The embedding of a cyclic permutable subgroup in a finite group. II

In two previous papers we established the structure of the normal closure of a cyclic permutable subgroup $A$ of a finite group, first when $A$ has odd order and second when $A$ has even order, but with an extra hypothesis that was unnecessary in the odd case. Here we describe the most general situation without any restrictions on $A$

Cyclic permutable subgroups of finite groups

The authors describe the structure of the normal closure of a cyclic permutable subgroup of odd order in a finite group

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

Graphs and classes of finite groups

There are different ways to associate to a finite group a certain graph. An interesting question is to analyse the relations between the structure of the group, given in group-theoretical terms, and the structure of the graph, given in the language of graph theory. This survey paper presents some contributions to this research line

On the abnormal structure of finite groups

We study finite groups in which every maximal subgroup is supersoluble or normal. Our results answer some questions arising from papers of Asaad and Rose.Ballester Bolinches, A.; Cossey, J.; Esteban Romero, R. (2014). On the abnormal structure of finite groups. Revista Matemática Iberoamericana. 30(1):13-24. doi:10.4171/rmi/767S132430

On the exponent of mutually permutable products of two abelian groups

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.2016.05.027In 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.2016.05.027S344346

On two questions from the Kourovka Notebook

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

On the Prufer rank of mutually permutable products of abelian groups

[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.Ballester-Bolinches, A.; Cossey, J.; Meng, H.; Pedraza Aguilera, MC. (2019). On the Prufer rank of mutually permutable products of abelian groups. Annali di Matematica Pura ed Applicata (1923 -). 198(3):811-819. https://doi.org/10.1007/s10231-018-0800-6S8118191983Amberg, B., Franciosi, S., De Giovanni, F.: Products of Groups, vol. 992. Clarendon Press, Oxford (1992)Amberg, B., Kazarin, L.S.: On the rank of a product of two finite $$p$$ p -groups and nilpotent $$p$$ p -algebras. Commun. Algebra 27(8), 3895–3907 (1999)Amberg, B., Sysak, Y.P.: Locally soluble products of two subgroups with finite rank. Commun. Algebra 24(7), 2421–2445 (1996)Ballester-Bolinches, A., Esteban-Romero, R., Asaad, M.: Products of Finite Groups, vol. 53. Walter de Gruyter, Berlin/New York (2010)Baumslag, G., Bieri, R.: Constructable solvable groups. Math. Z. 151(3), 249–257 (1976)Beidleman, J., Heineken, H.: Totally permutable torsion subgroups. J. Group Theory 2, 377–392 (1999)Beidleman, J., Heineken, H.: A survey of mutually and totally permutable products in infinite groups, topics in infinite groups. Quad. Mat 8, 45–62 (2001)Cooper, C.D.: Power automorphisms of a group. Math. Z. 107(5), 335–356 (1968)Dixon, M.R.: Sylow Theory, Formations, and Fitting Classes in Locally Finite Groups, vol. 2. World Scientific, Singapore (1994)Huppert, B.: Endliche Gruppen I, vol. 134. Springer, Berlin, Heidelberg (1967). https://doi.org/10.1007/978-3-642-64981-3Janko, Z.: Finite 2-groups with exactly one nonmetacyclic maximal subgroup. Isr. J. Math. 166(1), 313–347 (2008)Linnell, P.A., Warhurst, D.: Bounding the number of generators of a polycyclic group. Arch. Math. 37(1), 7–17 (1981)Lucchini, A.: A bound on the number of generators of a finite group. Arch. Math. 53(4), 313–317 (1989)Lucchini, A.: A bound on the presentation rank of a finite group. Bull. Lond. Math. Soc. 29(4), 389–394 (1997

Permutable subnormal subgroups of finite groups

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 conjugate-permutability 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

A characterisation via graphs of the soluble groups in which permutability is transitive

[EN] There are different ways to associate to a group a certain graph. In this context, it is interesting to ask for the relations between the structure of the group, given in group-theoretical terms, and the structure of the graphs, given in the language of graph theory. In this paper we recall some properties of the groups in which permutability is a transitive relation and present a new characterisation of the class of soluble groups in which permutability is a transitive relation in graph-theoretical terms.This paper has been suported by the research grants MTM2004-08219-C02-02 and MTM2007-68010-C03-02 from MEC (Spain) and FEDER (European Union), and GV/2007/243 from Generalitat (València).Ballester Bolinches, A.; Cossey, J.; Esteban Romero, R. (2009). A characterisation via graphs of the soluble groups in which permutability is transitive. Algebra and Discrete Mathematics. 4:10-17. http://hdl.handle.net/10251/73630S1017