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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 semi-simple 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 encyclopedic, 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.
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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.
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A broad range of topics is covered here, including commutative monoid rings, the Jacobson radical of semigroup rings, blocks of modular group algebras, nilpotency index of the radical of group algebras, the isomorphism problem for group rings, inverse semigroup algebras and the Picard group of an abelian group ring. The survey lectures provide an up-to-date account of the current state of the subject and form a comprehensive introduction for intending researchers.
Group rings --- Semigroup rings --- Congresses
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Commutative rings --- Group rings --- Unit groups (Ring theory)
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Group theory --- Ordered algebraic structures --- 512.552.7 --- Semigroup and group rings --- Group rings. --- Ideals (Algebra) --- Ideals (Algebra). --- 512.552.7 Semigroup and group rings --- Algèbres associatives --- Algèbres associatives --- Représentations de groupes
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Ordered algebraic structures --- 512.54 --- Group rings --- #WWIS:ALTO --- Group theory --- Rings (Algebra) --- Groups. Group theory --- Group rings. --- 512.54 Groups. Group theory --- Groupes, Théorie des
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