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Group Ring Groups Volume 1 Orders and Generic Constructions of Units 1st Edition by Eric Jespers, Ángel del Río ISBN 3110386178 9783110386172

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Group Ring Groups Volume 1 Orders and Generic Constructions of Units 1st Edition Eric Jespers Digital Instant Download

Author(s): Eric Jespers, Ángel del Río
ISBN(s): 9783110386172, 3110386178
Edition: 1
File Details: PDF, 2.83 MB
Year: 2016
Language: english
SKU: EB-50338844 Category: Tags: ,

Group Ring Groups Volume 1 Orders and Generic Constructions of Units 1st Edition by Eric Jespers, Ángel del Río – Ebook PDF Instant Download/Delivery:3110386178, 9783110386172
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Product details: 

ISBN 10: 3110386178 

ISBN 13: 9783110386172

Author: Eric Jespers, Ángel del Río

This two-volume graduate textbook gives a comprehensive, state-of-the-art account of describing large subgroups of the unit group of the integral group ring of a finite group and, more generally, of the unit group of an order in a finite dimensional semisimple rational algebra. Since the book is addressed to graduate students as well as young researchers, all required background on these diverse areas, both old and new, is included. Supporting problems illustrate the results and complete some of the proofs. Volume 1 contains all the details on describing generic constructions of units and the subgroup they generate. Volume 2 mainly is about structure theorems and geometric methods. Without being encyclopaedic, all main results and techniques used to achieve these results are included. Basic courses in group theory, ring theory and field theory are assumed as background.

Table of contents: 

1 Units in group rings: an introduction
1.1 Constructions of units: elementary matrices and bicyclic units
1.2 Construction of units: cyclotomic units and Bass units
1.3 Examples: unit groups of some orders in number fields
1.4 Examples: unit groups of some non-commutative orders
1.5 Examples: group rings of groups of small order
1.6 Finite rings
2 Representations of algebras
2.1 Semisimple algebras
2.2 Splitting fields
2.3 Characteristic polynomial, trace and norm
2.4 Brauer group
2.5 Cohomology
2.6 Crossed products
3 Wedderburn decomposition of semisimple group algebras
3.1 Representations and characters of finite groups
3.2 Some operations with characters
3.3 Wedderburn components from character tables
3.4 Wedderburn components from monomial characters
3.5 Strongly monomial characters
3.6 Induction theorems
3.7 Brauer-Witt Theorem
3.8 Examples
4 Dedekind domains, valuations and orders
4.1 Localization and algebraic integers
4.2 Dedekind domains
4.3 Finitely generated modules over Dedekind domains
4.4 Extensions of Dedekind domains
4.5 Valuations
4.6 Orders
4.7 The discriminant
4.8 Brauer group of a number field
5 Thegroupofunitsofanorder
5.1 Lattices in real vector spaces
5.2 Hey’s Theorem and Dirichlet’s Unit Theorem
5.3 The group of units of an order is finitely generated
5.4 The group of units of an order is finitely presented
5.5 Subgroups of finite index
6 Cyclotomic integers
6.1 Cyclotomic fields
6.2 Cyclotomic units
7 Central units
7.1 Thegroupofcentralunitsofanorder
7.2 Large subgroups of central units: an algorithm
7.3 Bass units as generators of large groups of units
8 Generic units
8.1 Shifted cyclotomic polynomials
8.2 The group of generic units
8.3 A logarithm function
8.4 A basis of generic units for a subgroup of finite index in
8.5 Polynomials of small degree defining units
9 K-theory
9.1 Grothendieck group
9.2 The Whitehead group
9.3 Stable range condition
9.4 Whitehead group and the stable range condition
9.5 Applications of K-theory to units
10 General linear groups of degree 2
10.1 Number theoretical results
10.2 Normality of in
10.3 The factor group by
10.4 The group E2(I) is of finite index in SL2(R)
11 Generators of unit groups of group rings
11.1 Bass Unit Theorem
11.2 Generalized bicyclic units and Bass units I
11.3 Bicyclic units and Bass units
11.4 Fixed point free groups and Frobenius complements
11.5 Group rings of nilpotent groups
12 Exceptional simple components
12.1 Components of index one
12.2 Components of index two
12.3 Generalized bicyclic units and Bass units II
12.4 Normal closure of the trivial units
12.5 Normal complements
12.6 Examples: metacyclic groups
12.7 Examples with insufficient Bass units and bicyclic units
13 Idempotents and central units in group rings
13.1 Central subgroups and abelian-by-supersolvable groups
13.2 Independent units and abelian-by-supersolvable groups
13.3 Central subgroups and strongly monomial groups
13.4 Independent units and strongly monomial groups
13.5 Primitive idempotents and nilpotent groups
13.6 Primitive idempotents and strongly monomial groups
13.7 Some metacyclic groups 420

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Eric Jespers, Ángel del Río,Groups,Generic