  In this paper we show that a normal affine toric variety X different from the
algebraic torus is uniquely determined by its automorphism group in the
category of affine irreducible, not necessarily normal, algebraic varieties if
and only if X is isomorphic to the product of the affine line and another
affine toric variety. In the case where X is the algebraic torus T, we reach
the same conclusion if we restrict the category to only include irreducible
varieties of dimension at most the dimension of T. There are examples of
varieties of dimension one higher than T having the same automorphism group of
T. Hence, this last result is optimal.
