Forum Mathematicum ( IF 0.733 ) Pub Date : 2020-12-12 , DOI: 10.1515/forum-2019-0228
Giovanni Falcone; Marco Pavone

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 $ℬkx$ 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 $ℬk0$ 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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