On the Unitriangular Groups over Rational Numbers Field

Let U(n, DOUBLE-STRUCK CAPITAL Q) be the group of all n x n (upper) unitriangular matrices over rational numbers field DOUBLE-STRUCK CAPITAL Q. Let S be a subset of U(n, DOUBLE-STRUCK CAPITAL Q). In this paper, we prove that S is a subgroup of U(n, DOUBLE-STRUCK CAPITAL Q) if and only if the (i, j)-th entry S-ij satisfies some condition (see Theorem 3.5). Furthermore, we compute the upper central series and the lower central series for S, and obtain the condition that the upper central series and the lower central series of S coincide.