Permutations of zero-sumsets in a finite vector space

In this paper, we consider a finite-dimensional vector space P over the Galois field GF(p), with p being an odd prime, and the family Bxk of all k-sets of elements of P summing up to a given element x. The main result of the paper is the characterization, for x=0, of the permutations of P inducing permutations of B0k as the invertible linear mappings of the vector space P if p does not divide k, and as the invertible affinities of the affine space P if p divides k. The same question is answered also in the case where the elements of the k-sets are required to be all nonzero, and, in fact, the two cases prove to be intrinsically inseparable.