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[EN] Braces were introduced by Rump in 2007 as a useful tool in the study of the set-theoretic solutions of the Yang¿Baxter equation. In fact, several aspects of the theory of finite left braces and their applications in the context of the Yang¿Baxter equation have been extensively investigated recently. The main aim of this paper is to introduce and study two finite brace theoretical properties associated with nilpotency, and to analyse their impact on the finite solutions of the Yang¿Baxter equation. ; This work was supported by the research grant MTM2014-54707-C3-1-P from the Ministerio de Economia y Competitividad, Spanish Government, and FEDER, European Union, and PROMETEO/2017/057 from Generalitat (Valencian Community, Spain). The first author was supported by grant number 201606890006 of the China Scholarship Council. ; Meng, H.; Ballester Bolinches, A.; Esteban Romero, R. (2019). Left braces and the quantum Yang-Baxter equation. Proceedings of the Edinburgh Mathematical Society. 62(2):595-608. https://doi.org/10.1017/S0013091518000664 ; S ; 595 ; 608 ; 62 ; 2 ; Yang, C. N. (1967). Some Exact Results for the Many-Body Problem in one Dimension with Repulsive Delta-Function Interaction. Physical Review Letters, 19(23), 1312-1315. doi:10.1103/physrevlett.19.1312 ; Smoktunowicz, A. (2018). On Engel groups, nilpotent groups, rings, braces and the Yang-Baxter equation. Transactions of the American Mathematical Society, 370(9), 6535-6564. doi:10.1090/tran/7179 ; Etingof, P., Schedler, T., & Soloviev, A. (1999). Set-theoretical solutions to the quantum Yang-Baxter equation. Duke Mathematical Journal, 100(2), 169-209. doi:10.1215/s0012-7094-99-10007-x ; Cedó, F., Jespers, E., & Okniński, J. (2014). Braces and the Yang–Baxter Equation. Communications in Mathematical Physics, 327(1), 101-116. doi:10.1007/s00220-014-1935-y ; Cedó, F., Gateva-Ivanova, T., & Smoktunowicz, A. (2017). On the Yang–Baxter equation and left nilpotent left braces. Journal of Pure and Applied Algebra, 221(4), 751-756. ...

The final publication is available at Springer via http://dx.doi.org/10.1007/s00013-016-0901-7 ; In this note, global information about a finite group is obtained by assuming that certain subgroups of some given order are S-semipermutable. Recall that a subgroup H of a finite group G is said to be S-semipermutable if H permutes with all Sylow subgroups of G of order coprime to . We prove that for a fixed prime p, a given Sylow p-subgroup P of a finite group G, and a power d of p dividing such that , if is S-semipermutable in for all normal subgroups H of P with , then either G is p-supersoluble or else . This extends the main result of Guo and Isaacs in (Arch. Math. 105:215-222 2015). We derive some theorems that extend some known results concerning S-semipermutable subgroups. ; The first and the second authors have been supported by the grant MTM2014-54707-C3-1-P from the Ministerio de Economia y Competitividad, Spain, and FEDER, European Union. The first author has been also supported by NSC of China (No. 11271085) and a project of Natural Science Foundation of Guangdong Province (No. 2015A030313791). The third author was supported by NSF of China (No. 11201082), Cultivation Program for Outstanding Young College Teachers (Yq2013061) and Project (2013B051000075) of Guangdong Province, Pei Ying Yu Cai Project of GDUT. The third author also thanks the China Scholarship Council and the Departament d'Algebra of the Universitat de Valencia for its hospitality. ; Ballester-Bolinches, A.; Esteban Romero, R.; Qiao, S. (2016). A note on a result of Guo and Isaacs about p-supersolubility of finite groups. Archiv der Mathematik. 106(6):501-506. https://doi.org/10.1007/s00013-016-0901-7 ; S ; 501 ; 506 ; 106 ; 6 ; Ballester-Bolinches A., Esteban-Romero R., Asaad M.: Products of finite groups, volume 53 of de Gruyter Expositions in Mathematics. Walter de Gruyter, Berlin (2010) ; Guo Y., Isaacs I. M.: Conditions on p-subgroups implying p-nilpotence or p-supersolvability. Arch. Math. 105, 215–222 (2015) ; B. Huppert, Endliche ...

