# Paper «On Sylow permutable subgroups of finite groups» published in Forum Math.

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Adolfo Ballester-Bolinches, Hermann Heineken and Francesca Spagnuolo.
On Sylow permutable subgroups of finite groups.
Forum Math., 29(6):1307-1310, 2017.

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.

2020 Mathematics Subject Classification: 20D20, 20D35, 20D40, 20E15.

Keywords: Finite groups, subnormal subgroups, permutability, S-permutability.

# Paper «On the intersection of certain maximal subgroups of a finite group» published in J. Group Theory

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Adolfo Ballester-Bolinches, James C. Beidleman, Hermann Heineken, Matthew F. Ragland, Jack Schmidt

On the intersection of certain maximal subgroups of a finite group

J. Group Theory, 17 (2014), 705–715

http://dx.doi.org/10.1515/jgt-2013-0052

Abstract:  Let \$\Delta(G)\$ denote the intersection of all non-normal maximal subgroups of a group G. We introduce the class of T2-groups which are defined as the groups G for which \$G/\Delta(G)\$ is a T-group, that is, a group in which normality is a transitive relation. Several results concerning the class T2 are discussed. In particular, if G is a solvable group, then Sylow permutability is a transitive relation in G if and only if every subgroup H of G is a T2-group such that the nilpotent residual of H is a Hall subgroup of H.

# Paper «On the intersection of certain maximal subgroups of a finite group» to appear in J. Group Theory

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Adolfo Ballester-Bolinches, James C. Beidleman, Hermann Heineken, Matthew F. Ragland, Jack Schmidt

On the intersection of certain maximal subgroups of a finite group

J. Group Theory, in press

http://dx.doi.org/10.1515/jgt-2013-0052

Abstract:  Let \$\Delta(G)\$ denote the intersection of all non-normal maximal subgroups of a group G. We introduce the class of T2-groups which are defined as the groups G for which \$G/\Delta(G)\$ is a T-group, that is, a group in which normality is a transitive relation. Several results concerning the class T2 are discussed. In particular, if G is a solvable group, then Sylow permutability is a transitive relation in G if and only if every subgroup H of G is a T2-group such that the nilpotent residual of H is a Hall subgroup of H.

# Paper «Prefactorized subgroups in pairwise mutually permutable products» published in Ann. Mat. Pura Appl.

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A. Ballester-Bolinches, J. C. Beidleman, H. Heineken, M. C. Pedraza-Aguilera

Prefactorized subgroups in pairwise mutually permutable subgroups

Ann. Math .Pura Appl., 192(6), 1043-1057 (2013)

http://dx.doi.org/10.1007/s10231-012-0257-y

Abstract

We continue here our study of pairwise mutually and pairwise totally permutable products. We are looking for subgroups of the product in which the given factorization induces a factorization of the subgroup. In the case of soluble groups, it is shown that a prefactorized Carter subgroup and a prefactorized system normalizer exist. A less stringent property have F-residual, F-projector and F-normalizer for any saturated formation F including the supersoluble groups.

MSC: 20D10, 20D20

Keywords: Finite group, Permutability, Factorization, Saturated formation.

# 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.