We consider a compact Lie group with bi-invariant metric, coming from the Killing form. In this paper, we study Einstein warped product space, M = M-1 x(f1) M-2 for the cases, (i) M-1 is a Lie group (ii) M-2 is a Lie group and (iii) both M-1 and M-2 are Lie groups. Moreover, we obtain the conditions for an Einstein warped product of Lie groups to become a simple product manifold. Then, we characterize the warping function for generalized Robertson-Walker spacetime, (M = I x(f1) G2,-dt(2) + f(1)(2)g(2)) whose fiber G2, being semi-simple compact Lie group of dimG(2) > 2, having bi-invariant metric, coming from the Killing form.