Abstract: A Z-group is a group that can be embedded in a cartesian power of Z and a topological Z-group is one that can be embedded topologically and algebraically in a power of Z. If D and C denote the categories of Z-groups and topological Z-groups, resp., then Hom(-,Z): C → D is adjoint on the left to Hom(-,Z): D → C. According to a well-known theorem of Lambek & Rattray, the fixed categories for each composite are equivalent (dual in this case, since the functors are contravariant). We identify the fixed groups of C as the ones that are a kernel of a map between powers of Z and the fixed groups of D as those that have a pure embedding (meaning the cokernel is torsion-free) into a power of Z.