This paper has been published in Science China Mathematics, 55(5):961-966 (2012). Copyright 2012 by Science China Press and Springer-Verlag. The final publication is available at www.springerlink.com. http://link.springer.com/article/10.1007/s11425-011-4356-9 http://dx.doi.org/10.1007/s11425-011-4356-9 ; A subgroup H of a finite group G is a partial CAP-subgroup of G if there is a chief series of G such that H either covers or avoids every chief factor of the series. The structural impact of the partial cover and avoidance property of some distinguished subgroups of a group has been studied by many authors. However there are still some open questions which deserve an answer. The purpose of the present paper is to give a complete answer to one of these questions. ; This work was supported by MEC of Spain, FEDER of European Union (Grant No. MTM-2007-68010-C03-02), MICINN of Spain (Grant No. MTM-2010-19938-C03-01), National Natural Science Foundation of China (Grant No. 11171353/A010201) and Natural Science Fund of Guangdong (Grant No. S2011010004447). Part of this research was carried out during a visit of the third author to the Departament d'Algebra, Universitat de Valencia, Burjassot, Valencia, Spain, and the Institut Universitari de Matematica Pura i Aplicada, Universitat Politecnica de Valencia, Valencia, Spain, between September, 2009 and August, 2010. He is grateful to both institutions for their warm hospitality and, in particular, to the Universitat Politecnica de Valencia for the financial support given via its Programme of Support to Research and Development 2010. ; http://link.springer.com/article/10.1007/s11425-011-4356-9 ; Ballester Bolinches, A.; Esteban Romero, R.; Li, Y. (2012). A question on partial CAP-subgroups of finite groups. Science China Mathematics. 5(55). doi:10.1007/s11425-011-4356-9 ; S ; 5 ; 55 ; Ballester-Bolinches A, Ezquerro L M. Classes of Finite Groups. In: Mathematics and its Applications, vol. 584. New York: Springer, 2006 ; Ballester-Bolinches A, Ezquerro L M, Skiba A N. Local ...

This paper has been published in Bulletin of the Australian Mathematical Society, 86(1):22-28 (2012). Copyright 2012 by Australian Mathematical Publishing Association Inc and Cambridge University Press Journals. The final publication is available at http://journals.cambridge.org/abstract_S0004972711003418 ; In this paper, solubility of groups factorised as a product of two subgroups which are connected by certain permutability properties is studied. ; The research of the second and the third authors has been supported by the grant MTM2010-19938-C03-01 from the Ministerio de Ciencia e Innovacion (Spanish government). ; http://dx.doi.org/10.1017/S0004972711003418 ; Asaad, M.; Ballester Bolinches, A.; Esteban Romero, R. (2012). Some solubility criteria in factorised groups. Bulletin of the Australian Mathematical Society. (86). doi:10.1017/S0004972711003418 ; S ; 86 ; Maier, R. (1989). Zur Vertauschbarkeit und Subnormalit�t von Untergruppen. Archiv der Mathematik, 53(2), 110-120. doi:10.1007/bf01198559 ; Huppert, B. (1967). Endliche Gruppen I. Grundlehren der mathematischen Wissenschaften. doi:10.1007/978-3-642-64981-3 ; Doerk, K., & Hawkes, T. O. (1992). Finite Soluble Groups. doi:10.1515/9783110870138 ; Deskins, W. E. (1959). On maximal subgroups. Finite Groups, 100-104. doi:10.1090/pspum/001/0125157 ; Ballester-Bolinches, A., Esteban-Romero, R., & Asaad, M. (2010). Products of Finite Groups. de Gruyter Expositions in Mathematics. doi:10.1515/9783110220612 ; Ballester-Bolinches, A., Guo, X., & Pedraza-Aguilera, M. C. (2000). A note on m-permutable products of finite groups. Journal of Group Theory, 3(4). doi:10.1515/jgth.2000.030 ; Ballester-Bolinches, A., Cossey, J., & Pedraza-Aguilera, M. C. (2001). ON PRODUCTS OF FINITE SUPERSOLUBLE GROUPS. Communications in Algebra, 29(7), 3145-3152. doi:10.1081/agb-5013 ; Ezquerro, L. M., & Soler-Escrivà, X. (2003). On MutuallyM-Permutable Products of Finite Groups. Communications in Algebra, 31(4), 1949-1960. doi:10.1081/agb-120018515

