On The Unitriangular Groups Of Order 4 Over The Rational Field

Nilpotent groups is a basic research object in algebra.We know that U(n,R)is a group of all unitriangular matrices over the commutative ring R with identity,and is a nilpotent group whose class is equal to n-1.The upper and lower central series of U(n,R)are coincidence,but the upper and lower central columns of subgroups of U(n,R)are quite different.We consider classification of subgroups on the unitriangular group over Q,let(?) be a set of U(n,Q),Where Gij is a subgroup of(Q,+),and 1 ?i<j ?n.In general,G is not a subgroup of U(n,Q).In particular,when some Gij is a trivial subgroup of(Q,+),it is not easy to determine whether G is a group or not.In this paper,When n=4 and at least one Gij is a trivial subgroup,we get a necessary and sufficient condition for G is a subgroup of U(n,Q).Then clculate the upper and lower central series of G,and then get the necessary and sufficient condition for the upper and lower central seies of G to be coincide.