El papel de los coordinadores en la Olimpiada Matemática Internacional

En esta nota se describe el papel de los coordinadores en la Olimpiada Matemática Internacional

Some characterisations of groups in which normality is a transitive relation by means of subgroup embedding properties

In this survey we highlight the relations between some subgroup embedding properties that characterise groups in which normality is a transitive relation in certain universes of groups with some finiteness properties

On minimal non-supersoluble groups

This paper has been published in Revista Matemática Iberoamericana, 23(1):127-142 (2007). Copyright 2007 by Real Sociedad Matemática Española and European Mathematical Society Publishing House. The final publication is available at http://rmi.rsme.es http://www.ems-ph.org/journals/journal.php?jrn=rmi http://projecteuclid.org/euclid.rmi/1180728887[EN] The aim of this paper is to classify the finite minimal non-p-supersoluble groups, p a prime number, in the p-soluble universe.Supported by Proyecto BFM2001-1667-C03-03 (MCyT) and FEDER (European Union)Ballester Bolinches, A.; Esteban Romero, R. (2007). On minimal non-supersoluble groups. Revista Matemática Iberoamericana. 1(23). http://hdl.handle.net/10251/1847712

On finite soluble groups in which Sylow permutability is a transitive relation

This paper has been published by Springer-Verlag and Akadémiai Kiadó in Acta Mathematica Hungarica, 101(3):193-202 (2003). The final publication is available at http://www.springerlink.com http://dx.doi.org/10.1023/B:AMHU.0000003903.71033.fcA characterisation of finite soluble groups in which Sylow permutability is a transitive relation by means of subgroup embedding properties enjoyed by all the subgroups is proved in the paper. The key point is an extension of a subnormality criterion due to Wielandt.http://dx.doi.org/10.1023/B:AMHU.0000003903.71033.fcBallester Bolinches, A.; Esteban Romero, R. (2003). On finite soluble groups in which Sylow permutability is a transitive relation. 3(101). doi:10.1023/B:AMHU.0000003903.71033.fc310

p-grupos finitos

En la primera parte se obtienen nuevas cotas para el número de clases de conjugación de longitud máxima de un $p$-grupo finito $G$, $r(G)$, relacionándolo con la longitud de estas clases, $b$. En el caso en que $r(G)=p^m-b-1$, existe un único subgrupo normal de orden $p^b$, $N_b$, que es característico y se estudian propiedades estructurales de estos grupos cuando $b\le 3$, prestando atención especial a la relación entre $N_b$, $Z(G)$ y $G$. En la seguna parte se establecen nuevas cotas para el grado de conmutatividad $c$ de un $p$-grupo de clase maxdimal. En el capítulo 2 se hace un repaso de las cotas conocidas para $c$. En el capítulo 3 se extienden los resultados obtenidos por Shepherd para $c_0\le 4$ hasta $c_0\le 10$ mediante el desarrollo de nuevas técnicas computacionales. En el capítulo 4 se presentan tablas que dan las cotas para $c$ en función de $c_0$ y $l$ para $p\le 43$, obtenidas con la ayuda de dichas técnicas computacionales, y se conjeturan las cotas más finas posibles para $c$ para la mayor parte de los valores de $c_0$ y $l$ y para cualquier primo $p$. Se resuelven la mayoría de dichas conjeturas, mejorando de esta manera las cotas dadas por Shepherd, Leedham-Green y McKay y Fernández Alcober. Se muestra también la optimalidad de dichas cotas mediante la construcción de las álgebras de Lie para las que se alcanzan las cotas.In the first part, new bounds for the number of conjugacy classes of maximal length of a finite $p$-group $G$, $r(G)$, are obtained, and they are related with the length of these classes. If $r(G)=p^m-b-1$, there exists a unique normal subgroup of order $p^b$, $N_b$, which is characteristic, and structural properties of these groups are studied when $b\le 3$, by paying special attention to the relation between $N_b$, $Z(G)$ and $G$. In the second part, new bounds for the degree of commutativity of a $p$-group of maximal class are established. In Chapter 2, the bounds known for $c$ are reviewed. In Chapter 3, the reuslts obtained by Shepherd for $c_0\le 4$ are extended to $c_0\le 10$ by means of the development of new computational techniques. In Chapter 4, we present some tables which give the bounds for $c$ as a function of $c_0$ and $l$ for $p\le 3$, obtained with the help of the mentioned computational techniques, and the finest possible bounds for $c$ for the majority of the values of $c_0$ and $l$ and every prime $p$ are conjectured. The majority of these conjectures are solved, and hence the bounds given by Shepherd, Leedham-Green and McKay, and Fernández-Alcober are improved. It is also shown the optimality of these bounds by means of the construction of the Lie algebras for which these bounds are attained