5 55 ; S Huppert B. Endliche Gruppen I. In: Grund Math Wiss, vol. 134. Berlin-Heidelberg-New York: Springer-Verlag, 1967 Huppert B, Blackburn N. Finite Groups III. In: Grund Math Wiss, vol. 243. Berlin: Springer-Verlag, 1982 ; This paper has been published in Science China Mathematics, 55(5):961-966 (2012). Copyright 2012 by Science China Press and Springer-Verlag. The final publication is available at www.springerlink.com. http://link.springer.com/article/10.1007/s11425-011-4356-9 http://dx.doi.org/10.1007/s11425-011-4356-9 A subgroup H of a finite group G is a partial CAP-subgroup of G if there is a chief series of G such that H either covers or avoids every chief factor of the series. The structural impact of the partial cover and avoidance property of some distinguished subgroups of a group has been studied by many authors. However there are still some open questions which deserve an answer. The purpose of the present paper is to give a complete answer to one of these questions. This work was supported by MEC of Spain, FEDER of European Union (Grant No. MTM-2007-68010-C03-02), MICINN of Spain (Grant No. MTM-2010-19938-C03-01), National Natural Science Foundation of China (Grant No. 11171353/A010201) and Natural Science Fund of Guangdong (Grant No. S2011010004447). Part of this research was carried out during a visit of the third author to the Departament d'Algebra, Universitat de Valencia, Burjassot, Valencia, Spain, and the Institut Universitari de Matematica Pura i Aplicada, Universitat Politecnica de Valencia, Valencia, Spain, between September, 2009 and August, 2010. He is grateful to both institutions for their warm hospitality and, in particular, to the Universitat Politecnica de Valencia for the financial support given via its Programme of Support to Research and Development 2010. http://link.springer.com/article/10.1007/s11425-011-4356-9 Ballester Bolinches, A.; Esteban Romero, R.; Li, Y. (2012). A question on partial CAP-subgroups of finite groups. Science China Mathematics. 5(55). ...

This paper has been published in Journal of Group Theory, 13(1):143-149 (2010). Copyright 2010 by Walter de Gruyter. The final publication is available at www.degruyter.com. http://dx.doi.org/10.1515/jgt.2009.038 http://www.degruyter.com/view/j/jgth.2010.13.issue-1/jgt.2009.038/jgt.2009.038.xml ; [EN] The aim of this paper is to characterise the classes of groups in which every subnormal subgroup is normal, permutable, or S-permutable by the embedding of the subgroups (respectively, subgroups of prime power order) in their normal, permutable, or S-permutable closure, respectively. ; This research was supported by the grants MTM2004-08219C02-02 and MTM2007-68010-C03-02 from MEC (Spanish Government) and FEDER (European Union) and GV/2007/243 from Generalitat (Valencian Community). ; http://dx.doi.org/10.1515/jgt.2009.038 ; Ballester Bolinches, A.; Esteban Romero, R.; Li, Y. (2010). On self-normalizing subgroups of finite groups. Journal of Group Theory. 1(13). https://doi.org/10.1515/jgt.2009.038 ; 1 ; 13 ; Agrawal, R. K. (1975). Finite Groups whose Subnormal Subgroups Permute with all Sylow Subgroups. Proceedings of the American Mathematical Society, 47(1), 77. doi:10.2307/2040211 ; Alejandre, M. J., Ballester-Bolinches, A., & Pedraza-Aguilera, M. . (2001). Finite Soluble Groups with Permutable Subnormal Subgroups. Journal of Algebra, 240(2), 705-722. doi:10.1006/jabr.2001.8732 ; Ballester-Bolinches, A., & Esteban-Romero, R. (2001). Sylow permutable subnormal subgroups of finite groups II. Bulletin of the Australian Mathematical Society, 64(3), 479-486. doi:10.1017/s0004972700019948 ; Ballester-Bolinches, A., & Esteban-Romero, R. (2002). Sylow Permutable Subnormal Subgroups of Finite Groups. Journal of Algebra, 251(2), 727-738. doi:10.1006/jabr.2001.9138 ; Beidleman, J. C., Brewster, B., & Robinson, D. J. S. (1999). Criteria for Permutability to Be Transitive in Finite Groups. Journal of Algebra, 222(2), 400-412. doi:10.1006/jabr.1998.7964 ; Beidleman, J. C., Heineken, H., & Ragland, M. F. ...

