Group Identities on Units and Symmetric Units of Group Rings [electronic resource] / by Gregory T. Lee.
Contributor(s): SpringerLink (Online service)Material type: TextSeries: Algebra and Applications: 12Publisher: London : Springer London, 2010Description: XII, 196 p. online resourceContent type: text Media type: computer Carrier type: online resourceISBN: 9781849965040Subject(s): Mathematics | Associative rings | Rings (Algebra) | Group theory | Mathematics | Associative Rings and Algebras | Group Theory and GeneralizationsAdditional physical formats: Printed edition:: No titleDDC classification: 512.46 LOC classification: QA251.5Online resources: Click here to access online
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Group Identities on Units of Group Rings -- Group Identities on Symmetric Units -- Lie Identities on Symmetric Elements -- Nilpotence of and.
Let FG be the group ring of a group G over a field F. Write U(FG) for the group of units of FG. It is an important problem to determine the conditions under which U(FG) satisfies a group identity. In the mid 1990s, a conjecture of Hartley was verified, namely, if U(FG) satisfies a group identity, and G is torsion, then FG satisfies a polynomial identity. Necessary and sufficient conditions for U(FG) to satisfy a group identity soon followed. Since the late 1990s, many papers have been devoted to the study of the symmetric units; that is, those units u satisfying u* = u, where * is the involution on FG defined by sending each element of G to its inverse. The conditions under which these symmetric units satisfy a group identity have now been determined. This book presents these results for arbitrary group identities, as well as the conditions under which the unit group or the set of symmetric units satisfies several particular group identities of interest.