p-grupos finitos

Tesis doctoral del autor, dirigida por Antonio Vera López en el Departament d'Àlgebra de la Universitat de València y defendida en julio de 1997. Author's PhD thesis, supervised by Antonio Vera López in the Department d'Àlgebra of the Universitat de València and presented in July, 1997. Tesi doctoral de l'autor, dirigida per Antonio Vera López al Departament d'Àlgebra de la Universitat de València i defensada en juliol de 1997.In the first part, new bounds for the number of conjugacy classes of maximal length of a finite $p$-group $G$, $r(G)$, are obtained, and they are related with the length of these classes. If $r(G)=p^m-b-1$, there exists a unique normal subgroup of order $p^b$, $N_b$, which is characteristic, and structural properties of these groups are studied when $b\le 3$, by paying special attention to the relation between $N_b$, $Z(G)$ and $G$. In the second part, new bounds for the degree of commutativity of a $p$-group of maximal class are established. In Chapter 2, the bounds known for $c$ are reviewed. In Chapter 3, the reuslts obtained by Shepherd for $c_0\le 4$ are extended to $c_0\le 10$ by means of the development of new computational techniques. In Chapter 4, we present some tables which give the bounds for $c$ as a function of $c_0$ and $l$ for $p\le 3$, obtained with the help of the mentioned computational techniques, and the finest possible bounds for $c$ for the majority of the values of $c_0$ and $l$ and every prime $p$ are conjectured. The majority of these conjectures are solved, and hence the bounds given by Shepherd, Leedham-Green and McKay, and Fernández-Alcober are improved. It is also shown the optimality of these bounds by means of the construction of the Lie algebras for which these bounds are attained.En la primera part s'obtenen noves fites per al nombre de classes de conjugació de longitud màxima d'un $p$-grup finit $G$, $r(G)$, relacionant-lo amb la longitud d'aquestes classes, $b$. En el cas en el qual $r(G)=p^m-b-1$, existeix un únic subgrup normal d'ordre $p^b$, $N_b$, que és característic, i s'estudien propietats estructurals d'aquests grups quan $b\le 3$, parant especial atenció a la relació entre $N_b$, $Z(G)$ i $G$. En la segona part s'estableixen noves fites per al grau de commutativitat $c$ d'un $p$-grup de classe maximal. En el capítol 2 es fa un repàs de les fites conegudes per a $c$. En el capítol 3 s'estenen els resultats obtinguts per Shepherd per a $c_0\le 4$ fins a $c_0\le 10$ mitjançant el desenvolupament de noves tècniques computacionals. En el capítol 4 es presenten quadres que dónen les fites per a $c$ en funció de $c_0$ i $l$ per a $p\le 43$, obtingudes amb l'ajut de dites tècniques computacionals, i es conjecturen les cotes més fines possibles per a $c$ per a la major part dels valors de $c_0$ i $l$ i per a qualsevol primer $p$. Es resolen la majoria d'aquestes conjectures, millorant d'aquesta manera les fites donades per Shepherd, Leedham-Green i McKay i Fernández Alcober. Es mostra també l'optimalitat d'aquestes fites mitjançant la construcció de les àlgebres de Lie per a les quals s'assoleixen les fites.En la primera parte se obtienen nuevas cotas para el número de clases de conjugación de longitud máxima de un $p$-grupo finito $G$, $r(G)$, relacionándolo con la longitud de estas clases, $b$. En el caso en que $r(G)=p^m-b-1$, existe un único subgrupo normal de orden $p^b$, $N_b$, que es característico y se estudian propiedades estructurales de estos grupos cuando $b\le 3$, prestando atención especial a la relación entre $N_b$, $Z(G)$ y $G$. En la seguna parte se establecen nuevas cotas para el grado de conmutatividad $c$ de un $p$-grupo de clase maxdimal. En el capítulo 2 se hace un repaso de las cotas conocidas para $c$. En el capítulo 3 se extienden los resultados obtenidos por Shepherd para $c_0\le 4$ hasta $c_0\le 10$ mediante el desarrollo de nuevas técnicas computacionales. En el capítulo 4 se presentan tablas que dan las cotas para $c$ en función de $c_0$ y $l$ para $p\le 43$, obtenidas con la ayuda de dichas técnicas computacionales, y se conjeturan las cotas más finas posibles para $c$ para la mayor parte de los valores de $c_0$ y $l$ y para cualquier primo $p$. Se resuelven la mayoría de dichas conjeturas, mejorando de esta manera las cotas dadas por Shepherd, Leedham-Green y McKay y Fernández Alcober. Se muestra también la optimalidad de dichas cotas mediante la construcción de las álgebras de Lie para las que se alcanzan las cotas.Esteban Romero, R. (1999). p-grupos finitos. Universitat de València - Servei de Publicacions. http://hdl.handle.net/10251/1834

