Let be a finite group. If is a positive integer and is a normal subgroup of , then a function from the Cartesian power to is said to be compatible with if and for implies . A function that is compatible with all normal subgroups of is said to be compatible. A group is called -affine complete if every compatible function is a polynomial function as defined by W. Nöbauer and the reviewer [Algebra of polynomials. North-Holland Mathematical Library. Vol. 5. (1973; Zbl 0283.12101)], and is called affine complete if is -affine complete for every positive integer .
Let be a finite Abelian group. Then is defined as the semidirect product of with the group of order 2 where the conjugation action of its non-identity element on is inversion. Theorem 1.1: is 1-affine complete if and only is 1-affine complete. Theorem 1.2: If is the Sylow 2-subgroup of and its complement, then the following conditions are equivalent: (i) is affine complete, (ii) is 2-affine complete, (iii) is elementary-Abelian and is affine complete.