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 “A note on Sylow permutable subgroups of infinite groups” published in J. Algebra

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A. Ballester-Bolinches, S. Camp-Mora, L. A. Kurdachenko

A note on Sylow permutable subgroups of infinite groups

J. Algebra, 398, 156-161 (2014)

http://dx.doi.org/10.1016/j.jalgebra.2013.08.042

Abstract: A subgroup A of a periodic group G is said to be Sylow permutable,
or S-permutable, subgroup of G if A P = P A for all Sylow subgroups
P of G. The aim of this paper is to establish the local nilpotency
of the section A^G /Core_G( A) for an S-permutable subgroup A of a
locally finite group G.
MSC: 20E15, 20F19, 20F22
Keywords: Locally finite group, Hyperfinite group, Sylow permutability, Ascendant subgroup

Paper “Groups with every subgroup ascendant-by-finite” published in Cent. Eur. J. Math.

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Sergio Camp-Mora

Groups with every subgroup ascendant-by-finite

Cent. Eur. J. Math., 11(12), 2182-2185 (2013)

http://dx.doi.org/10.2478/s11533-013-0312-y

Abstract: A subgroup H of a group G is called ascendant-by-finite in G if there exists a subgroup K of H such that K is ascendant in G and the index of K in H is finite. It is proved that a locally finite group with every subgroup ascendant-by-finite is locally nilpotent-by-finite. As a consequence, it is shown that the Gruenberg radical has finite index in the whole group.

MSC:  20F19, 20F22, 20F50
Keywords: Ascendant subgroup, Locally nilpotent, Radical, Locally finite group

Paper “Finite solvable groups in which semi-normality is a transitive relation” published in Beitr. Algebra Geom.

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A. Ballester-Bolinches, J. C. Beidleman, A. D. Feldman, H. Heineken, M. F. Ragland

Finite solvable groups in which semi-normality is a transitive relation

Beitr. Algebra Geom., 54(2), 549-558 (2013)

http://dx.doi.org/10.1016/j.jalgebra.2013.08.042

Abstract: A subgroup H of a finite group G is said to be seminormal in G if every Sylow p-subgroup of G, p a prime, with (|H|, p) = 1 normalizes H. A group G is called an SNT-group if seminormality is a transitive relation in G. Properties of solvable SNT-groups are studied. For example, subgroups of solvable SNT-groups are SNT-groups.
MSC: 20D05, 20D10, 20F16
Keywords: Finite groups, S-permutability, S-semipermutability, seminormal.

Paper “Algorithms for permutability in finite groups” published in Cent. Eur. J. Math.

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

Algorithms for permutability in finite groups

Cent. Eur. J. Math., 11 (11), 1914-1922 (2013).

Abstract: In this paper we describe some algorithms to identify permutable and Sylow-permutable subgroups of finite groups, Dedekind and Iwasawa finite groups, and finite T-groups (groups in which normality is transitive), PT-groups (groups in which permutability is transitive), and PST-groups (groups in which Sylow permutability is transitive). These algorithms have been implemented in a package for the computer algebra system GAP.

Keywords:  Finite group • Permutable subgroup • S-permutable subgroup • Dedekind group • Iwasawa group • T-group • PT-group • PST-group • Algorithm
Mathematics Subject Classification (2010):  20D10, 20D20, 20-04

http://dx.doi.org/10.2478/s11533-013-0299-4

 

Paper “Extension of Schur theorem to groups with a central factor with a bounded section rank” published in J. Algebra

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A. Ballester-Bolinches, S. Camp-Mora, L. A. Kurdachenko, J. Otal

Extension of Schur theorem to groups with a central factor with a bounded section rank

J. Algebra, 393, 1-15 (2013)

http://dx.doi.org/10.1016/j.jalgebra.2013.06.035

Abstract: A well-known result reported by Schur states that the derived subgroup of a group is finite provided its central factor is finite. Here we show that if the p-section rank of the central factor of a locally generalized radical group is bounded, then so is the p-section rank of its derived subgroup. We also give an explicit expression for this bound.

MSC: 20F14, 20F19, 20F99

Keywords: Schur class, Schur multiplier, Special rank of a group, p-section rank of a group, 0-rank of a group, Generalized radical group

Paper “Primitive subgroups and PST-groups” to appear in Bull. Aust. Math. Soc.

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

Primitive groups and PST-groups

Bull. Aust. Math. Soc.

http://dx.doi.org/10.1017/S0004972713000592

Abstract

All groups considered in this paper are finite. A subgroup H of a group G is called a primitive subgroup if it is a proper subgroup in the intersection of all subgroups of G containing H as a proper subgroup. He et al. [‘A note on primitive subgroups of finite groups’, Commun. Korean Math. Soc. 28(1) (2013), 55–62] proved that every primitive subgroup of G has index a power of a prime if and only if G/Φ(G) is a solvable PST-group. Let X denote the class of groups G all of whose primitive subgroups have prime power index. It is established here that a group G is a solvable PST-group if and only if every subgroup of G is an X-group.

2010 Mathematics subject classification: primary 20D10; secondary 20D15, 20D20

Keywords and phrases: finite groups, primitive subgroups, solvable PST-groups, T0-groups

Paper “On conjugacy of supplements of normal subgroups of finite groups” to appear in Bull. Aust. Math. Soc.

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A. Ballester-Bolinches, Luis M. Ezquerro,

On conjugacy of supplements of normal subgroups of finite groups

Bull. Aust. Math. Soc.

http://dx.doi.org/10.1017/S0004972713000506

Abstract: The objective of this paper is to find some sufficient conditions to ensure the conjugacy of supplements of a normal subgroup of a soluble group.

2010 Mathematics subject classification: primary 20D10; secondary 20D20.
Keywords and phrases: finite groups, supplements, Schunck classes, formations.

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 “On subgroups of hypercentral type of finite groups” to appear in Israel J. Math.

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A. Ballester-Bolinches, Luis M. Ezquerro, Alexander N. Skiba

On subgroups of hypercentral type of finite groups

Israel J. Math.

http://dx.doi.org/10.1007/s11856-013-0030-y

Abstract

The main purpose of this paper is to analyze the influence on the structure of a finite group of some subgroups lying in the hypercenter. More precisely, we prove the following: Let F be a Baer-local formation. Given a group G and a normal subgroup E of G, let ZF (G) contain a p-subgroup A of E which is maximal being abelian and of exponent dividing pk, where k is some natural number, k = 1 if p = 2 and the Sylow 2-subgroups of E are non-abelian. Then E/ O p (E) ≤ ZF(G/ Op (E)) (Theorem 1). Some well-known results turn out to be consequences of this theorem.