Group Identities on Units and Symmetric Units of Group Rings (Algebra and Applications, Vol. 12) (Algebra and Applications, 12)

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. Group Identities On Units Of Group Rings -- Group Identities On Symmetric Units -- Lie Identities On Symmetric Elements -- Nilpotence Of And. Gregory T. Lee. Includes Bibliographical References (p. 187-190) And Index. "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."--pub. desc. Front Matter....Pages i-xii Group Identities on Units of Group Rings....Pages 1-43 Group Identities on Symmetric Units....Pages 45-75 Lie Identities on Symmetric Elements....Pages 77-101 Nilpotence of $$\mathcal{U}(FG)$$ and $${\mathcal{U}}^{+}(FG)$$ ....Pages 103-135 The Bounded Engel Property....Pages 137-147 Solvability of $$\mathcal{U}(FG)$$ and $${\mathcal{U}}^{+}(FG)$$ ....Pages 149-159 Further Reading....Pages 161-169 Back Matter....Pages 171-194 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-- Source other than Library of Congress