Gorenstein tiled orders

Let A = {O, epsilon(Lambda)} be a reduced tiled Gorenstein order with Jacob son radical R and J a two-sided ideal of Lambda such that A superset of R-2 superset of J superset of R-n (n greater than or equal to 2). The quotient ring Lambda /J is quasi-Frobenius (QF) if and only if there exists p is an element of R-2 such that J = p Lambda = Lambdap. We prove that an adjacenc y matrix of a quiver of a cyclic Gorenstein tiled order is a multiple of a double stochastic matrix. A requirement for a Gorenstein tiled order to be a cyclic order cannot be omitted. It is proved that a Cayley table of a fin ite group G is an exponent matrix of a reduced Gorenstein tiled order if an d only if G = G(k) = (2) x (...) x (2).