Some secrets of the Spanish national identitydocument: an application of modulararithmetics to error detecting codes

[EN] We present some control characters and digits present in the Spanish national identity document as an example of the application of modular arithmetics to the design of error correcting codes. This activity has been developed inside the ESTALMAT programme for the stimulation of mathematical talent and in the summer scientific campuses organised by VLC/Campus[ES] Presentamos algunos caracteres y dígitos de control que aparecen en el documento nacional de identidad español como ejemplo de aplicación de la aritmética modular al diseño de códigos detectores de errores. Esta actividad se ha desarrollado dentro del programa ESTALMAT de estímulo del talento matemático y en los campus científicos de verano organizados por VLC/CampusEl autor agradece la financiaci´on del proyecto MTM2014-54707-C3-1-P del Ministerio de Econom´ıa y Competitividad (Espa˜na) y FEDER (Uni´on Europea).Esteban Romero, R. (2016). Algunos secretos del documento nacional de identidad español: una aplicación de la aritmética modular a códigos detectores de errores. Modelling in Science Education and Learning. 9(2):59-66. https://doi.org/10.4995/msel.2016.6338SWORD59669

A generalization to Sylow permutability of pronormal subgroups of finite groups

Electronic version of an article published as Journal of Algebra and Its Applications, 2020, 19:03 https://doi.org/10.1142/S0219498820500528 © World Scientific Publishing Company.[EN] In this note, we present a new subgroup embedding property that can be considered as an analogue of pronormality in the scope of permutability and Sylow permutability in finite groups. We prove that finite PST-groups, or groups in which Sylow permutability is a transitive relation, can be characterized in terms of this property, in a similar way as T-groups, or groups in which normality is transitive, can be characterized in terms of pronormality.The research of the first author has been supported by the research grants MTM2014-54707-C3-1-P by the "Ministerio de Economia y Competitividad" (Spain) and FEDER (European Union) and PROMETEO/2017/057 from "Generalitat" (Valencian Community, Spain). Part of the work of this paper has been done during some visits of the first author to the Dipartimento di Matematica of the Universita degli Studi di Salerno supported by the "National Group for Algebraic and Geometric Structures, and their Applications" (GNSAGA - INdAM), Italy.Esteban Romero, R.; Longobardi, P.; Maj, M. (2020). A generalization to Sylow permutability of pronormal subgroups of finite groups. Journal of Algebra and Its Applications. 