For a standard Artinian k-algebra A = R/I , we give equivalent conditions for A to have the weak (or strong) Lefschetz property or the strong Stanley property in terms of the minimal system of generators of gin(I ). Using the equivalent condition for the weak Lefschetz property, we show that some graded Betti numbers of gin(I ) are determined just by the Hilbert function of I if A has the weak Lefschetz property. Furthermore, for the case that A is a standard Artinian k-algebra of codimension 3, we show that every graded Betti number of gin(I ) is determined by the graded Betti numbers of I if A has the weak Lefschetz property. And if A has the strong Lefschetz (respectively Stanley) property, then we show that the minimal system of generators of gin(I ) is determined by the graded Betti numbers (respectively by the Hilbert function) of I .