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Open Mathematics

formerly Central European Journal of Mathematics

Editor-in-Chief: Vespri, Vincenzo / Marano, Salvatore Angelo

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Volume 13, Issue 1


Volume 13 (2015)

Invariance groups of finite functions and orbit equivalence of permutation groups

Eszter K. Horváth / Géza Makay / Reinhard Pöschel / Tamás Waldhauser
Published Online: 2014-10-28 | DOI: https://doi.org/10.1515/math-2015-0010


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.

Keywords : Invariance groups; Symmetry groups; Galois connections; Orbit equivalence of permutation groups; Symmetric and alternating groups; Functions of several variables


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About the article

Received: 2013-07-25

Accepted: 2014-07-31

Published Online: 2014-10-28

Published in Print: 2015-01-01

Citation Information: Open Mathematics, Volume 13, Issue 1, ISSN (Online) 2391-5455, DOI: https://doi.org/10.1515/math-2015-0010.

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© 2015 Eszter K. Horváth et al.. This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 3.0 License. BY-NC-ND 3.0

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