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BY-NC-ND 3.0 license Open Access Published by De Gruyter Open Access October 28, 2014

Invariance groups of finite functions and orbit equivalence of permutation groups

  • Eszter K. Horváth EMAIL logo , Géza Makay , Reinhard Pöschel and Tamás Waldhauser
From the journal Open Mathematics

Abstract

Which subgroups of the symmetric group Sn 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 Sn 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 Sn, 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.

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Received: 2013-7-25
Accepted: 2014-7-31
Published Online: 2014-10-28
Published in Print: 2015-1-1

© 2015 Eszter K. Horváth et al.

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License.

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