# Invariance groups of finite functions and orbit equivalence of permutation groups

• Eszter K. Horváth , 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.

## References

[1] Bochert A., Ueber die Zahl der verschiedenen Werthe, die eine Function gegebener Buchstaben durch Vertauschung derselben erlangen kann, Math. Ann., 1889, 33, 584-59010.1007/BF01444035Search in Google Scholar

[2] Clote P., Kranakis E., Boolean functions, invariance groups, and parallel complexity, SIAM J. Comput., 1991, 20, 553-59010.1137/0220036Search in Google Scholar

[3] Crama Y., Hammer P.L., Boolean functions. Theory, algorithms, and applications., Encyclopedia of Mathematics and its Applications 142. Cambridge University Press, 201110.1017/CBO9780511852008Search in Google Scholar

[4] Dixon J. D., Mortimer B., Permutation groups, Graduate Texts in Mathematics, 163, Springer-Verlag, 199610.1007/978-1-4612-0731-3Search in Google Scholar

[5] Hall M.,The theory of groups, Chelsea Publishing Company, New York, 1976Search in Google Scholar

[6] Inglis N.F.J., On orbit equivalent permutation groups, Arch. Math., 1984, 43, 297-30010.1007/BF01196650Search in Google Scholar

[7] Kisielewicz A., Symmetry groups of Boolean functions and constructions of permutation groups, J. Algebra, 1998, 199, 379-40310.1006/jabr.1997.7198Search in Google Scholar

[8] Klein F., Vorlesungen über die Theorie der elliptischen Modulfunctionen. Ausgearbeitet und vervollständigt von Dr. Robert Fricke, Teubner, Leipzig, 1890Search in Google Scholar

[9] Kearnes K., personal communication, 2010Search in Google Scholar

[10] Pöschel R., Galois connections for operations and relations, In: K. Denecke, M. Erné, and S.L. Wismath (Eds.), Galois connections and applications, Mathematics and its Applications, 565, Kluwer Academic Publishers, Dordrecht, 2004, 231-25810.1007/978-1-4020-1898-5_5Search in Google Scholar

[11] Pöschel R. and Kalužnin L. A., Funktionen- und Relationenalgebren, Deutscher Verlag der Wissenschaften, Berlin, 1979, Birkhäuser Verlag Basel, Math. Reihe Bd. 67, 197910.1007/978-3-0348-5547-1Search in Google Scholar

[12] Remak R., Über die Darstellung der endlichen Gruppen als Untergruppen direkter Produkte, J. Reine Angew. Math., 1930, 163, 1-4410.1515/crll.1930.163.1Search in Google Scholar

[13] Seress Á., Primitive groups with no regular orbits on the set of subsets, Bull. Lond. Math. Soc., 1997, 29, 697-70410.1112/S0024609397003536Search in Google Scholar

[14] Seress Á., Yang K., On orbit-equivalent, two-step imprimitive permutation groups, Computational Group Theory and the Theory of Groups, Contemp. Math., 2008, 470, 271-28510.1090/conm/470/09194Search in Google Scholar

[15] Siemons J., Wagner A., On finite permutation groups with the same orbits on unordered sets, Arch. Math. 1985, 45, 492-50010.1007/BF01194888Search in Google Scholar

[16] Wielandt H., Finite permutation groups, Academic Press, New York and London, 1964Search in Google Scholar

[17] Wielandt H., Permutation groups through invariant relations and invariant functions, Dept. of Mathematics, Ohio State University Columbus, Ohio, 1969Search in Google Scholar