Min-wise independent families with respect to any linear order

A set of permutations S on a finite linearly ordered set Ω is said to be k-min-wise independent, k-MWI for short, if Pr(minπ(X) = π(x)) = 1/|X | for every X ⊆Ω such that |X | ≤ k and for every x ∈ X . (Here π(x) and π(X) denote the image of the element x or subset X of Ω under the permutation π , and Pr refers to a probability distribution on S , which we take to be the uniform distribution.) We are concerned with sets of permutations which are k-MWI families for any linear order. Indeed, we characterize such families in a way that does not involve the underlying order. As an application of this result, and using the Classification of Finite Simple Groups, we deduce a complete classification of the k-MWI families that are groups, for k ≥ 3.