Paper “Finite groups with all minimal subgroups solitary” published in J. Algebra Appl.

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R. Esteban-Romero and Orieta Liriano.

Finite groups with all minimal subgroups solitary.

J. Algebra Appl., 15(8):1650140, 9, 2016

https://doi.org/10.1142/S0219498816501401

Abstract 

We give a complete classification of the finite groups with a unique subgroup of order p for each prime p dividing its order.

2010 Mathematical Subject Classification: 20D10, 20D30

Keywords: Finite group; solitary subgroup; minimal subgroup

Paper “Some Local Properties Defining T₀-Groups and Related Classes of Groups” published in Publ. Mat.

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A. Ballester-Bolinches, J. C. Beidleman, R. Esteban-Romero, and M. F. Ragland

Some local properties defining T0-groups and related classes of groups

Publ. Mat., 60(1):265–272, 2016

http://projecteuclid.org/euclid.pm/1450818490

Abstract

We call G a Hall_χ-group if there exists a normal nilpotent subgroup N of G for which G/N is an χ-group. We call G a T-group provided G/Φ(G) is a T-group, that is, one in which normality is a transitive relation. We present several new local classes of groups which locally define Hall_χ-groups and T-groups where χ{T, PT, PST}; the classes PT and PST denote, respectively, the classes of groups in which permutability and S-permutability are transitive relations.

2010 Mathematical Subject Classification: 20D10, 20D20, 20D35

Keywords: Subnormal subgroup, T-group, PST-group, finite solvable group

 

Paper “A Note on Solitary Subgroups of Finite Groups” published in Comm. Algebra

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R. Esteban-Romero and Orieta Liriano

A note on solitary subgroups of finite groups.

Comm. Algebra, 44(7):2945–2952, 2016

https://doi.org/10.1080/00927872.2015.1065855

Abstract

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.

2010 Mathematics Subject Classification: 20D10, 20D30, 20F16

Keywords: Finite group, Fitting class, Formation, Quotient solitary subgroup, Solitary subgroup

Paper “On finite groups with many supersoluble subgroups” published in Arch. Math. (Basel)

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A. Ballester-Bolinches, R. Esteban-Romero, and Jiakuan Lu

On finite groups with many supersoluble subgroups

Arch. Math. (Basel), 109(1):3–8, 2017

https://doi.org/10.1007/s00013-017-1041-4

Abstract

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 A_5 or SL_2 (5). Furthermore, it is shown that a finite insoluble group has exactly 22 non-nilpotent subgroups if and only if it is isomorphic to A_5 or SL_2(5). This confirms a conjecture of Zarrin (Arch Math (Basel) 99:201–206, 2012).

2010 Mathematics Subject Classification: 20D10, 20D20

Keywords: Finite group, Supersoluble subgroup, Soluble group

 

Paper “On the supersoluble hypercentre of a finite group” published in Monatsh. Math.

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Liyun Miao, Adolfo Ballester-Bolinches, Ramón Esteban-Romero, and Yangming Li

On the supersoluble hypercentre of a finite group

Monatsh. Math., 184(4):641–648, 2017

https://doi.org/10.1515/forum-2016-0262

Abstract

A subgroup H of a group G is called Sylow permutable, or S-permutable, in G if H permutes with all Sylow p-subgroups of G for all primes p. A group G is said to be a PST-group if Sylow permutability is a transitive relation in G. We show that a group G which is factorised by a normal subgroup and a subnormal PST-subgroup of odd order is supersoluble. As a consequence, the normal closure S^G of a subnormal PST-subgroup S of odd order of a group G is supersoluble, and the subgroup generated by subnormal PST-subgroups of G of odd order is supersoluble as well.

2010 Mathematics Subject Classification: 20D10, 20D20

Keywords: Finite group, p-Supersoluble group, S-semipermutable subgroup

Paper “Group extensions and graphs” published in Expo. Math.

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A. Ballester-Bolinches, E. Cosme-Llópez, and R. Esteban-Romero.

Group extensions and graphs.

Expo. Math., 34(3):327–334, 2016

https://doi.org/10.1016/j.exmath.2015.07.005

Abstract

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 φ: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.

2010 Mathematical Subject Classification: Primary 20F65; Secondary 05C25, 20D20, 20E22, 20F05, 20F10

Keywords: Group, Group extension, Graph

Paper “On generalised FC-groups in which normality is a transitive relation” published in J. Aust. Math. Soc.

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R. Esteban-Romero and G. Vincenzi.

On generalised FC-groups in which normality is a transitive relation.

 J. Aust. Math. Soc., 100(2):192–198, 2016

https://doi.org/10.1017/S1446788715000397

Abstract

We extend to soluble FC-groups, the class of generalised FC-groups introduced in de Giovanni et al. [‘Groups with restricted conjugacy classes’, Serdica Math. J. 28(3) (2002), 241–254], the characterisation of finite soluble T-groups obtained recently in Kaplan [‘On T-groups, supersolvable groups, and maximal subgroups’, Arch. Math. (Basel) 96(1) (2011), 19–25].

2010 Mathematical Subject Classification: Primary 20F24; Secondary 20E34, 20F14, 20F19

Paper “A note on a result of Guo and Isaacs about p-supersolubility of finite groups” published in Arch. Math. (Basel)

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Adolfo Ballester-Bolinches, Ramón Esteban-Romero, and ShouHong Qiao.

A note on a result of Guo and Isaacs about p-supersolubility of finite groups.

Arch. Math. (Basel), 106(6):501–506, 2016.

https://doi.org/10.1007/s00013-016-0901-7

Abstract

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 |H|. 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 |G| such that 1d<|P|, if HO^p(G) is S-semipermutable in O^p(G) for all normal subgroups H of P with |H|=d, then either G is p-supersoluble or else |PO^p(G)|>d. 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.

2010 Mathematical Subject Classification: 20D10, 20D20

Keywords: Finite group, p-supersoluble group, S-semipermutable subgroup

Paper “Z-permutable subgroups of finite groups” published in Monatsh. Math.

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A. A. Heliel, A. Ballester-Bolinches, R. Esteban-Romero, and M. O. Almestady.

Ζ-permutable subgroups of finite groups.

Monatsh. Math., 179(4):523–534, 2016

https://doi.org/10.1007/s00605-015-0756-1

Abstract

Let be a complete set of Sylow subgroups of a finite group G, that is, a set composed of a Sylow p-subgroup of G for each p dividing the order of G. A subgroup H of G is called -permutable if H permutes with all members of . The main goal of this paper is to study the embedding of the -permutable subgroups and the influence of -permutability on the group structure.

2010 Mathematics Subject Classification: 20D10, 20D20, 20D35, 20D40

Keywords: Finite group, p-soluble group, p-supersoluble, ℨ-permutable subgroup, Subnormal subgroup

Paper “Some subgroup embeddings in finite groups: A mini-review” published in J. Adv. Res.

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A. Ballester-Bolinches, J. C. Beidleman, R. Esteban-Romero, M. F. Ragland

Some subgroup embeddings in finite groups: A mini-review

J. Adv. Res., 6(3) (2015), 359–362

http://dx.doi.org/10.1016/j.jare.2014.04.004

Abstract

In this survey paper several subgroup embedding properties related to some types of permutability are introduced and studied.

Keywords and phrases: Finite group; Permutability; S-permutability; Semipermutability; Primitive subgroup; Quasipermutable subgroup