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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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"The authors have succeeded in giving an appealing treatment of units in group rings managing to convey the beauty as well as the hardship of the subject."János Kurdics in: Zenralblatt MATH 1338 "This is a book written by two of the most prominent experts in the eld of group rings, covering most recent developments and organized in a modular and easily understandable way. So, enjoy the ride!"Stanley Orlando Juriaans in: Mathematical Reviews University of Michigan, MR3618092 (05.2018)

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Eric Jespers, Vrije Universiteit Brussel, Belgium; µngel del Río Mateos, Universidad de Murcia, Espa¤a.

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