On finite p-groups with few normal subgroups

A p-group G is called a CCts-group if |G'/(G' boolean AND N)| <= p(s) or |N/(N boolean AND Z(G))| <= p(t) for every normal subgroup N in G. In this paper, we first investigate the properties of CCts-groups, in particular, we give the upper bounds of the exponent of G' with G' <= Z(G) and the order of G'Z(G)/Z(G) for a CCts-group G. Then, we try to describe the structure of CC11-groups, and a necessary condition for a p-group to be a CC11-group is given. We also give some elementary properties of p- groups with very small derived subgroups by using the properties of capable p-groups, which maybe could apply some other research of p-groups.