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.
@article{bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1131,
author = {P\'eter P. P\'alfy},
title = {Maximal clones and maximal permutation groups},
journal = {Discussiones Mathematicae - General Algebra and Applications},
volume = {27},
year = {2007},
pages = {277-291},
zbl = {1147.08004},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1131}
}
Péter P. Pálfy. Maximal clones and maximal permutation groups. Discussiones Mathematicae - General Algebra and Applications, Tome 27 (2007) pp. 277-291. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmgaa_1131/
[000] [1] M. Aschbacher and L. Scott, Maximal subgroups of finite groups, J. Algebra 92 (1985), 44-80. | Zbl 0549.20011
[001] [2] F. Buekenhout, On a theorem of O'Nan and Scott, Bull. Soc. Math. Belg. 40 (1988), 1-9. | Zbl 0653.20003
[002] [3] P.J. Cameron, Permutation Groups, Cambridge University Press, 1999. | Zbl 0922.20003
[003] [4] J.D. Dixon and B. Mortimer, Permutation Groups, Springer 1996. | Zbl 0951.20001
[004] [5] M.W. Liebeck, C.E. Praeger and J. Saxl, A classification of the maximal subgroups of the finite alternating and symmetric groups, J. Algebra 111 (1987), 365-383. | Zbl 0632.20011
[005] [6] M.W. Liebeck, C.E. Praeger and J. Saxl, The O'Nan-Scott theorem for finite primitive permutation groups, J. Austral. Math. Soc., Ser. A 44 (1988), 389-396. | Zbl 0647.20005
[006] [7] D. Mašulović and M. Pech, On traces of maximal clones, Novi Sad J. Math. 35 (2005), 161-185. | Zbl 1103.08001
[007] [8] M. Ponjavić and D. Mašulović, On chains and antichains in the partially ordered set of traces of maximal clones, pp. 119-134 in: 'Contributions to General Algebra', Vol. 15 (Proc. Conf. Klagenfurt 2003), Heyn, Klagenfurt 2004.
[008] [9] R. Pöschel and L.A. Kalužnin, Funktionen- und Relationenalgebren, Deutscher Verlag der Wissenschaften, Berlin 1979.
[009] [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.
[010] [11] I. Rosenberg, La structure des fonctions de plusieurs variables sur un ensemble fini, C. R. Acad. Sc. Paris 260 (1965), 3817-3819. | Zbl 0144.01002
[011] [12] I. Rosenberg, Über die Verschiedenheit maximaler Klassen in , Rev. Roumaine Math. Pures Appl. 14 (1969), 431-438. | Zbl 0193.29101
[012] [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.
[013] [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. | Zbl 0257.05006
[014] [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.