We say that a group \(G\) is a CIP group if \(H^G\cap K^G=(H\cap K)^G\) for all subgroups \(H\) and \(K\) in \(G\). Clearly, every Dedekind group is a CIP group. Whether the converse holds in general is still an open question. In this paper it is shown that the converse is true when the group is either finitely generated or a torsion group. More generally the following is proved: if \(G\) is a non-Dedekind CIP group, then \(G\) has a torsion-free simple factor with the property that every two (nontrivial) subgroups intersect nontrivially.