Department of applied mathematics, Faculty of Electrical Engineering and Computing , University of Zagreb , 10 000 Zagreb, Croatia
Abstract. We describe some aspects of the structure of nonabelian p-groups G for which every nonabelian subgroup has a trivial centralizer in G, i.e. only it's center. We call such groups CZ-groups. The problem of describing the structure of all CZ-groups was posted as one of the first research problems in the open problems list in Yakov Berkovich's book 'Groups of prime power order' Vol 1 (). Among other features of such groups, we prove that a minimal CZ-group must contain at least p5 elements. The structure of maximal abelian subgroups of these groups is described as well.
2010 Mathematics Subject Classification. 20D15, 20D25.
Key words and phrases. p-group, center, centralizer, Frattini subgroup, minimal nonabelian subgroup.