Units in integral group rings

In this thesis, we study several related problems concerning the unit group {dollar}{lcub}cal U{rcub}(doubz G){dollar} of the integral group ring {dollar}doubz G{dollar} of a periodic group G. Although Chapter 1 involves a lengthy calculation, the most important results appear in Chapter 2 through Chapter 5.; Chapter 1 describes constructively {dollar}{lcub}cal U{rcub}(doubz(Gtimes Csb2)),{dollar} where {dollar}{lcub}cal U{rcub}(doubz G){dollar} has been described in some way. We are also interested in the following question: If G has a normal complement generated by bicyclic units, does {dollar}Gtimes Csb2{dollar} also have a normal complement generated by bicyclic units? We show that none of the normal complements of {dollar}Dsb8times Csb2times Csb2{dollar} is generated by bicyclic units by explicitly constructing a set of generators for a normal complement of {dollar}Dsb8times Csb2times Csb2,{dollar} although a normal complement of {dollar}Dsb8times Csb2{dollar} is indeed generated by bicyclic units.; In chapter 2, we first study the subgroup of all unitary units {dollar}{lcub}cal U{rcub}sb{lcub}f{rcub}(doubz G).{dollar} We prove that if G has a normal complement generated by unitary units, then it is also true for {dollar}Gtimes Csb2.{dollar} Then we investigate generalized unitary units and prove that all of these units form a subgroup {dollar}{lcub}cal U{rcub}sb{lcub}g,f{rcub}(doubz G){dollar} of the unit group. Furthermore, we show that this subgroup is exactly the normalizer of the subgroup of unitary units. One of our main results is that the normalizer of {dollar}{lcub}cal U{rcub}sb{lcub}g,f{rcub}(doubz G){dollar} is equal to itself when G is a periodic group. We also obtain some other interesting results on {dollar}{lcub}cal U{rcub}sb{lcub}g,f{rcub}(doubz G).{dollar}; Chapter 3 investigates central units of {dollar}doubz Asb5.{dollar} We show that the centre {dollar}{lcub}cal C{rcub}({lcub}cal U{rcub}(doubz Asb5))=pmlangle urangle,{dollar} where {dollar}langle urangle{dollar} is an infinite cyclic group and we explicitly find the generator u.; In chapter 4, we study the hypercentral units in the integral group ring of a periodic group G. We prove that the central height is at most 2. We also discuss the relationship between hypercentral units and generalized unitary units.; Chapter 5 characterizes the n-centre of the unit group of the integral group ring of a periodic group. It is proved that the n-centre is either the centre or the second centre of the unit group for all {dollar}nge2.{dollar}...