# 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 “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 “Some Characterisations of Soluble SST-Groups” published in Comm. Algebra

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

Some characterisations of soluble SST-groups.

Comm. Algebra, 44(4):1821–1827, 2016.

https://doi.org/10.1080/00927872.2015.1027397

Abstract

All groups considered in this paper are finite. A subgroup H of a group G is said to be SS-permutable or SS-quasinormal in G if H has a supplement K in G such that H permutes with every Sylow subgroup of K. Following [6 Chen, X. Y., Guo, W. B. (2014). Finite groups in which SS-permutability is a transitive relation. Acta Math. Hungar. 143(2):466–479.], we call a group G an SST-group provided that SS-permutability is a transitive relation in G, that is, if A is an SS-permutable subgroup of B and B is an SS-permutable subgroup of G, then A is an SS-permutable subgroup of G. The main aim of this paper is to present several characterisations of soluble SST-groups.

2010 Mathematics Subject Classification: Primary 20D10; Secondary 20D15, 20D20

Keywords: BT-group, Finite group, Soluble PST-group, SST-group

# Paper “On partial CAP-subgroups of finite groups” published in J. Algebra

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

On partial CAP-subgroups of finite groups

J. Algebra, 431 (2015), 196–208

http://dx.doi.org/10.1016/j.jalgebra.2015.01.035

Abstract

Given a chief factor H/K of a finite group G, we say that a subgroup A of G avoids H/K if H∩A=K∩A; if HA=KA, then we say that A covers H/K. If A either covers or avoids the chief factors of some given chief series of G, we say that A is a partial CAP-subgroup of G. Assume that G has a Sylow p-subgroup of order exceeding pk. If every subgroup of order pk, where k≥1, and every subgroup of order 4 (when pk=2 and the Sylow 2-subgroups are non-abelian) are partial CAP-subgroups of G, then G is p-soluble of p-length at most 1.

2010 Mathematics subject classification: 20D10; 20D20

Keywords: Finite group; Partial CAP-subgroup; p-soluble group; p-length

# Paper “A note on solubly saturated formations of finite groups” published in J. Algebra Appl.

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A. Ballester-Bolinches, S. F. Kamornikov

A note on solubly saturated formations of finite groups

J. Algebra Appl., 14(4) (2015), 1550047, 4

http://dx.doi.org/10.1142/S0219498815500474

Abstract

The main aim of this note is to give a criterion for a subgroup-closed formation to be solubly saturated, which we hope may provide a useful proving ground for outstanding questions about this family of formations.

2010 Mathematics subject classification20D10, 20D20

KeywordsFinite group; formation; saturation; soluble saturation

# 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

# Paper “On a class of generalised Schmidt groups” published in Ann. Mat. Pura Appl.

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A. Ballester-Bolinches, R. Esteban-Romero, Qinhui Jiang, and Xianhua Li

On a class of generalised Schmidt groups

Ann. Mat. Pura Appl. (4), 194(1) (2015), 77–86

http://dx.doi.org/10.1007/s10231-013-0365-3

Abstract

In this paper families of non-nilpotent subgroups covering the non-nilpotent part of a finite group are considered. An A_5-free group possessing one of these families is soluble, and soluble groups with this property have Fitting length at most three. A bound on the number of primes dividing the order of the group is also obtained.

2010 Mathematics subject classification: 20D05; 20D10; 20F16

Keywords: finite groups; nilpotent groups; maximal subgroups

# Paper “Subgroup embedding properties and the structure of finite groups” published in Note Mat.

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A. Ballester-Bolinches

Subgroup embedding properties and the structure of finite groups

Note Mat., 34(1) (2014), 35–52

http://dx.doi.org/10.1285/i15900932v34n1p35

Abstract

Our main aim in this paper is to present some results to help us better understand some different ways a subgroup can be embedded in a finite group and their impact on the group structure

2010 Mathematics subject classification: 20D10; 20D20

Keywords and phrasesfinite group; subgroup embedding property; supplements; Schunck classes; formations; hypercentral groups; p-length; p-nilpotency

# Paper “On the p-length of some finite p-soluble groups” published in Israel J. Math.

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Adolfo Ballester-Bolinches, Ramón Esteban-Romero, Luis M. Ezquerro

On the p-length of some finite p-soluble groups

Israel J. Math., 204(1) (2014), 359–371

http://dx.doi.org/10.1007/s11856-014-1095-y

Abstract

The main aim of this paper is to give structural information of a finite group of minimal order belonging to a subgroup-closed class of finite groups and whose p-length is greater than 1, p a prime number. Alternative proofs and improvements of recent results about the influence of minimal p-subgroups on the p-nilpotence and p-length of a finite group arise as consequences of our study.