On The Unitriangular Groups Of Order 5 Over The Rational Field

Nilpotent group is a basic research object in algebra.Let R be a commutative ring with identity,and let U(n,R)be the group of all triangular matrices over R.it is a nilpotent group with Nilpotent class equal to n-1,which is the most basic group example of nilpotent groups.It is well known that the upper and lower central columns of U(n,R)are coincident,but the upper and lower central columns of subgroups of U(n,R)can be quite different.Let(?)be a subset of U(n,Q),where Gij be a subgroup of rational add group Q+.In general,G is not a subgroup of U(n,Q).In particular,when some Gij is a trivial sub-group of(Q,+),it is not easy to determine whether G is a group or not.In this paper,a sufficient and necessary condition is given for G group when n=5 and at least one Gij is a trivial sub-group.The upper and lower central columns of G are calculated for G group,and the necessary and sufficient condition for the upper and lower central columns of G to be coincidence is ob-tained.Finally,under the condition that G is a subgroup of U(n,Q),we discuss the number of possible structures of G.