Units OF Z(G × C ₂)

Abstract

In this paper, we are mainly interested in describing constructively u(Z(G × C ₂)), where u(ZG) has been described in some way. We first consider unitary units and prove that if u(ZG) is generated by unitary units, then U(Z(G × C₂)) is also generated by unitary units. Then we consider bicyclic units and ask the following question: If G has a normal complement generated by bicyclic units, does G × C ₂ also have a normal complement generated by bicyclic units? We give a negative answer to the above in general, showing that none of the normal complements of D ₈ × C ₂ × C ₂ is generated by bicyclic units, although a normal complement of D ₈ × C ₂ is indeed generated by bicyclic units.