非交换商群的中心均循环的有限p群

Finite p-groups Whose Non-abelian Quotient Groups Have the Cyclic Center

称有限群G为QCC群,若对G中的任意非交换商群G/N,均有Z(G/N)循环,其中1≠N■G.本文对QCC p群进行了研究,证明了非交换QCC p群或为特殊群、或换位子群的阶为p、或生成元的个数为2.进一步,证明了二元生成且|G|≠3^(5)的非交换QCC 2群(QCC 3群)G为内交换群或极大类群.

A finite group G is called a QCC-group if whenever 1≠N■G with non-abelian G/N,then Z(G/N)is cyclic.In this paper,we study the QCC-p-groups.We prove that such p-groups G are special or|G′|=p or d(G)=2.Moreover,we prove 2-generator QCC-2-groups(QCC-3-groups)G with|G|≠3^(5)are inner abelian or of maximal class.