中文摘要：

完整地确定了Frattini子群是无限循环群的有限生成幂零群的结构，证明了下面的定理．设G是有限生成幂零群，则G的Frattini子群是无限循环群当且仅当G可以分解为G=S×F×T，其中F是秩为s的自由Abel群，T=Zm1¤Zm2¤…¤Zmu,m1,m2...,mu都是大于1的没有平方因子的自然数，m1｜m2｜…｜mu,S={[10…000/d1α121…000/d2α130...000.../drα1r＋10…010/α1r＋2α2r＋2…αrr＋2αr＋1r＋21]｜αij∈Z},式中d1，d2,...,dr都是正整数，d1｜d2｜…｜dr.进一步，（d1，d2,...，dr;s;m1,m2,...,mu）是群G的同构不变量，即若群H也是Frattini子群是无限循环群的有限生成幂零群，那么G同构于H的充要条件是它们有相同的不变量．

英文摘要：

The structure of the finitely generated nilpotent groups with infinite cyclic Frattini subgroups are completely determined. More exactly, the following theorem is proved. Let G be a finitely generated nilpotent group. Then the Prattini subgroup of G is infinite cyclic if and only if G has a decompositionG=S×F×T, where F is a flee abelian group of rank s, T=Zm1¤Zm2¤…¤Zmu,m1,m2...,mu are square free integers greater than 1,m1｜m2｜…｜mu,S={[10…000/d1α121…000/d2α130...000.../drα1r＋10…010/α1r＋2α2r＋2…αrr＋2αr＋1r＋21]｜αij∈Z} where d1,d2,...,dr are integers and d1｜d2｜…｜dr. Moreover, （d1,d2,...,dr;s;m1,m2,...,mu） is an isomorphic invariant of G. That is to say, if H is also a finitely generated nilpotent group with infinite cyclic Prattini subgroup, then G is isomorphic to H if and only if they have the same invariants.