Given a finite abelian group G, consider a uniformly random permutation of the set of all elements of G. Compute the difference of each pair of consecutive elements along the permutation. What is the number of occurrences of \(h\in G\setminus \{0\}\) in this sequence of differences? How do these numbers of occurrences behave for several group elements simultaneously? Can we get similar results for non-abelian G? How do the answers change if differences are replaced by sums? In this paper, we answer these questions. Moreover, we formulate analogous results in a general combinatorial setting.

中文翻译：

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关于有限群的随机排列

给定一个有限阿贝尔群G ^，考虑组的所有元件的均匀随机排列ģ。计算沿着排列的每对连续元素的差。在此差异序列中\（h \ in G \ setminus \ {0 \} \）出现的次数是多少？这些出现次数如何同时作用于多个组元素？对于非阿贝尔G我们可以获得类似的结果吗？如果用和代替差异，答案将如何改变？在本文中，我们回答了这些问题。此外，我们在一般的组合环境中制定类似的结果。