C-Groups

In this paper we explore the structure and properties of C-groups. We define a C-group as a group $G$ with $rk(G) < rk(Z(G))$ (where $rk(G)$ is the minimal cardinal of a generating set for a group $G$). Using GAP (a group theory program) and traditional methods, we identified an interesting infinite class of C-groups. In particular, we have proved that there is always a C-group of order an integer multiple of a fifth power of a prime. One way to obtain C-groups is take the direct product of certain C-groups, mentioned in this paper, with other appropriate groups. We, at this time, do not know whether the C-groups discussed in this paper are the building block of all finite C-groups. But a complete classification of finite or infinite C-groups is an interesting problem. We have also formulated a number of open questions relating to C-groups: Are they all solvable? What is the structure of the C-groups that are not in our class? Is the minimal number of generators of the center always polynomially bounded by the minimal number of generators of the group? What are the isoperimetric inequalities of infinite C-groups?