A binary mean operation m(x; y) is said to be compatible with a semigroup law , if  satises the Gauss' functional equation m(x; y) m(x; y) = x  y for all x; y. Thus the arithmetic mean is compatible with the group addition in the set of real numbers, while the geometric mean is compatible with the group multiplication in the set of all positive real numbers. Using one of the Jacobi theta functions, Tanimoto [6], [7] has constructed a novel binary operation  compatible with the arithmetico-geometric mean agm(x; y) of Gauss. Tanimoto shows that it is only a loop operation, but not associative. A natural question is to ask if there exists a group law  compatible with arithmetic-geometric mean. In this paper we prove that there is no semigroup law compatible with agm and hence, in particular, no group law either. Among other things, this explains why Tanimoto's operation  using theta functions must be non-associative.