A finite non-abelian group G is called an \({\cal{M}}{\cal{C}}\) -group if all non-abelian subgroups H of G have minimum centralizers (i.e., CG(H) = Z(G)). In this paper, the authors give some characterizations of \({\cal{M}}{\cal{C}}\) -groups, and it is proved that \({\cal{M}}{\cal{C}}\) -groups are just the finite groups with modular centralizer lattice of length 2 depicted by Schmidt, which leads to a classification of \({\cal{M}}{\cal{C}}\) -groups. However, Schmidt’s depiction said nothing for \({\cal{M}}{\cal{C}}\) -p-groups. They give a characterization of \({\cal{M}}{\cal{C}}\) -p-groups. In particular, they characterize special \({\cal{M}}{\cal{C}}\) -p-groups by means of the commutator matrices, and provide a method to determine or classify special \({\cal{M}}{\cal{C}}\) -p-groups. As applications, some examples are given, and special \({\cal{M}}{\cal{C}}\) -p-groups with an abelian maximal subgroup are classified up to isoclinism.
