Invariance groups of finite functions and orbit equivalence of permutation groups

TL;DR

This paper characterizes the subgroups of the symmetric group S_n that can serve as invariance groups of n-variable functions on a k-element domain, providing complete classifications for certain cases and partial results for others.

Contribution

It offers a complete classification of invariance groups for k=n-1 and k=n-2, and introduces a generalized orbit equivalence concept using Galois connections.

Findings

All subgroups of S_n are invariance groups for k>=n.

Complete classification for k=n-1 and k=n-2 cases.

Primitive groups except for alternating groups arise as invariance groups for functions on a three-element domain.

Abstract

Which subgroups of the symmetric group S_n arise as invariance groups of n-variable functions defined on a k-element domain? It appears that the higher the 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 a complete answer in the cases k=n-1 and k=n-2, and we also give a partial answer in the general case: we describe invariance groups when n is much larger than n-k. The proof utilizes Galois connections and the corresponding closure operators on S_n, which turn out to provide a generalization of orbit equivalence of permutation groups. We also present some computational results, which show that all primitive groups except for the alternating groups arise as invariance groups of functions defined on a three-element domain.