| Assume p is a prime number and G is a finite p-group.c(G)(in brief,c)denotes the nilpotent class of G and Zi(G)the i term of upper central series of G.As we well know,p-groups of maximal class play an important role in p-groups.In this paper we introduce a new concept,that is,UCC(c)-groupsG is called a UCC(c)-group if Zi(G)/Zi-1(G)is cyclic for all 1 ≤i≤c-1.It is easy to see that Z(G)is cyclic if G is UCC(c)-group.Obviously,if G is a UCC(c)-group of order pc+1 then G is a p-groups of maximal class.So UCC(c)-groups are a wider class of p-groups than that of p-groups of maximal class.When c=2,G is UCC(c)-group if and only if the nilpotent class of G is 2 and Z(G)is cyclic.In term of UCC(c)-groups,UCC(2)-groups were classified by Brady、Bryce、Cossey and Leong.So we need only to consider the case of c>3.In this paper we prove some properties of UCC(c)-groups,where the most basic properties among them are as follows:1.G/Zc-i(G)is UCC(i)-groups,where 2<i<c.2.d(G/Zc-2(G))=2Based on the above properties and the method of central extension,UCC(3)-groups and UCC(4)-groups are classified for p=2. |