19(3):1-13. https://doi.org/10.1142/S0219498820500528S113193Ballester-Bolinches, A., & Esteban-Romero, R. (2002). Sylow Permutable Subnormal Subgroups of Finite Groups. Journal of Algebra, 251(2), 727-738. doi:10.1006/jabr.2001.9138Ballester-Bolinches, A., & Esteban-Romero, R. (2003). On finite J-groups. Journal of the Australian Mathematical Society, 75(2), 181-192. doi:10.1017/s1446788700003712Ballester-Bolinches, A., & Esteban-Romero, R. (2003). On finite soluble groups in which Sylow permutability is a transitive relation. Acta Mathematica Hungarica, 101(3), 193-202. doi:10.1023/b:amhu.0000003903.71033.fcBallester-Bolinches, A., Esteban-Romero, R., & Asaad, M. (2010). Products of Finite Groups. de Gruyter Expositions in Mathematics. doi:10.1515/9783110220612Ballester-Bolinches, A., Feldman, A. D., Pedraza-Aguilera, M. C., & Ragland, M. F. (2011). A class of generalised finite T-groups. Journal of Algebra, 333(1), 128-138. doi:10.1016/j.jalgebra.2011.02.018Deskins, W. E. (1963). On quasinormal subgroups of finite groups. Mathematische Zeitschrift, 82(2), 125-132. doi:10.1007/bf01111801Doerk, K., & Hawkes, T. O. (1992). Finite Soluble Groups. doi:10.1515/9783110870138Huppert, B. (1967). Endliche Gruppen I. Grundlehren der mathematischen Wissenschaften. doi:10.1007/978-3-642-64981-3Kaplan, G. (2010). On T-groups, supersolvable groups, and maximal subgroups. Archiv der Mathematik, 96(1), 19-25. doi:10.1007/s00013-010-0207-0Kegel, O. H. (1962). Sylow-Gruppen und Subnormalteiler endlicher Gruppen. Mathematische Zeitschrift, 78(1), 205-221. doi:10.1007/bf01195169Mysovskikh, V. I. (1999). Investigation of Subgroup Embeddings by the Computer Algebra Package GAP. Computer Algebra in Scientific Computing CASC’99, 309-315. doi:10.1007/978-3-642-60218-4_24Peng, T. A. (1969). Finite groups with pro-normal subgroups. Proceedings of the American Mathematical Society, 20(1), 232-232. doi:10.1090/s0002-9939-1969-0232850-1Peng, T. A. (1971). Pronormality in Finite Groups. Journal of the London Mathematical Society, s2-3(2), 301-306. doi:10.1112/jlms/s2-3.2.301Schmid, P. (1998). Subgroups Permutable with All Sylow Subgroups. Journal of Algebra, 207(1), 285-293. doi:10.1006/jabr.1998.7429Wielandt, H. (1939). Eine Verallgemeinerung der invarianten Untergruppen. Mathematische Zeitschrift, 45(1), 209-244. doi:10.1007/bf01580283Yi, X., & Skiba, A. N. (2014). On $$S$$ S -propermutable Subgroups of Finite Groups. Bulletin of the Malaysian Mathematical Sciences Society, 38(2), 605-616. doi:10.1007/s40840-014-0038-

