BUSCADOR RECOLECTA

[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, El 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).

[EN] Kang and Liu ['On supersolvability of factorized finite groups', Bull. Math. Sci. 3 (2013), 205-210] investigate the structure of finite groups that are products of two supersoluble groups. The goal of this note is to give a correct proof of their main theorem., The first author was supported by the grant MTM2014-54707-C3-1-P from the Ministerio de Economia y Competitividad, Spain, and FEDER, European Union, and a project of Natural Science Foundation of Guangdong Province (No. 2015A030313791).

We extend to soluble FC∗ -groups, the class of generalised FC-groups introduced in [F. de Giovanni, A. Russo, G. Vincenzi, Groups with restricted conjugacy classes , Serdica Math. J. 28(3) (2002), 241 254], the characterisation of finite soluble T-groups obtained recently in [G. Kaplan, On T-groups, supersolvable groups and maximal subgroups , Arch. Math. 96 (2011), 19 25]., The first author has been supported by the research grant MTM2014-54707-C3-1-P from the Ministerio de Economia y Competitividad, Spain and FEDER, European Union.

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.

A classical result of Gaschütz affirms that given a finite A-generated group G and a prime p, there exists a group G# and an epimorphism phi: G# ---> G whose kernel is an elementary abelian p-group which is universal among all groups satisfying this property. This Gaschütz universal extension has also been described in the mathematical literature with the help of the Cayley graph. We give an elementary and self-contained proof of the fact that this description corresponds to the Gaschütz universal extension. Our proof depends on another elementary proof of the Nielsen-Schreier theorem, which states that a subgroup of a free group is free., This work has been supported by the grant MTM-2014-54707-C3-1-P of the Ministerio de Economia y Competitividad (Spain). The first author is also supported by Project No. 11271085 from the National Natural Science Foundation of China. The second author is supported by the predoctoral grant AP2010-2764 (Programa FPU, Ministerio de Educacion, Spain).

[EN] We obtain some structural properties of a factorised group G = AB, given that the conjugacy class sizes of certain elements in A boolean OR B are not divisible by p(2), for some prime p. The case when G = AB is a mutually permutable product is especially considered., The first author is supported by Proyecto Prometeo II/2015/011, Generalitat Valenciana (Spain), and the second author is supported by Proyecto MTM2014-54707-C3-1-P, Ministerio de Economia, Industria y Competitividad (Spain). The results in this paper are part of the third author's Ph.D. thesis, and he acknowledges the predoctoral grant ACIF/2016/170, Generalitat Valenciana (Spain).

[EN] The main purpose of this paper is to study mutually permutable products G = AB in which the subgroups of prime order p and cyclic of order 4 (if p = 2) of the largest normal subgroup of G contained in A boolean AND B are well situated in G. Our results confirm once again the important role of the intersection of the factors in the structural study of mutually permutable products., The first author is supported by the grant MTM2014-54707-C3-1-P from the Ministerio de Economia y Competitividad, Spain, and FEDER, European Union. The first and second authors are supported by Major Projects in Basic Research and Applied Research (Natural Science) of Guangdong Province (2017KZDXM058). The first and third authors are supported by Prometeo/2017/057 of Generalitat (Valencian Community, Spain). The second author is supported by Funds of Guangzhou Science and Technology (No: 201804010088). The fourth author is supported by the project of NSF China (11401597).

[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.

[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.

We give a complete classification of the finite groups with a unique subgroup of order p for each prime p dividing its order. All the groups considered in this paper will be finite.
One of the most fruitful lines in the research in abstract group theory during
the last years has been the study of groups in which the members of a certain
family of subgroups satisfy a certain subgroup embedding property. The family
of the subgroups of prime order (also called minimal subgroups) has attracted
the interest of many mathematicians. For example, a well-known result of Itˆo (see
[8, Kapitel III, Satz 5.3; 9]) states that a group of odd order with all minimal
subgroups in the center is nilpotent. The following result of Gasch¨utz and Itˆo (see
[5, Kapitel IV, Satz 5.7; 9]) gives interesting properties of groups with all minimal
subgroups normal., The first author has been supported by the research grant MTM2014-54707-C03-1-P from the Ministerio de Economia y Competitividad, Spain and FEDER, European Union. The research of the second author has been done during some visits to the Departament de Matematica Aplicada and the Institut de Matematica Pura i Aplicada of the Universitat Politecnica de Valencia and the Departament d' Algebra of the Universitat de Valencia. She expresses her gratitude to these institutions for the use of their facilities and, especially to the first university for the financial support of some of the visits.

[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.

[EN] We provide structural criteria for some finite factorised groups G = AB when the conjugacy class sizes in G of certain pi-elements in A boolean OR B are either pi-numbers or pi'-numbers, for a set of primes pi. In particular, we extend for products of groups some earlier results, M. J. Felipe is supported by Proyecto Prometeo II/2015/011, Generalitat Valenciana (Spain). A. Martínez-Pastor is supported by Proyecto MTM 2014-54707-C3-1-P, Ministerio de Economía, Industria y Competitividad (Spain), and by Proyecto Prometeo/2017/057, Generalitat Valenciana (Spain). The results in this paper are part of the V. M. Ortiz Sotomayor Ph.D. thesis, and he acknowledges the predoctoral grant ACIF/2016/170, Generalitat Valenciana (Spain). The authors are also supported by Proyecto PGC2018-096872-B-I00, Ministerio de Ciencia, Innovación y Universidades

[EN] In this note, we give an easy algorithm to construct the rational canonical form of a square matrix or an endomorphism h of a finite dimensional vector space which does not depend on either the structure theorem for finitely generated modules over principal ideal domains or matrices over the polynomial ring. The algorithm is based on the construction of an element whose minimum polynomial coincides with the minimum polynomial of the endomorphism and on the fact that the h-invariant subspace generated by such an element admits an h-invariant complement. It is also shown that this element can be easily obtained without the factorisation of a polynomial as a product of irreducible polynomials., This work has been supported by the grants MTM2014-54707-C3-1-P from the Ministerio de Economia y Competitividad, Spain, and FEDER, European Union, and PROMETEO/2017/057 from the Generalitat (Valencian Community, Spain). 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). The third author has been also supported by a predoctoral grant from the programme Atracció del talent of the Universitat de València.

