LAUSR

Abstract In this paper, we consider a finite-dimensional vector space ????{{\mathcal{P}}} over the Galois field GF⁡(p){\operatorname{GF}(p)}, with p being an odd prime, and the family ℬkx{{\mathcal{B}}_{k}^{x}} of all k-sets of elements of ????{\mathcal{P}} summing up to a given element x. The main result of the paper is the characterization, for x=0{x=0}, of the permutations of ????{\mathcal{P}} inducing permutations of ℬk0{{\mathcal{B}}_{k}^{0}} as the invertible linear mappings of the vector space ????{\mathcal{P}} if p does not divide k, and as the invertible affinities of the affine space ????{\mathcal{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.