Paper “On the focal subgroup of a saturated fusion system” published in J. Algebra

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Adolfo Ballester-Bolinches, Luis M. Ezquerro, Ning Su, and Yanming Wang.

On the focal subgroup of a saturated fusion system.

J. Algebra, 468:72–79, 2016

https://doi.org/10.1016/j.jalgebra.2016.07.034

Abstract

The influence of the cyclic subgroups of order p or 4 of the focal subgroup of a saturated fusion system F over a p-group S is investigated in this paper. Some criteria for normality of S in F as well as necessary and sufficient conditions for nilpotency of F are given. The resistance of a p-group in which every cyclic subgroup of order p or 4 is normal, and earlier results about p-nilpotence of finite groups and nilpotency of saturated fusion systems are consequences of our study.

2010 Mathematical Subject Classification: 20C20

Keywords: Saturated fusion system, Control of fusion

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 “On finite p-nilpotent groups” published in Monatsh. Math.

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Adolfo Ballester-Bolinches, Xiuyun Guo, Yangming Li, and Ning Su.

On finite p-nilpotent groups.

Monatsh. Math., 181(1):63–70, 2016

https://doi.org/10.1007/s00605-015-0803-y

Abstract

In this paper the structure of a minimal counterexample among the non-p-nilpotent groups having p-nilpotent p-Sylow normalisers is analysed. Several p-nilpotency criteria and many earlier results follow from our main theorem.

2010 Mathematics Subject Classification: 20D10, 20D20

Keywords: Finite groups, p-nilpotency, Minimal subgroups, Sylow normalisers

Paper “On bounds for the p-length of finite p-soluble groups” published in Collect. Math.

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Adolfo Ballester-Bolinches, Luis M. Ezquerro, and Ning Su.

On bounds for the p-length of finite p-soluble groups.

Collect. Math., 67(3):373–378, 2016.

https://doi.org/10.1007/s13348-015-0144-0

Abstract

The aim of this paper is to obtain a bound for the p-length of a p-soluble group G whose elements of order p or order 4 (if p=2) of a Sylow p-subgroup of a residual subgroup of G are contained in the k-th term of the upper central series of a Sylow p-subgroup of G.

2010 Mathematics Subject Classification: 20D10, 20D20

Keywords: Finite group, p-soluble group, p-length

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 groups whose subgroups of infinite rank are Sylow permutable” published in Ann. Mat. Pura Appl.

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A. Ballester-Bolinches, S. Camp-Mora, L. A. Kurdachenko, and F. Spagnuolo.

On groups whose subgroups of infinite rank are Sylow permutable.

Ann. Mat. Pura Appl. (4), 195(3):717–723, 2016.

https://doi.org/10.1007/s10231-015-0485-z

Abstract

In this paper, we investigate the structure of locally finite groups of infinite section rank (respectively, special rank) whose subgroups of infinite section rank (respectively, special rank) are Sylow permutable, permutable or normal. Some earlier results for locally finite groups appear as consequences of our study.

2010 Mathematics Subject Classification: 20E15, 20F19, 20F22

Keywords: Locally finite group, Section p-rank, Section rank, Special rank, Permutable, Sylow permutable, Normal

Paper “Triple Factorizations and Supersolubility of Finite Groups” published on Proc. Edinb. Math. Soc.

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Adolfo Ballester-Bolinches and Luis M. Ezquerro.

Triple factorizations and supersolubility of finite groups.

Proc. Edinb. Math. Soc. (2), 59(2):301–309, 2016.

https://doi.org/10.1017/S0013091515000231

Abstract

In this paper we analyse the structure of a finite group of minimal order among the finite non-supersoluble groups possessing a triple factorization by supersoluble subgroups of pairwise relatively prime indices. As an application we obtain some sufficient conditions for a triple factorized group by supersoluble subgroups of pairwise relatively prime indices to be supersoluble. Many results appear as consequences of our analysis.

Keywords: Finite group, Supersoluble group, Factorization

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