Characterization of the unit group of Z[G x Cn 2]

Let G be a group and C-2 be the cyclic group of order 2. We characterize the unit group U(DOUBLE-STRUCK CAPITAL Z[G x C-2(n)]) of the integral group ring DOUBLE-STRUCK CAPITAL Z[G x C-2(n)] in terms of the unit group U(DOUBLE-STRUCK CAPITAL ZG), for every positive integer n. As a consequence, for each n = 2, 3, horizontal ellipsis , we determine conditions on some class of matrices A'(n)s, to be in GL(2(n-1), ZG). Using this we calculate the integer solutions of some specific class of symmetric polynomial f(n) with integer coefficients.