  For an automorphism group G on an n-dimensional (n > 2) normal projective
variety or a compact K\"ahler manifold X so that G modulo its subgroup N(G) of
null entropy elements is an abelian group of maximal rank n-1, we show that
N(G) is virtually contained in Aut_0(X), the X is a quotient of a complex torus
T and G is mostly descended from the symmetries on the torus T, provided that
both X and the pair (X, G) are minimal.
