The Radon Transform on Abelian Groups

The Randon transform on a group A is a linear operator on the space of functions f : A -> C. It is shown that if A = ZZnp then the Radon transform with respect to a subset B C A is not invertible if and only if B has the same number of elements in every coset of some maximal subgroup of A. The same does not hold in general for arbitrary finite abelian groups.