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Tytuł artykułu

Invariance groups of finite functions and orbit equivalence of permutation groups

Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Wydawca
Czasopismo
Rocznik
Tom
13
Numer
1
Opis fizyczny
Daty
wydano
2015-01-01
otrzymano
2013-07-25
zaakceptowano
2014-07-31
online
2014-10-28
Twórcy
autor
  • Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720, Szeged, Hungary,, makayg@math.u-szeged.hu
Bibliografia
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  • [2] Clote P., Kranakis E., Boolean functions, invariance groups, and parallel complexity, SIAM J. Comput., 1991, 20, 553-590
  • [3] Crama Y., Hammer P.L., Boolean functions. Theory, algorithms, and applications., Encyclopedia of Mathematics and its Applications 142. Cambridge University Press, 2011
  • [4] Dixon J. D., Mortimer B., Permutation groups, Graduate Texts in Mathematics, 163, Springer-Verlag, 1996
  • [5] Hall M.,The theory of groups, Chelsea Publishing Company, New York, 1976
  • [6] Inglis N.F.J., On orbit equivalent permutation groups, Arch. Math., 1984, 43, 297-300
  • [7] Kisielewicz A., Symmetry groups of Boolean functions and constructions of permutation groups, J. Algebra, 1998, 199, 379-403
  • [8] Klein F., Vorlesungen über die Theorie der elliptischen Modulfunctionen. Ausgearbeitet und vervollständigt von Dr. Robert Fricke, Teubner, Leipzig, 1890
  • [9] Kearnes K., personal communication, 2010
  • [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, 231-258
  • [11] Pöschel R. and Kalužnin L. A., Funktionen- und Relationenalgebren, Deutscher Verlag der Wissenschaften, Berlin, 1979, Birkhäuser Verlag Basel, Math. Reihe Bd. 67, 1979
  • [12] Remak R., Über die Darstellung der endlichen Gruppen als Untergruppen direkter Produkte, J. Reine Angew. Math., 1930, 163, 1-44
  • [13] Seress Á., Primitive groups with no regular orbits on the set of subsets, Bull. Lond. Math. Soc., 1997, 29, 697-704
  • [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, 271-285
  • [15] Siemons J., Wagner A., On finite permutation groups with the same orbits on unordered sets, Arch. Math. 1985, 45, 492-500
  • [16] Wielandt H., Finite permutation groups, Academic Press, New York and London, 1964
  • [17] Wielandt H., Permutation groups through invariant relations and invariant functions, Dept. of Mathematics, Ohio State University Columbus, Ohio, 1969
Typ dokumentu
Bibliografia
Identyfikatory
Identyfikator YADDA
bwmeta1.element.doi-10_1515_math-2015-0010
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