Polynomial Clones on Squarefree Groups

We prove that, on a set of size n, the number of clones that contain a group operation and all constant functions is finite if n is squarefree. This confirms a conjecture by Pawe l Idziak from [5] where the converse implication was shown. Our result follows from the observation that the polynomial clone of an expansion of a squarefree group is uniquely determined by its binary functions. We also note that, in general, such a clone is not determined by the congruence lattice and the commutator operation of the corresponding algebra. This refutes a second conjecture from [5].