NOTICE: this is the author's version of a work that was accepted for publication in Journal of Algebra. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in JOURNAL OF ALGEBRA [VOL 466, (15 November 2016)] DOI 10.1016/j.jalgebra.2016.05.027 ; In this paper we obtain some bounds for the exponent of a finite group, and its derived subgroup, which is a mutually permutable product of two abelian subgroups. They improve the ones known for products of finite abelian groups, and they are used to derive some interesting structural properties of such products. ; The first author has been supported by the grant MTM2014-54707-C3-1-P from the Ministerio de Economia y Competitividad, Spain, and FEDER, European Union. He has been also supported by a project from the National Natural Science Foundation of China (NSFC, No. 11271085) and a project of Natural Science Foundation of Guangdong Province (No. 2015A030313791). ; Ballester Bolinches, A.; Cossey, J.; Pedraza Aguilera, MC. (2016). On the exponent of mutually permutable products of two abelian groups. Journal of Algebra. 466:34-43. https://doi.org/10.1016/j.jalgebra.2016.05.027 ; S ; 34 ; 43 ; 466

[EN] This paper has been published in Journal of Group Theory, 12(6):961-963 (2009). Copyright 2009 by Walter de Gruyter. The final publication is available at www.degruyter.com. http://dx.doi.org/10.1515/JGT.2009.026 http://www.degruyter.com/view/j/jgth.2009.12.issue-6/jgt.2009.026/jgt.2009.026.xml ; This is a corrigendum to "A Note on Finite PST-Groups" (J. Group Theory 10 (2007), 205-210). ; This research has been supported by the grants MTM2004-08219-C02-02 and MTM2007-68010-C03-02 from MEC (Spanish Government) and FEDER (European Union) and GV/2007/243 from Generalitat (València) ; http://dx.doi.org/10.1515/JGT.2009.026 ; Ballester Bolinches, A.; Esteban Romero, R.; Ragland, MF. (2009). Corrigendum: A note on finite PST-groups [J. Group Theory 10 (2007), 205-210]. Journal of Group Theory. 6(12). https://doi.org/10.1515/JGT.2009.026 ; 6 ; 12

The aim of this paper is to give answers to some questions concerning intersections of system normalisers and prefrattini subgroups of finite soluble groups raised by the third author, Shemetkov and Vasil'ev in the Kourovka Notebook [10]. Our approach depends on results on regular orbits and it can be also used to extend a result of Mann [9] concerning intersections of injectors associated to Fitting classes. ; The first and fourth authors have been supported by the grant MTM2014- 54707-C3-1-P from the Ministerio de Economía y Competitividad, Spain, and FEDER, European Union.

