Permutations of zero-sumsets in a finite vector space

  • 1 Dipartimento di Matematica e Informatica, Università degli Studi di Palermo, Via Archirafi 34, 90123, Palermo, Italy
  • 2 Dipartimento di Ingegneria, Università degli Studi di Palermo, Viale delle Scienze, 90128, Palermo, Italy
Giovanni FalconeORCID iD: and Marco PavoneORCID iD:


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 kx 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 k0 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.

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