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Which subgroups of the symmetric group S_n arise as invariance groups ofn-variable functions defined on a k-element domain? It appears that the higherthe difference n-k, the more difficult it is to answer this question. For k>=n,the answer is easy: all subgroups of S_n are invariance groups. We give acomplete answer in the cases k=n-1 and k=n-2, and we also give a partial answerin the general case: we describe invariance groups when n is much larger thann-k. The proof utilizes Galois connections and the corresponding closureoperators on S_n, which turn out to provide a generalization of orbitequivalence of permutation groups. We also present some computational results,which show that all primitive groups except for the alternating groups arise asinvariance groups of functions defined on a three-element domain.

Invariance groups of finite functions and orbit equivalence of permutation groups