[EN] We give some sufficient conditions for a normal p-subgroup P of a finite group G to have every G-chief factor below it cyclic. The S-permutability of some p-subgroups of O^p(G)plays an important role. Some known results can be reproved and some others appear as corollaries of our main theorems. ; A. Ballester-Bolinches and R. Esteban-Romero have been supported by the Grant MTM2014-54707-C3-1-P from the Ministerio de Economia y Competitividad, Spain, and FEDER, European Union. A. Ballester-Bolinches and Y. Li have been supported by a project from the National Natural Science Foundation of China (NSFC, No. 11271085) and a project of Natural Science Foundation of Guangdong Province (No. 2015A030313791). L. Miao thanks the China Scholarship Council and for its financial support and the Department of Mathematics of the University of Valencia for its hospitality. ; Miao, L.; Ballester-Bolinches, A.; Esteban Romero, R.; Li, Y. (2017). On the supersoluble hypercentre of a finite group. Monatshefte für Mathematik. 184(4):641-648. https://doi.org/10.1007/s00605-016-0987-9 ; S ; 641 ; 648 ; 184 ; 4 ; Ballester-Bolinches, A., Esteban-Romero, R., Asaad, M.: Products of finite groups. In: de Gruyter Expositions in Mathematics, vol. 53. Walter de Gruyter, Berlin (2010). doi:10.1515/9783110220612 ; Ballester-Bolinches, A., Esteban-Romero, R., Qiao, S.H.: A note on a result of Guo and Isaacs about $$p$$ p -supersolubility of finite groups. Arch. Math. (Basel) 106, 501–506 (2016). doi:10.1007/s00013-016-0901-7 ; Berkovich, Y., Isaacs, I.M.: $$p$$ p -Groups stabilizing certain subgroups. J. Algebra 414, 82–94 (2014). doi:10.1016/j.jalgebra.2014.04.026 ; Chen, X., Guo, W., Skiba, A.N.: Some conditions under which a finite group belongs to a Baer-local formation. Commun. Algebra 42(10), 4188–4203 (2014). doi:10.1080/00927872.2013.806519 ; Doerk, K., Hawkes, T.: Finite soluble groups. In: De Gruyter Expositions in Mathematics, vol. 4. Walter de Gruyter, Berlin (1992). doi:10.1515/9783110870138 ; Gorenstein, D.: Finite Groups. ...

Our first main result proves that every element of order 4 of a Sylow 2-subgroup S of a minimal non-2-nilpotent group G, is a real element of S. This allows to give a character-free proof of a theorem due to Isaacs and Navarro (see [9, Theorem B]). As an application, the authors show a common extension of the p-nilpotence criteria proved in [3] and [9]. ; The first and the second authors have been supported by the grant MTM2014-54707-C3-1-P from the Ministerio de Economía y Competitividad, Spain, and FEDER, European Union. The first author has been also supported by a project from the National Natural Science Foundation of China (NSFC, No. 11271085) and a project of Natural Science Foundation of Guangdong Province, China (No. 2015A030313791)

[EN] In this survey we present some significant bounds for the number of maximal subgroups of a given index of a finite group. As a consequence, new bounds for the number of random generators needed to generate a finite d-generated group with high probability which are significantly tighter than the ones obtained in the paper of Jaikin-Zapirain and Pyber (Random generation of finite and profinite groups and group enumeration, Ann. Math., 183 (2011) 769-814) are obtained. The results of Jaikin-Zapirain and Pyber, as well as other results of Lubotzky, Detomi, and Lucchini, appear as particular cases of our theorems. ; This work has been supported by the research grants MTM2014-54707-C3-1-P from the Ministerio de Economa y Competitividad (Spanish Government) and FEDER (European Union) and PRO-METEO/2017/057 from Generalitat (Valencian Community, Spain). The fourth author has been supported by the grant number 201606890006 of the China Scholarship Council. ; Ballester-Bolinches, A.; Esteban Romero, R.; Jiménez-Seral, P.; Meng, H. (2020). The number of maximal subgroups and probabilistic generation of finite groups. International Journal of Group Theory. 9(1):31-42. https://doi.org/10.22108/ijgt.2019.114469.1521 ; S ; 31 ; 42 ; 9 ; 1

