On random permutations of finite groups

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∈ G\ { 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.