Contents

1 | |

3 | |

8 | |

10 | |

14 | |

17 | |

21 | |

Representations and Modules | 25 |

The Tetrahedral and Octahedral Groups | 329 |

Representations of Metacyclic Groups | 333 |

Multiplicityfree Representations | 340 |

The Restriction of Irreducible Modules to Normal Subgroups | 342 |

Imprimitive Modules | 346 |

Projective Representations | 348 |

Applications | 355 |

Schurs Theory of Projective Representations | 358 |

Linear Transformations | 26 |

Definitions and Examples of Representations | 30 |

Representations of Groups and Algebras | 38 |

Modules | 50 |

Tensor Products | 59 |

Composition Series | 76 |

Indecomposable Modules | 81 |

Completely Reducible Modules | 86 |

Algebraic Number Theory | 91 |

Algebraic Integers | 102 |

Ideals | 107 |

Valuations Padic Numbers | 115 |

Norms of Ideals Ideal Classes | 123 |

Cyclotomic Fields | 135 |

Modules over Dedekind Domains | 144 |

Semisimple Rings and Group Algebras | 157 |

The Radical of a Ring with Minimum Condition | 159 |

Semisimple Rings and Completely Reducible Modules | 163 |

The Structure of Simple Rings | 173 |

Theorems of Burnside Frobenius and Schur | 179 |

Irreducible Representations of the Symmetric Group | 190 |

Extension of the Ground Field | 198 |

Group Characters | 207 |

Orthogonality Relations | 217 |

Simple Applications of the Orthogonality Relations | 224 |

Central Idempotents | 233 |

Burnsides Criterion for Solvable Groups | 239 |

The FrobeniusWielandt theorem on the Existence of Normal Subgroups in a Group | 241 |

Theorems of Jordan Burnside and Schur on Linear Groups | 250 |

Units in a Group Ring | 262 |

Induced Characters | 265 |

Rational Characters | 279 |

Brauers Theorem on Induced Characters | 283 |

Applications | 292 |

The Generalized Induction Theorem | 301 |

Induced Representations | 313 |

Induced Representations and Modules | 314 |

The Tensor Product Theorem and the Intertwining Number Theorem | 323 |

Irreducibility and Equivalence of Induced Modules | 328 |

NonSemiSimple Rings | 367 |

The Classification of the Principal Indecomposable Modules into Blocks | 377 |

Projective Modules | 380 |

Injective Modules | 384 |

QuasiFrobenius Rings | 393 |

Modules over QuasiFrobenius Rings | 403 |

Frobenius Algebras | 409 |

Frobenius and QuasiFrobenius Algebras | 413 |

Projective and Injective Modules for a Frobenius Algebra | 420 |

Group Algebras of Finite Representation Type | 431 |

The Vertex and Source of an Indecomposable Module | 435 |

Centralizers of Modules over Symmetric Algebras | 440 |

Irreducible Tensor Representations of GLV | 449 |

Splitting Fields and Separable Algebras | 453 |

Separable Extensions of the Base Field | 459 |

The Schur Index | 463 |

Separable Algebras | 480 |

The WedderburnMalcev Theorem | 485 |

Integral Representations | 493 |

Introduction | 494 |

The Cyclic Group of Prime Order | 506 |

Modules over Orders | 515 |

PIntegral Equivalence | 531 |

Local Theory | 542 |

Global Theory | 550 |

The JordanZassenhaus Theorem | 558 |

Order Ideals | 563 |

Genus | 567 |

Modular Representations | 583 |

Introduction | 584 |

Cartan Invariants and Decomposition Numbers | 590 |

Orthogonality Relations | 598 |

Blocks | 604 |

The Defect of a Block | 611 |

Defect Groups | 618 |

Block Theory for Groups with Normal PSubgroups | 627 |

655 | |

673 | |

### Other editions - View all

Representation Theory of Finite Groups and Associative Algebras Charles W. Curtis,Irving Reiner No preview available - 1988 |

### Common terms and phrases

absolutely irreducible algebraic integers algebraic number field automorphism Brauer Chapter char characteristic roots completely reducible completes the proof component composition factors composition series conjugate classes contains Corollary cosets cyclic group decomposition defect group define Definition denote direct sum direct summand elements of G equivalent Exercise exists field for G finite group finite-dimensional follows Frobenius algebra full set g e G group algebra group G hence homomorphism i?-module idempotent implies indecomposable modules induced injective irreducible characters irreducible representations isomorphic left A-module left ideal Lemma Let G Let H linear transformation Math matrix representation minimal left ideal minimum condition multiplication nilpotent non-zero normal subgroup obtain permutation polynomial prime ideal projective proof of Theorem prove rational representation of G result ring with minimum satisfy splitting field subgroup of G submodule Suppose Sylow subgroup tensor product theory two-sided ideal vector space x e G

### Popular passages

Page 4 - G, the number of distinct right (left) cosets of H in G is called the index of H in G and is denoted by [G : H] or by ic (H).

Page 668 - K. MORITA. On group rings over a modular field which possess radicals expressible as principal ideals, Sci.