Paper “On complements of F-residuals of finite groups” published in Comm. Algebra

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A. Ballester-Bolinches, S. F. Kamornikov, and V. Pérez-Calabuig

On Complements of F-residuals of finite groups

Comm. Algebra, 45(2):878–882, 2017.

https://doi.org/10.1080/00927872.2016.1175615

Abstract

A formation F of finite groups has the generalized Wielandt property for residuals, or is a GWP-formation, if the F-residual of a group generated by two F-subnormal subgroups is the subgroup generated by their F-residuals. The main aim of the paper is to determine some sufficient conditions for a finite group to split over its F-residual.

2010 Mathematics subject classification: 20D10; 20D20

Keywords: Finite group; formation; residual; subnormality

Paper “Normalisers of residuals of finite groups” published in Arch. Math. (Basel)

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A. Ballester-Bolinches, S. F. Kamornikov, and H. Meng

Normalisers of residuals of finite groups

Arch. Math. (Basel), 109(4):305–310, 2017

https://doi.org/10.1007/s00013-017-1074-8

Abstract:

Let F be a subgroup-closed saturated formation of finite groups containing all finite nilpotent groups, and let M be a subgroup of a finite group G normalising the F-residual of every non-subnormal subgroup of G. We show that M normalises the F-residual of every subgroup of G. This answers a question posed by Gong and Isaacs (Arch Math 108:1–7, 2017) when F is the formation of all finite supersoluble groups.

2010 Mathematics Subject Classification: 20D10, 20D35

Keywords: Finite group, Formation, Residual, Subnormality

Paper “On locally finite groups whose subgroups of infinite rank have some permutable property” published in Ann. Mat. Pura Appl.

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A. Ballester-Bolinches, S. Camp-Mora, M. R. Dixon, R. Ialenti, and F. Spagnuolo

On locally finite groups whose subgroups of infinite rank have some permutable property

Ann. Mat. Pura Appl. (4), 196(5):1855–1862, 2017

https://doi.org/10.1007/s10231-017-0642-7

Abstract

In this paper, we study the behavior of locally finite groups of infinite rank whose proper subgroups of infinite rank have one of the three following properties, which are generalizations of permutability: S-permutability, semipermutability and S-semipermutability. In particular, it is proved that if G is a locally finite group of infinite rank whose proper subgroups of infinite rank are S-permutable (resp. semipermutable), then G is locally nilpotent (resp. all subgroups are semipermutable). For locally finite groups whose proper subgroups of infinite rank are S-semipermutable, the same statement can be proved only for groups with min-p for every prime p. A counterexample is given for the general case.

2010 Mathematical Subject Classification: 20F19, 20F50

Keywords: Locally finite group, Section p-rank, Section rank, Special rank, Permutable, Sylow permutable, Semipermutable, S-semipermutable

 

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 “Some Results on Products of Finite Groups” published in Bull. Malays. Math. Sci. Soc.

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Adolfo Ballester-Bolinches, Luis M. Ezquerro, A. A. Heliel, and M. M. Al-Shomrani

Some results on products of finite groups

Bull. Malays. Math. Sci. Soc., 40(3):1341–1351, 2017

https://doi.org/10.1007/s40840-015-0111-7

Abstract

Subgroups A and B of a finite group are said to be mutually permutable (respectively, M-permutable and sn-permutable) if A permutes with every subgroup (respectively, every maximal subgroup and every subnormal subgroup) of B and viceversa. If every subgroup of A permutes with every subgroup of B, then the product is said to be totally permutable. These kinds of products have received much attention in the last twenty years. The aim of this paper is to analyse the behaviour of finite pairwise mutually permutable, mutually M-permutable, mutually sn-permutable and totally permutable products with respect to certain classes of groups including the supersoluble groups, widely supersoluble groups, and also the classes of PST-, PT– and T-groups.

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

Keywords: Finite group, Permutability, Products of groups,  Supersoluble group

 

 

 

Paper “On seminormal subgroups of finite groups” published in Rocky Mountain J. Math.

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A. Ballester-Bolinches, J. C. Beidleman, V. Pérez-Calabuig, and M. F. Ragland

On seminormal subgroups of finite groups

Rocky Mountain J. Math., 47(2):419–427, 2017

https://doi.org/10.1216/RMJ-2017-47-2-419

Abstract

All groups considered in this paper are finite. A subgroup H of a group G is said to be seminormal in G if H is normalized by all subgroups K of G such that gcd(|H|,|K|)=1 . We call a group G an MSN-group if the maximal subgroups of all the Sylow subgroups of G are seminormal in G. In this paper, we classify all MSN-groups.

2010 Mathematics Subject Classification: 20D10, 20D15, 20D20

Keywords: Finite group, soluble PST-group,T₀-group, MS-group, MSN-group

 

Paper “On S-Semipermutable Subgroups and Soluble PST-Groups” published in Mediterr. J. Math.

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R. A. Hijazi, W. M. Fakieh, A. Ballester-Bolinches, and J. C. Beidleman

On S-semipermutable subgroups and soluble PST-groups

Mediterr. J. Math., 14(2):Art. 87, 6, 2017

https://doi.org/10.1007/s00009-017-0893-y

Abstract

All groups presented in this article are finite. Using several permutability embedding properties, a number of new characterisations of soluble PST-groups are studied.

2010 Mathematics subject classification:  20D10; 20D20; 20F16

Keywords: Finite group; Permutability; S-Semipermutability

 

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 subgroup functors of finite soluble groups” published in Sci. China Math.

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Adolfo Ballester-Bolinches, Enric Cosme-Llópez, Sergey Fedorovich Kamornikov

On subgroup functors of finite soluble groups.

Sci. China Math., 60(3):439–448, 2017

https://doi.org/10.1007/s11425-015-0330-9

Abstract

The principal aim of this paper is to study the regular and transitive subgroup functors in the universe of all finite soluble groups. We prove that they form a complemented and non-modular lattice containing two relevant sublattices. This is the answer to a question (Question 1.2.12) proposed by Skiba (1997) in the finite soluble universe.

2010 Mathematics subject classification: 20D10; 20D30

Keywords: finite group; soluble group; lattices of subgroups; subgroup functors; formations

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