Some results on FC-hypercentral units of integral group rings

We investigate the upper FC-central series of the unit group of an integral group ring ZG of a periodic group G. We prove that U = U-1(ZG) has FC-central height one if and only if the FC-hypercenter of U-1(ZG) is contained in the normalizer of the trivial units. Further, in these conditions, the FC-hypercenter of the unit group is non-central if and only if G is a Q*-group. Let H (sic) U, H contained in the normalizer of the trivial units, suppose that either the elements of finite order form a subgroup or H is a polycyclic-by-finite (polycyclic) subgroup, then H is contained in the finite conjugacy center of U-1(ZG).