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Nilpotent groups of class two which underly a unique regular dessin. (English) Zbl 1330.14058

On peut associer à chaque dessin d’enfants son groupe d’automorphismes. Un problème important consiste à caractériser ces groupes. Dans cet article, les \(p\)-groupes \(G\) finis de longueur 2 (i.e. tels qu’il existe une suite non triviale \(\left<e\right>\triangleleft G'\triangleleft G\)) qui apparaissent comme groupe d’automorphismes d’un unique dessin d’enfant sont classifiés. Ces groupes appartiennent à trois familles infinies. La preuve est entièrement algébrique et repose principalement sur des considérations élémentaires sur les groupes.

MSC:

14H57 Dessins d’enfants theory
14H37 Automorphisms of curves
20B25 Finite automorphism groups of algebraic, geometric, or combinatorial structures
30F10 Compact Riemann surfaces and uniformization
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[1] Belyǐ, G.V.: Galois extensions of a maximal cyclotomic field. Izv. Akad. Nauk. SSSR Ser. Mat. 43, 267-276 (1979) · Zbl 0409.12012
[2] Breda d’Azevedo, A., Nedela, R.: Join and intersection of hypermaps. Acta. Univ. M. Belii. 9, 13-28 (2001) · Zbl 1043.05039
[3] Conder, M.D.E.: All proper orientable regular hypermaps on surfaces of genus 2 to 101. URL: https://www.math.auckland.ac.nz/ conder/OrientableProperHypermaps101.txt · Zbl 1262.14034
[4] Conder, M.D.E., Du, S.-F., Nedela, R., Škoviera, M.; Bounding the size of a regular map with nilpotent automorphism group, in preparation · Zbl 1362.57004
[5] Conder, M.D.E., Jones, G.A., Streit, M., Wolfart, J.: Galois actions and regular dessins of small genera. Rev. Mat. Iberoam. 29, 163-181 (2013) · Zbl 1262.14034 · doi:10.4171/RMI/717
[6] Corn, D., Singerman, D.: Regular hypermaps. Eur. J. Comb. 9, 337-351 (1988) · Zbl 0665.57002 · doi:10.1016/S0195-6698(88)80064-7
[7] Coste, A.D., Jones, G.A., Streit, M., Wolfart, J.: Generalised Fermat hypermaps and Galois orbits. Glasgow Math. J. 51(2), 289-299 (2009) · Zbl 1185.14027 · doi:10.1017/S0017089509004972
[8] González-Diez, G., Jaikin-Zapirain, A.: The absolute Galois group acts faithfully on regular dessins and on Beauville surfaces, preprint, (2013) · Zbl 1333.11061
[9] Grothendieck, A.: Esquisse d’un programme, in: L. Schneps and P. Lochak (eds.), Geometric Galois Actions 1. The Inverse Galois Problem, Moduli spaces and Mapping Class Groups, London Math. Soc. Lecture Notes Ser. 242, pp. 5-48, Cambridge Univ. Press, Cambridge (1997) · Zbl 0901.14001
[10] Hidalgo, R.A.: The bipartite graphs of abelian dessins d’enfants. Ars Math. Contemp. 6, 301-304 (2013) · Zbl 1288.11065
[11] Hu, K., Nedela, R., Wang, N.-E.: Nilpotent dessins: Decomposition theorem and classification of the abelian dessins, preprint · Zbl 0665.57002
[12] Hu, K., Nedela, R., Wang, N.-E.: Classification of \[p\] p-groups of class three which underly a unique regular dessin, in preparation · Zbl 1330.14058
[13] Huppert, B.: Endliche Gruppen, vol. 1. Springer-Verlag, Berlin (1967) · Zbl 0217.07201 · doi:10.1007/978-3-642-64981-3
[14] James, L.D.: Operations on hypermaps, and outer automorphisms. Eur. J. Comb. 9, 551-560 (1988) · Zbl 0698.05028 · doi:10.1016/S0195-6698(88)80052-0
[15] Jones, G.A.: Regular embeddings of complete bipartite graphs: classification and enumeration. Proc. Lond. Math. Soc. 101(3), 427-453 (2010) · Zbl 1202.05028 · doi:10.1112/plms/pdp061
[16] Jones, G.A.: Regular dessins with a given automorphism group, arXiv:1309.5219 [math.GR], (2013) · Zbl 1346.14091
[17] Jones, G.A., Pinto, D.: Hypermap operations of finite order. Discrete Math. 310, 1820-1827 (2010) · Zbl 1210.05060 · doi:10.1016/j.disc.2009.12.019
[18] Jones, G.A., Singerman, D.: Belyıfunctions, hypermaps and Galois groups. Bull. Lond. Math. Soc. 28, 561-590 (1996) · Zbl 0853.14017
[19] Jones, G.A., Streit, M., Wolfart, J.: Galois action on families of generalised Fermat curves. J. Algebra 307, 829-840 (2007) · Zbl 1112.14034 · doi:10.1016/j.jalgebra.2006.10.009
[20] Malnič, A., Nedela, R., Škoviera, M.: Regular maps with nilpotent automorphism groups. Eur. J. Comb. 33(8), 1974-1986 (2012) · Zbl 1252.51013 · doi:10.1016/j.ejc.2012.06.001
[21] Wang, N.-E.: Regular bipartite maps, PhD thesis, Matej Bel University, Banská Bystrica, (2014) · Zbl 0836.20032
[22] Wilson, S.E.: Parallel products in groups and maps. J. Algebra 167, 539-546 (1994) · Zbl 0836.20032 · doi:10.1006/jabr.1994.1200
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