Representation Theory of Finite Groups and Associative Algebras

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American Mathematical Soc., 1966 - Mathematics - 689 pages
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Contents

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
Bibliography
655
Index
673
Copyright

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Common terms and phrases

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.
Page 12 - Vaandrager shows, [15, 5.1], that two deterministic event structures are isomorphic if, and only if, they have the same set of step sequences.
Page 673 - L. Solomon, The representation of finite groups in algebraic number fields, J. Math. Soc. Japan, 13 (1961), 144-164.

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