On Finite NDC-Groups

A finite p-group G is called a NDC-group if G /G n N is cyclic or N/ N n Z( G) is cyclic for every normal subgroup N in G. In this paper, firstwe give some properties of NDC-groups and in particular, we give the order of G Z(G)/Z(G) for a NDC-group G so that we could investigate the structure of NDC-groups. Next, some necessary and sufficient conditions for a p-group G whose quotient group G/Z(G) is generated by two elements to be a NDC-group are given. Finally, we also give some further properties of p-groups with small derived subgroups by using the properties of capable p-groups with small generating systems, which maybe could apply some other research of p-groups.