Paper “On hall subnormally embedded subgroups of finite groups” published in Monatsh. Math.

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Adolfo Ballester-Bolinches, John Cossey, and ShouHong Qiao.

On Hall subnormally embedded subgroups of finite groups.

Monatsh. Math., 181(4):753–760, 2016

https://doi.org/10.1007/s00605-015-0838-0

Abstract

A subgroup H of a finite group G is said to be Hall subnormally (respectively normally) embedded in G if there is a subnormal (respectively normal) subgroup N of G such that H is a Hall subgroup of N. The aim of this paper is to characterise the groups G having a Hall subnormally embedded subgroup of order |B| for each subgroup B of G. Some earlier results are consequences of our main theorem.

2010 Mathematical Subject Classification: 20D10 20D20

Keywords: Finite group, Soluble group, Hall subgroup, Subnormal subgroup

Paper “On the exponent of mutually permutable products of two abelian groups” published in J. Algebra

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A. Ballester-Bolinches, John Cossey, and M. C. Pedraza-Aguilera.

On the exponent of mutually permutable products of two abelian groups.

J. Algebra, 466:34–43, 2016.

https://doi.org/10.1016/j.jalgebra.2016.05.027

Abstract

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.

2010 Mathematical Subject Classification: 20D10, 20D20

Keywords: Finite group, Abelian group, Exponent, Factorisations, p-Supersolubility, p-Length

Paper “A note on finite groups with the maximal permutiser condition” published in Rev. R. Acad. Cienc. Exactas Fí s. Nat. Ser. A Math. RACSAM

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Adolfo Ballester-Bolinches, John Cossey, and ShouHong Qiao.

A note on finite groups with the maximal permutiser condition.

Rev. R. Acad. Cienc. Exactas Fí s. Nat. Ser. A Math. RACSAM, 110(1):247–250, 2016

https://doi.org/10.1007/s13398-015-0232-8

Abstract

A finite group G is said to satisfy the maximal permutiser condition, or G is an MPC-group, if for any maximal subgroup M of G, there is an element xGM such that G=Mx⟩. In this note, we show that the class of MPC-groups is not residually closed and so it is not a formation. It answers a question posed in Qiao et al. (J Algebra Appl 12(5):1250217, 2013). Following Ballester-Bolinches and Esteban-Romero (Commun Algebra 30(12):5757–5770, 2002), a finite group G is said to be a QP-group if G is soluble and if F is a non-cyclic chief factor of G, then F has order 4 and G induces the full automorphism group in F. We prove that the class of all QP-groups is the unique largest formation contained in the class of all MPC-groups. A detailed description of the MPC-groups is also given.

2010 Mathematics Subject Classification: 20D10, 20D15

Keywords: Finite group, Soluble group, Permutability, Formations

Paper “On the abnormal structure of finite groups” published in Revista Matemática Iberoamericana

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Adolfo Ballester-Bolinches, John Cossey, Ramón Esteban-Romero

On the abnormal structure of finite groups

Rev. Mat. Iberoamericana., 30, 13-24 (2014)

http://dx.doi.org/10.4171/rmi/767

Abstract: 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.


MSC: 20D10, 20D05, 20F16
Keywords: Finite group, supersoluble group, maximal subgroup

Paper “Generalised norms in finite soluble groups” published in J. Algebra

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A. Ballester-Bolinches, John Cossey, Liangcai Zhang

Generalised norms in finite soluble groups

J. Algebra, 402, 392-405 (2014)

http://dx.doi.org/10.1016/j.jalgebra.2013.12.012

Abstract: We give a framework for a number of generalisations of Baer’s norm that have appeared recently. For a class C of finite nilpotent groups we define the C-norm κC(G) of a finite group G to be the intersection of the normalisers of the subgroups of G that are not in C. We show that those groups for which the C-norm is not hypercentral have a very restricted structure. The non-nilpotent groups G for which G = κC (G) have been classified for some classes. We give a classification for nilpotent classes closed under subgroups, quotients and direct products of groups of coprime order and show the known classifications can be deduced from our classification.


MSC: 20D10, 20D20, 20D25
Keywords: Finite group, normaliser, norm

Paper “Graphs and classes of finite groups” published in Note Mat.

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A. Ballester-Bolinches, John Cossey, R. Esteban-Romero

Graphs and classes of finite groups

Note Mat., 33 (1), 89-94 (2013)

http://dx.doi.org/10.1285/i15900932v33n1p89

Abstract. 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.
Keywords: finite groups, classes of groups, graphs
MSC 2000 classification: primary 20D10, secondary 05C25

Paper “Mutually permutable products and conjugacy classes” to appear in Monatsh. Math.

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A. Ballester-Bolinches, John Cossey, Yangming Li

Mutually permutable products and conjugacy classes

Monatsh. Math.

http://dx.doi.org/10.1007/s00605-012-0411-z

Abstract: The question of how certain arithmetical conditions on the lengths of the conjugacy classes of a finite group G influence the group structure has been studied by several authors with many results available. The purpose of this paper is to analyse the restrictions imposed by the lengths of the conjugacy classes of some elements of the factors of a finite group G = G 1G2 · · · Gr , which is the product of the pairwise mutually permutable subgroups G 1, G 2, . . . , Gr , on its structure. Some earlier results appear as corollaries of our main theorems.

Keywords: Finite groups, Mutually permutable products, Conjugacy classes.
Mathematics Subject Classification: 20D10, 20D20, 20D40, 20E45