On The Unitriangular Groups Of Order 5 Or 6 Over Q_π

Let Tr1(n,Qπ)be a group of all upper triangular matrices whose main diagonal elements are all 1 on Qπ.Where Qπ={n/m|(m,n)=1,n∈ Z,m is π-number}and Qπ is a subring of(Q,+,·).Here π is a set of some prime numbers.If all prime factors of the integer m belong to the set π,then m is said to be a π-number.Let kij(1≤i≤j≤n)is a given positive integers,kij can be written as the product of prime powers.i.e.kij=p1e1p2e2…pnen and each pi(?)πij,where πij is the set of prime numbers of the denominator at the position of the matrix(ij)andThis paper studies the subgroup structure of the unit upper triangular matrix group on Qπ when n-5 and n=6.It contains the conditions when G is a Tr1(5,Qπ)、Tr1(6,Qπ)subgroup;When G is grouped,the conditions of the upper and lower center series and the conditions under which their upper and lower center series overlap.In the first and second chapters,we give the basic concepts to be used.Secondly,a brief overview of the research background and main progress involved in this article are given.In the third and fourth chapters,we extend the structure of the subgroups of the triangular matrix groups Tr1(5,Qπ)、Tr1(6,Qπ)on the fifth and sixth order units on Qπ,and obtain the conditions when G is a subgroup and a special method is used to obtain the upper and lower center series of G when they are grouped,and discuss the conditions under which the upper and lower center series coincide.