31 42 9 1 ; S ; [EN] In this survey we present some significant bounds for the number of maximal subgroups of a given index of a finite group. As a consequence, new bounds for the number of random generators needed to generate a finite d-generated group with high probability which are significantly tighter than the ones obtained in the paper of Jaikin-Zapirain and Pyber (Random generation of finite and profinite groups and group enumeration, Ann. Math., 183 (2011) 769-814) are obtained. The results of Jaikin-Zapirain and Pyber, as well as other results of Lubotzky, Detomi, and Lucchini, appear as particular cases of our theorems. This work has been supported by the research grants MTM2014-54707-C3-1-P from the Ministerio de Economa y Competitividad (Spanish Government) and FEDER (European Union) and PRO-METEO/2017/057 from Generalitat (Valencian Community, Spain). The fourth author has been supported by the grant number 201606890006 of the China Scholarship Council. Ballester-Bolinches, A.; Esteban Romero, R.; Jiménez-Seral, P.; Meng, H. (2020). The number of maximal subgroups and probabilistic generation of finite groups. International Journal of Group Theory. 9(1):31-42. https://doi.org/10.22108/ijgt.2019.114469.1521

The main goal in this paper is to use a dual equivalence in automata theory started in [25] and developed in [3] to prove a general version of the Eilenberg-type theorem presented in [4]. Our principal results confirm the existence of a bijective correspondence between three concepts; formations of monoids, formations of languages and formations of congruences. The result does not require finiteness on monoids, nor regularity on languages nor finite index conditions on congruences. We relate our work to other results in the field and we include applications to non-r-disjunctive languages, Reiterman s equational description of pseudovarieties and varieties of monoids. ; The authors gratefully acknowledge various discussions with Jean-Eric Pin. This work has been supported by the grants MTM2010-19938-C03-01 from the Ministerio de Ciencia e Innovacion (Spanish Government) and MTM2014-54707-C3-1-P from the Ministerio de Economia y Competitividad (Spanish Government) and FEDER (European Union). The first author has been supported by the grant No. 11271085 from the National Natural Science Foundation of China. The second author has been supported by the predoctoral grant AP2010-2764 from the Ministeriode Educacion (Spanish Government) and by an internship from CWI. ; Ballester Bolinches, A.; Cosme-Llópez, E.; Esteban Romero, R.; Rutten, J. (2015). Formations of monoids, congruences, and formal languages. Scientific Annals of Computer Science. 25(2):171-209. https://doi.org/10.7561/SACS.2015.2.171 ; S ; 171 ; 209 ; 25 ; 2

The final publication is available at Springer via http://dx.doi.org/10.1007/s00013-009-2976-x ; The aim of this paper is to prove certain characterization theorems for groups in which permutability is a transitive relation, the so called PT -groups. In particular, it is shown that the finite solvable PT -groups, the finite solvable groups in which every subnormal subgroup of defect two is permutable, the finite solvable groups in which every normal subgroup is permutable sensitive, and the finite solvable groups in which conjugatepermutability and permutability coincide are all one and the same class. This follows from our main result which says that the finite modular p-groups, p a prime, are those p-groups in which every subnormal subgroup of defect two is permutable or, equivalently, in which every normal subgroup is permutable sensitive. However, there exist finite insolvable groups which are not PT -groups but all subnormal subgroups of defect two are permutable. ; This paper was started and completed in two visits of the first author to the Department of Mathematics of the Australian National University, Canberra, Australia. He is grateful to the Department of Mathematics for its warm hospitality and financial support while this investigation was carried out. He also gratefully acknowledges financial support by Grant MTM2007-68010-C03-02 from MEC (Spanish Government) and FEDER. The research of the fourth author has been supported by the Grants MTM2007-68010-C03-02 from MEC (Spanish Government) and FEDER, and GV/2007/243 from Generalitat (Valencian Government). He thanks all these institutions for their financial support. ; Ballester Bolinches, A.; Beidleman, JC.; Cossey, J.; Esteban Romero, R.; Rangland, MF.; Schmidt, J. (2009). Permutable subnormal subgroups of finite groups. Archiv der Mathematik. 92(6):549-557. https://doi.org/10.1007/s00013-009-2976-x ; S ; 549 ; 557 ; 92 ; 6