The Study Of Some Classes Of Infinite Groups

Infinite soluble groups and infinite nilpotent groups are the basic research objects in infinite groups,and their research methods and results have important model significance.This paper studies the following three questions:(1)Residual finiteness is a basic finiteness condition,and it is an important object in infinite group theory.One of the important aspects of the research on the residual finiteness of infinite soluble groups is to prove that some special groups have residual finiteness and make it more accurate.In this paper,we start from a class of monic irreducible polynomials λn-mλn-1+1 over the ring of integers,we get an automorphism of the completely decomposable homogeneous torsion-free Abelian group of rank n through the adj oint matrix of the polynomials,and construct a class of infinite soluble groups by group extension.We study the residual finiteness of this class of groups,which is a precise complement to the results of Seksenbaev,Robinson et al.(2)We apply the research method of studying the automorphism group of generalized extra-special p-groups to a class of infinite (?)ernikov p-groups and study its automorphism group.Let G be an infinite (?)ernikov p-group,and G is not abelian but every proper quotient group of G is abelian.In this paper,we first determine the structure of this class of groups,then we give the structure of automorphism groups by group extension.Moreover,we calculate the detailed structure of automorphism group of a class of (?)ernikov 2-group.(3)The unitriangular matrix group U(n,Z)is a basic example in nilpotent groups,which is of fundamental importance.Many properties of such groups have been studied.In this paper,we start with a subset G of the unitriangular matrix group U(n,Q).First,we obtain the necessary and sufficient conditions for G to be a subgroup of U(n,Q).Then we calculate the upper and lower central series of G when G is a group,and give the necessary and sufficient conditions for the upper and lower central series to be coincide,which fully generalizes the results on U(n,Z).