Finitely Generated Nilpotent Groups With Infinite Cyclic Derived Subgroup

In this dissertation,we,study the following three classe,s of questions:the first class of problems is concerned with the structure of a central extension of an infinite cyclic group by a finitely generated abelian group;the second is on an isomorphic invariant of the finitely generated nilpotent group with infinite cyclic derived subgroup;the third is on the automorphism group of an extraspecial Z-group.In fact,the central results we obtained can be considered as natural extensions to the fundamental theorem for finitely generated abelian groups.This dissertation is divided into fifth chapters.In the first chapter we introduce the background and the contents of this dissertation.In the second chapter we concern the structure of a central extension of an infinite cyclic group by a finitely generated abelian group,and we have found an isomorphic invariant of this groups.Suppose that G is a central extension of an infinite cyclic group by a finitely generated abelian group,and T is the torsion subgroup of G.If |T| is prime to the order of the torsion subgroup of ?G/(G'? T),then G has a decomposition G = S×F×T,where and di are positive integers satisfying d1 | d2 |…| dr,and F is a free abelian group with rank s,T is a finite abelian group such that T=Ze1? Ze2?…?Zet with e1>1,e1|e2|…et,and(d1,et)=1.Moreover,(d1,d2,…,dr;s;e1,e2,…,et,)is an isomorphic invariant of G,that is to say,if H is also a central extension of an infinite cyclic group by a finitely generated abelian group and the order of the torsion subgroup TH of H is prime to that of the torsion subgroup of ?H/(H' ? TH),then G is isomorphic to H if and only if they have the same invariants.In the third chapter we study the isomorphic invariant of the finitely generated nilpo-tent group with infinite cyclic derived subgroup.Suppose that G is a finitely generated nilpotent group such that the derived subgroup G' is infinite cyclic.Then G has a unique decomposition G = SG·?G,where the center ?G of G is a finitely generated abelian group,where di are positive integers satisfying d1| d2 |…|dr,and ?SG = S'G= FratSG = G' =(t1 r+3(1))??G.If H is also a finitely generated nilpotent group with infinite cyclic derived subgroup,then G is isomorphic to H if and only if there are two isomorphisms a:SG? SH,?:?G??H,and ?|G'=?|G'.Then we study the isomorphic invariant of the finitely generated nilpotent groups with infinite cyclic Frattini subgroups.Let G be a finitely generated nilpotent group.Then the Frattini subgroup of G is infinite cyclic if and only if G has a decomposition G = S× F×T,where F is a free abelian group of rank s,T = Zm1,?Zm2 ?…?Zmu,m1,m2,…,mu are square free integers greater than 1,m1 | m2|…|mu,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 nilpo-tent group with infinite cyclic Frattini subgroup,then G is isomorphic to H if and only if they have the same invariants.In the fourth chapter we research the automorphism group of an extraspecial Z-group.Let G be an extraspecial Z-group,where let Autc G be the normal subgroup of AutG consisting of all elements of AultG which act trivially on ?G,then AutG = AutcG × Z2,and there is an exact sequence...