In this paper, we consider a finite-dimensional vector space 𝒫{{\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 𝒫{\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 𝒫{\mathcal{P}} inducing permutations of ℬk0{{\mathcal{B}}_{k}^{0}} as the invertible linear mappings of the vector space 𝒫{\mathcal{P}} if p does not divide k , and as the invertible affinities of the affine space 𝒫{\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.

中文翻译：

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有限向量空间中零和集的置换

在本文中，我们考虑了Galois场GF⁡（p）{\ operatorname {GF}（p）}上的有限维向量空间𝒫{{\\ mathcal {P}}}，其中p是奇质数，并且𝒫{\ mathcal {P}}的所有k个元素集的家庭ℬkx{{\ mathcal {B}} _ {k} ^ {x}}中，总和为给定元素x。本文的主要结果是针对x = 0 {x = 0}表征𝒫{\ mathcal {P}}的排列，从而诱导ℬk0{{\ mathcal {B}} _ {k} ^ { 0}}是向量空间𝒫{\ mathcal {P}}的可逆线性映射（如果p不将k除），以及向量空间𝒫{\ mathcal {P}}的可逆亲和力（如果p将k除以k）。在要求k个集合的元素全为非零的情况下，也回答了相同的问题，实际上，这两种情况本质上是不可分割的。