Maximal subgroups and PST-groups

AbstractA subgroup H of a group G is said to permute with a subgroup K of G if HK is a subgroup of G. H is said to be permutable (resp. S-permutable) if it permutes with all the subgroups (resp. Sylow subgroups) of G. Finite groups in which permutability (resp. S-permutability) is a transitive relation are called PT-groups (resp. PST-groups). PT-, PST- and T-groups, or groups in which normality is transitive, have been extensively studied and characterised. Kaplan [Kaplan G., On T-groups, supersolvable groups, and maximal subgroups, Arch. Math. (Basel), 2011, 96(1), 19–25] presented some new characterisations of soluble T-groups. The main goal of this paper is to establish PT- and PST-versions of Kaplan’s results, which enables a better understanding of the relationships between these classes.

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On weakly ℨ-permutable subgroups of finite groups

Journal of Algebra and Its Applications ◽

10.1142/s0219498815500620 ◽

2015 ◽

Vol 14 (05) ◽

pp. 1550062 ◽

Cited By ~ 1

Author(s):

A. A. Heliel ◽

M. M. Al-Shomrani ◽

T. M. Al-Gafri

Keyword(s):

Finite Groups ◽

Subnormal Subgroup ◽

Maximal Subgroups ◽

Supersolvable Groups ◽

Fitting Subgroup ◽

Permutable Subgroup ◽

Sylow Subgroups ◽

Permutable Subgroups ◽

Complete Set ◽

A Finite Group

Let ℨ be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G, ℨ contains exactly one and only one Sylow p-subgroup of G. A subgroup H of G is said to be ℨ-permutable of G if H permutes with every member of ℨ. A subgroup H of G is said to be a weakly ℨ-permutable subgroup of G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K ≤ Hℨ, where Hℨ is the subgroup of H generated by all those subgroups of H which are ℨ-permutable subgroups of G. In this paper, we prove that if p is the smallest prime dividing the order of G and the maximal subgroups of Gp ∈ ℨ are weakly ℨ-permutable subgroups of G, then G is p-nilpotent. Moreover, we prove that if 𝔉 is a saturated formation containing the class of all supersolvable groups, then G ∈ 𝔉 iff there is a solvable normal subgroup H in G such that G/H ∈ 𝔉 and the maximal subgroups of the Sylow subgroups of the Fitting subgroup F(H) are weakly ℨ-permutable subgroups of G. These two results generalize and unify several results in the literature.

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Finite groups with two supersoluble subgroups

Journal of Group Theory ◽

10.1515/jgth-2018-0150 ◽

2019 ◽

Vol 22 (2) ◽

pp. 297-312 ◽

Cited By ~ 6

Author(s):

Victor S. Monakhov ◽

Alexander A. Trofimuk

Keyword(s):

Finite Group ◽

Finite Groups ◽

Sufficient Conditions ◽

Maximal Subgroups ◽

Sylow Subgroups ◽

Derived Subgroup ◽

Nilpotent Residual ◽

A Finite Group

AbstractLetGbe a finite group. In this paper we obtain some sufficient conditions for the supersolubility ofGwith two supersoluble non-conjugate subgroupsHandKof prime index, not necessarily distinct. It is established that the supersoluble residual of such a group coincides with the nilpotent residual of the derived subgroup. We prove thatGis supersoluble in the following cases: one of the subgroupsHorKis nilpotent; the derived subgroup{G^{\prime}}ofGis nilpotent;{|G:H|=q>r=|G:K|}andHis normal inG. Also the supersolubility ofGwith two non-conjugate maximal subgroupsMandVis obtained in the following cases: all Sylow subgroups ofMand ofVare seminormal inG; all maximal subgroups ofMand ofVare seminormal inG.

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ON GENERALISED FC-GROUPS IN WHICH NORMALITY IS A TRANSITIVE RELATION

Journal of the Australian Mathematical Society ◽

10.1017/s1446788715000397 ◽

2015 ◽

Vol 100 (2) ◽

pp. 192-198

Author(s):

R. ESTEBAN-ROMERO ◽

G. VINCENZI

Keyword(s):

Conjugacy Classes ◽

Maximal Subgroups ◽

Transitive Relation ◽

Supersolvable Groups

We extend to soluble $\text{FC}^{\ast }$-groups, the class of generalised FC-groups introduced in de Giovanni et al. [‘Groups with restricted conjugacy classes’, Serdica Math. J. 28(3) (2002), 241–254], the characterisation of finite soluble T-groups obtained recently in Kaplan [‘On T-groups, supersolvable groups, and maximal subgroups’, Arch. Math. (Basel) 96(1) (2011), 19–25].

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The maximal subgroups of Sylow subgroups and the structure of finite groups

Boletim da Sociedade Paranaense de Matemática ◽

10.5269/bspm.v37i1.20200 ◽

2017 ◽

Vol 37 (1) ◽

pp. 113-124

Author(s):

Changwen Li ◽

Xuemei Zhang ◽

Jianhong Huang

Keyword(s):

Finite Groups ◽

Maximal Subgroups ◽

Sylow Subgroups

In this paper we investigate the influence of some subgroups of Sylow subgroups with semi cover-avoiding property and $E$-supplementation on the structure of finite groups. Some recent results are generalized and unified.

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A note on s-conditional permutability of maximal subgroups of Sylow subgroups of finite groups

Monatshefte für Mathematik ◽

10.1007/s00605-017-1113-3 ◽

2017 ◽

Vol 187 (4) ◽

pp. 729-733

Author(s):

Haoran Yu

Keyword(s):

Finite Groups ◽

Maximal Subgroups ◽

Sylow Subgroups

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A generalization to Sylow permutability of pronormal subgroups of finite groups

Journal of Algebra and Its Applications ◽

10.1142/s0219498820500528 ◽

2019 ◽

Vol 19 (03) ◽

pp. 2050052

Author(s):

R. Esteban-Romero ◽

P. Longobardi ◽

M. Maj

Keyword(s):

Finite Groups ◽

Transitive Relation ◽

Or Groups

In this note, we present a new subgroup embedding property that can be considered as an analogue of pronormality in the scope of permutability and Sylow permutability in finite groups. We prove that finite PST-groups, or groups in which Sylow permutability is a transitive relation, can be characterized in terms of this property, in a similar way as T-groups, or groups in which normality is transitive, can be characterized in terms of pronormality.

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