[EN] Let the group G = AB be the product of the subgroups A and B. We determine some structural properties of G when the p-elements in A. B have prime power indices in G, for some prime p. More generally, we also consider the case that all prime power order elements in A. B have prime power indices in G. In particular, when G = A = B, we obtain as a consequence some known results., The first author is supported by Proyecto Prometeo II/2015/011, Generalitat Valenciana (Spain), and the second author is supported by Proyecto MTM2014-54707-C3-1-P, Ministerio de Economia, Industria y Competitividad (Spain). The results in this paper are part of the third author's Ph.D. thesis, and he acknowledges the predoctoral grant ACIF/2016/170, Generalitat Valenciana (Spain).

We say that a subgroup H of a finite group G is solitary (respectively, normal solitary) when it is a subgroup (respectively, normal subgroup) of G such that no other subgroup (respectively, normal subgroup) of G is isomorphic to H. A normal subgroup N of a group G is said to be quotient solitary when no other normal subgroup K of G gives a quotient isomorphic to G/N . We show some new results about lattice properties of these subgroups and their relation with classes of groups and present examples showing a negative answer to some questions about these subgroups., The first author has been supported by the research grants MTM2010-19038-C03-01 from Ministerio de Ciencia e Innovacion, Spain, MTM2014-54707-C3-1-P from Ministerio de Economia y Competitividad, Spain, and FEDER, European Union. The research of the second author has been done during some visits to the Departament de Matematica Aplicada and the Institut Universitari de Matematica Pura i Aplicada of the Universitat Politecnica de Valencia and the Departament d'Algebra of the Universitat de Valencia. She wants to express her gratitude to these institutions for the use of their facilities and, especially to the first university for the financial support of some of the visits.

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.

[EN] The solubility of a finite group with less than 6 non-supersoluble subgroups is confirmed in the paper. Moreover we prove that a finite insoluble group has exactly 6 non-supersoluble subgroups if and only if it is isomorphic to A5 or SL2 (5). Furthermore, it is shown that a finite insoluble group has exactly 22 non-nilpotent subgroups if and only if it is isomorphic to A5 or SL2 (5). This confirms a conjecture of Zarrin (Arch Math (Basel) 99:201 206, 2012)., The first and second author are supported by the Grant MTM2014-54707-C3-1-P from the Ministerio de Economia y Competitividad, Spain, and FEDER, European Union. The first author is supported by the National Natural Science Foundation of China (11271085) and a Project of Natural Science Foundation of Guangdong Province (2015A030313791). The third author is supported by the National Natural Science Foundation of China (11461007), and the Guangxi Natural Science Foundation Program (2016GXNSFAA380156). This research has been done during a visit of the third author to the Departament de Matematiques of the Universitat de Valencia. He expresses his gratitude to this institution. We thank the anonymous referee for his/her comments that have helped us to improve the presentation of the paper.

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).

[EN] 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., The research of the first author has been supported by the research grant MTM2014-54707-C3-1-P by the Ministerio de Economia 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.

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 Chin

[EN] A group G has finite (or Prufer or special) rank if every finitely generated subgroup of G can be generated by r elements and r is the least integer with this property. The aim of this paper is to prove the following result: assume that G=AB is a group which is the mutually permutable product of the abelian subgroups A and B of Prufer ranks r and s, respectively. If G is locally finite, then the Prufer rank of G is at most r+s+3. If G is an arbitrary group, then the Prufer rank of G is at most r+s+4., The first and third authors are supported by the Grant MTM2014-54707-C3-1-P from the Ministerio de Economia y Competitividad, Spain, and FEDER, European Union. The first and fourth authors are supported by Prometeo/2017/057 of Generalitat, Valencian Community, Spain. The third author is also supported by the predoctoral Grant 201606890006 from the China Scholarship Council. We are grateful to the referee of an earlier version of this paper for comments and suggestions that have lead to improvements in the bounds and their proofs.

[EN] An element g of a finite group G is said to be vanishing inG if there exists an irreducible character of G such that (g)=0; in this case, g is also called a zero of G. The aim of this paper is to obtain structural properties of a factorised group G=AB when we impose some conditions on prime power order elements gAB which are (non-)vanishing in G., The first author is supported by Proyecto PrometeoII/2015/011, Generalitat Valenciana (Spain). The second author is supported by Proyecto MTM2014-54707-C3-1-P, Ministerio de Economia, Industria y Competitividad (Spain), and by Proyecto Prometeo/2017/057, Generalitat Valenciana (Spain). The third author acknowledges the predoctoral grant ACIF/2016/170, Generalitat Valenciana Spain).

The first author would like to thank the Universitat de Valencia for its warm hospitality and financial support during the preparation of this paper.

In this paper we study groups G generated by two subgroups A and B such that <. a, b> is nilpotent of class at most 2 for all a¿. A and b¿. B. A detailed description of the structure of such groups is obtained, generalizing the classical result of Hopkins and Levi on 2-Engel groups.

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.