Maximal clones and maximal permutation groups

Péter P. Pálfy

Discussiones Mathematicae - General Algebra and Applications (2007)

  • Volume: 27, Issue: 2, page 277-291
  • ISSN: 1509-9415

Abstract

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A fundamental result in universal algebra is the theorem of Rosenberg describing the maximal subclones in the clone of all operations over a finite set. In group theory, the maximal subgroups of the symmetric groups are classified by the O'Nan-Scott Theorem. We shall explore the similarities and differences between these two analogous major results. In addition, we show that a primitive permutation group of diagonal type can be maximal in the symmetric group only if its socle is the direct product of two isomorphic simple groups, because if the number of simple factors of the socle is greater than two, then the group is contained in the alternating group.

How to cite

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Péter P. Pálfy. "Maximal clones and maximal permutation groups." Discussiones Mathematicae - General Algebra and Applications 27.2 (2007): 277-291. <http://eudml.org/doc/276890>.

@article{PéterP2007,
abstract = {A fundamental result in universal algebra is the theorem of Rosenberg describing the maximal subclones in the clone of all operations over a finite set. In group theory, the maximal subgroups of the symmetric groups are classified by the O'Nan-Scott Theorem. We shall explore the similarities and differences between these two analogous major results. In addition, we show that a primitive permutation group of diagonal type can be maximal in the symmetric group only if its socle is the direct product of two isomorphic simple groups, because if the number of simple factors of the socle is greater than two, then the group is contained in the alternating group.},
author = {Péter P. Pálfy},
journal = {Discussiones Mathematicae - General Algebra and Applications},
keywords = {maximal clones; Rosenberg's primality criterion; O'Nan-Scott theorem; primitive permutation groups},
language = {eng},
number = {2},
pages = {277-291},
title = {Maximal clones and maximal permutation groups},
url = {http://eudml.org/doc/276890},
volume = {27},
year = {2007},
}

TY - JOUR
AU - Péter P. Pálfy
TI - Maximal clones and maximal permutation groups
JO - Discussiones Mathematicae - General Algebra and Applications
PY - 2007
VL - 27
IS - 2
SP - 277
EP - 291
AB - A fundamental result in universal algebra is the theorem of Rosenberg describing the maximal subclones in the clone of all operations over a finite set. In group theory, the maximal subgroups of the symmetric groups are classified by the O'Nan-Scott Theorem. We shall explore the similarities and differences between these two analogous major results. In addition, we show that a primitive permutation group of diagonal type can be maximal in the symmetric group only if its socle is the direct product of two isomorphic simple groups, because if the number of simple factors of the socle is greater than two, then the group is contained in the alternating group.
LA - eng
KW - maximal clones; Rosenberg's primality criterion; O'Nan-Scott theorem; primitive permutation groups
UR - http://eudml.org/doc/276890
ER -

References

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  1. [1] M. Aschbacher and L. Scott, Maximal subgroups of finite groups, J. Algebra 92 (1985), 44-80. Zbl0549.20011
  2. [2] F. Buekenhout, On a theorem of O'Nan and Scott, Bull. Soc. Math. Belg. 40 (1988), 1-9. Zbl0653.20003
  3. [3] P.J. Cameron, Permutation Groups, Cambridge University Press, 1999. Zbl0922.20003
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  9. [9] R. Pöschel and L.A. Kalužnin, Funktionen- und Relationenalgebren, Deutscher Verlag der Wissenschaften, Berlin 1979. 
  10. [10] R.W. Quackenbush, A new proof of Rosenberg's primal algebra characterization theorem, pp. 603-634 in: 'Finite Algebra and Multiple-Valued Logic' (Proc. Conf. Szeged 1979), Colloq. Math. Soc. J. Bolyai, Vol. 28, North-Holland, Amsterdam 1981. 
  11. [11] I. Rosenberg, La structure des fonctions de plusieurs variables sur un ensemble fini, C. R. Acad. Sc. Paris 260 (1965), 3817-3819. Zbl0144.01002
  12. [12] I. Rosenberg, Über die Verschiedenheit maximaler Klassen in P k , Rev. Roumaine Math. Pures Appl. 14 (1969), 431-438. Zbl0193.29101
  13. [13] I. Rosenberg, Über die funktionale Vollständigkeit in den mehrwertigen Logiken, Rozpravy Československé Akademie Věd, Řada Matematických a Přirodních Věd 80 (4) (1970), 3-93. 
  14. [14] I. Rosenberg, The number of maximal closed classes in the set of functions over a finite domain, J. Combinat. Theory, Ser. A 14 (1973), 1-7. Zbl0257.05006
  15. [15] L.L. Scott, Representations in characteristic p, pp. 319-331 in: 'The Santa Cruz Conference on Finite Groups' (Santa Cruz, 1979), Proc. Sympos. Pure Math., Vol. 37, Amer. Math. Soc., Providence, RI, 1980. 

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