Frattini子群是无限循环群的有限生成幂零群

Finitely Generated Nilpotent Groups with Infinite Cyclic Frattini Subgroups

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=Z_m_1⊕Zm_2⊕…⊕Z_m_u,m_1,m_2,…,m_u都是大于1的没有平方因子的自然数,m_1|m_2|…|m_u,■式中d_1,d_2,…,d_r都是正整数,d_1|d_2|…|d_r.进一步,(d_1,d2,…,d_r;s;m_1…,m_2,…,m_u)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.

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.