Invariance groups of finite functions and orbit equivalence of permutation groups

Eszter K. Horváth; Géza Makay; Reinhard Pöschel; Tamás Waldhauser

Open Mathematics (2015)

  • Volume: 13, Issue: 1, page 83-95, electronic only
  • ISSN: 2391-5455

Abstract

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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.

How to cite

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Eszter K. Horváth, et al. "Invariance groups of finite functions and orbit equivalence of permutation groups." Open Mathematics 13.1 (2015): 83-95, electronic only. <http://eudml.org/doc/268733>.

@article{EszterK2015,
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.},
author = {Eszter K. Horváth, Géza Makay, Reinhard Pöschel, Tamás Waldhauser},
journal = {Open Mathematics},
keywords = {Invariance groups; Symmetry groups; Galois connections; Orbit equivalence of permutation groups; Symmetric and alternating groups; Functions of several variables; invariance groups; symmetry groups; orbit equivalence of permutation groups; symmetric groups; alternating groups; functions of several variables},
language = {eng},
number = {1},
pages = {83-95, electronic only},
title = {Invariance groups of finite functions and orbit equivalence of permutation groups},
url = {http://eudml.org/doc/268733},
volume = {13},
year = {2015},
}

TY - JOUR
AU - Eszter K. Horváth
AU - Géza Makay
AU - Reinhard Pöschel
AU - Tamás Waldhauser
TI - Invariance groups of finite functions and orbit equivalence of permutation groups
JO - Open Mathematics
PY - 2015
VL - 13
IS - 1
SP - 83
EP - 95, electronic only
AB - 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.
LA - eng
KW - Invariance groups; Symmetry groups; Galois connections; Orbit equivalence of permutation groups; Symmetric and alternating groups; Functions of several variables; invariance groups; symmetry groups; orbit equivalence of permutation groups; symmetric groups; alternating groups; functions of several variables
UR - http://eudml.org/doc/268733
ER -

References

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