scholarly journals On supersolvable groups whose maximal subgroups of the Sylow subgroups are subnormal

2019 ◽  
pp. 315-322
Author(s):  
Pengfei Guo ◽  
Xingqiang Xiu ◽  
Guangjun Xu
Keyword(s):  
Maximal Subgroups ◽  
Supersolvable Groups ◽  
Sylow Subgroups

scholarly journals Maximal subgroups and PST-groups

2013 ◽  
Vol 11 (6) ◽  
Author(s):  
Adolfo Ballester-Bolinches ◽  
James Beidleman ◽  
Ramón Esteban-Romero ◽  
Vicent Pérez-Calabuig
Keyword(s):  
Finite Groups ◽  
Maximal Subgroups ◽  
Transitive Relation ◽  
Supersolvable Groups ◽  
Sylow Subgroups ◽  
Or 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.

On weakly ℨ-permutable subgroups of finite groups

2015 ◽  
Vol 14 (05) ◽  
pp. 1550062 ◽  
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.

scholarly journals ON A CLASS OF SUPERSOLUBLE GROUPS

2014 ◽  
Vol 90 (2) ◽  
pp. 220-226 ◽  
Author(s):  
A. BALLESTER-BOLINCHES ◽  
J. C. BEIDLEMAN ◽  
R. ESTEBAN-ROMERO ◽  
M. F. RAGLAND
Keyword(s):  
Finite Group ◽  
Maximal Subgroups ◽  
Supersoluble Groups ◽  
Sylow Subgroups ◽  
A Finite Group

AbstractA subgroup $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}H$ of a finite group $G$ is said to be S-semipermutable in $G$ if $H$ permutes with every Sylow $q$-subgroup of $G$ for all primes $q$ not dividing $|H |$. A finite group $G$ is an MS-group if the maximal subgroups of all the Sylow subgroups of $G$ are S-semipermutable in $G$. The aim of the present paper is to characterise the finite MS-groups.

Finite groups with two supersoluble subgroups

2019 ◽  
Vol 22 (2) ◽  
pp. 297-312 ◽  
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.

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

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].

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

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.

On the supersolubility of a group with some tcc-subgroups

2019 ◽  
pp. 2150020
Author(s):  
Alexander Trofimuk
Keyword(s):  
Sylow Subgroup ◽  
Prime Order ◽  
Maximal Subgroups ◽  
Sylow Subgroups

A subgroup [Formula: see text] of a group [Formula: see text] is called tcc-subgroup in [Formula: see text], if there is a subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and for any [Formula: see text] and for any [Formula: see text], there exists an element [Formula: see text] such that [Formula: see text]. The notation [Formula: see text] means that [Formula: see text] is a subgroup of a group [Formula: see text]. In this paper, we proved the supersolubility of a group [Formula: see text] in the following cases: [Formula: see text] and [Formula: see text] are supersoluble tcc-subgroups in [Formula: see text]; all Sylow subgroups of [Formula: see text] and of [Formula: see text] are tcc-subgroups in [Formula: see text]; all maximal subgroups of [Formula: see text] and of [Formula: see text] are tcc-subgroups in [Formula: see text]. Besides, the supersolubility of a group [Formula: see text] is obtained in each of the following cases: all maximal subgroups of every Sylow subgroup of [Formula: see text] are tcc-subgroups in [Formula: see text]; every subgroup of prime order or 4 is a tcc-subgroup in [Formula: see text]; all 2-maximal subgroups of [Formula: see text] are tcc-subgroups in [Formula: see text].

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