On large orbits of supersoluble subgroups of linear groups

The research of this paper has been supported by the grant MTM2014-54707-C3-1-P from the Ministerio de Economia y Competitividad, Spain, and FEDER, European Union, by the grant PGC2018-095140-B-I00 from the Ministerio de Ciencia, Innovacion y Universidades and the Agencia Estatal de Investigacion, Spain, and FEDER, European Union, and by the grant PROMETEO/2017/057 from the Generalitat, Valencian Community, Spain. The first author is supported by the predoctoral grant 201606890006 from the China Scholarship Council. The second author is supported by the grant 11401597 from the National Science Foundation of ChinMeng, H.; Ballester-Bolinches, A.; Esteban Romero, R. (2019). On large orbits of supersoluble subgroups of linear groups. Journal of the London Mathematical Society. 101(2):490-504. https://doi.org/10.1112/jlms.12266S4905041012Doerk, K., & Hawkes, T. O. (1992). Finite Soluble Groups. doi:10.1515/9783110870138Dolfi, S. (2008). Large orbits in coprime actions of solvable groups. Transactions of the American Mathematical Society, 360(01), 135-153. doi:10.1090/s0002-9947-07-04155-4Dolfi, S., & Jabara, E. (2007). Large character degrees of solvable groups with abelian Sylow 2-subgroups. Journal of Algebra, 313(2), 687-694. doi:10.1016/j.jalgebra.2006.12.004Espuelas, A. (1991). Large character degrees of groups of odd order. Illinois Journal of Mathematics, 35(3). doi:10.1215/ijm/1255987794The GAP group ‘GAP – groups algorithms and programming version 4.9.1’ 2018 http://www.gap‐system.org.Halasi, Z., & Maróti, A. (2015). The minimal base size for a -solvable linear group. Proceedings of the American Mathematical Society, 144(8), 3231-3242. doi:10.1090/proc/12974Huppert, B. (1967). Endliche Gruppen I. Grundlehren der mathematischen Wissenschaften. doi:10.1007/978-3-642-64981-3Keller, T. M., & Yang, Y. (2015). Abelian quotients and orbit sizes of solvable linear groups. Israel Journal of Mathematics, 211(1), 23-44. doi:10.1007/s11856-015-1259-4Manz, O., & Wolf, T. R. (1993). Representations of Solvable Groups. doi:10.1017/cbo9780511525971Meng, H., Ballester-Bolinches, A., & Esteban-Romero, R. (2019). On large orbits of subgroups of linear groups. Transactions of the American Mathematical Society, 372(4), 2589-2612. doi:10.1090/tran/7639Wolf, T. R. (1999). Large Orbits of Supersolvable Linear Groups. Journal of Algebra, 215(1), 235-247. doi:10.1006/jabr.1998.7730Yang, Y. (2009). Orbits of the actions of finite solvable groups. Journal of Algebra, 321(7), 2012-2021. doi:10.1016/j.jalgebra.2008.12.016Yang, Y. (2011). Large character degrees of solvable 3’-groups. Proceedings of the American Mathematical Society, 139(9), 3171-3173. doi:10.1090/s0002-9939-2011-10735-4Yang, Y. (2014). Large orbits of subgroups of solvable linear groups. Israel Journal of Mathematics, 199(1), 345-362. doi:10.1007/s11856-014-0002-

On finite groups generated by strongly cosubnormal subgroups

This paper has been published in Journal of Algebra, 259(1):226-234 (2003). Copyright 2003 by Elsevier. http://dx.doi.org/10.1016/S0021-8693(02)00535-5[EN] Two subgroups A and B of a group G are cosubnormal if A and B are subnormal in their join and are strongly cosubnormal if every subgroup of A is cosubnormal with every subgroup of B. We find necessary and sufficient conditions for A and B to be strongly cosubnormal in and, if Z is the hypercentre of G=, we show that A and B are strongly cosubnormal if and only if G/Z is the direct product of AZ/Z and BZ/Z. We also show that projectors and residuals for certain formations can easily be constructed in such a group. Two subgroups A and B of a group G are N-connected if every cyclic subgroup of A is cosubnormal with every cyclic subgroup of B (N denotes the class of nilpotent groups). Though the concepts of strong cosubnormality and N-connectedness are clearly closely related, we give an example to show that they are not equivalent. We note, however, that if G is the product of the N-connected subgroups A and B, then A and B are strongly cosubnormal.The first and the third authors have been supported by Proyecto BFM2001-1667-C03-03 from Ministerio de Ciencia y Tecnolog´ıa, Spain. The third author has been supported by a grant from the Program of Support of Research (Stays of Researchers in other academic institutions) of the Universitat Polit`ecnica de Val`encia. Part of this research has been carried out during a visit of the third author to the School of Mathematical Sciences of the Australian National University in Canberra (Australia), to whom he wants to express his gratitude for their kindness and financial support.Ballester Bolinches, A.; Cossey, J.; Esteban Romero, R. (2003). On finite groups generated by strongly cosubnormal subgroups. Journal of Algebra. 1(259):226-234. doi:10.1016/S0021-8693(02)00535-5226234125