Atoms and antiatoms in the lattice of group topologies (Q1822640)

It is known that the intersection of two topological group topologies for a group G need not be a topological group topology for G. Thus the lattice L(G) of topological group topologies for G need not be a sublattice of the lattice of topologies on the set G. Generalizing known results about atoms in L(G), the authors here show that if the center of G is nontrivial and is not isomorphic to a group of the form \({\mathbb{Z}}_ p={\mathbb{Z}}/p{\mathbb{Z}}\) (p prime), or if a copy of \({\mathbb{Z}}\) is normal in G, then L(G) contains no Hausdorff atoms.