Supersolubility of a Finite Group with Normally Embedded Maximal Subgroups in Sylow Subgroups

2018 ◽  
Vol 59 (5) ◽  
pp. 922-930 ◽  
Author(s):  
V. S. Monakhov ◽  
A. A. Trofimuk
Keyword(s):  
Finite Group ◽  
Maximal Subgroups ◽  
Sylow Subgroups ◽  
A Finite Group

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.

On the π-Quasinormality of 2-Maximal Subgroups of Sylow Subgroups of a Finite Group

2012 ◽  
Vol 19 (04) ◽  
pp. 657-664
Author(s):  
Songliang Chen ◽  
Yun Fan
Keyword(s):  
Finite Group ◽  
Maximal Subgroup ◽  
Sufficient Conditions ◽  
Maximal Subgroups ◽  
Sylow Subgroups ◽  
Group A ◽  
A Finite Group ◽  
Sylow Tower

Let G be a finite group. A subgroup H of G is called a 2-maximal subgroup of G if there exists a maximal subgroup M of G such that H is a maximal subgroup of M. In this paper, we discuss the influence of π-quasinormality of 2-maximal subgroups of Sylow subgroups on the structure of a finite group, and obtain some sufficient conditions under which the finite group is p-nilpotent, supersolvable, or possesses an ordered Sylow tower.

On the Intersection of a Family of Maximal Subgroups Containing the Sylow Subgroups of a Finite Group

1988 ◽  
Vol 40 (2) ◽  
pp. 352-359 ◽  
Author(s):  
N. P. Mukherjee ◽  
Prabir Bhattacharya
Keyword(s):  
Finite Group ◽  
Maximal Subgroups ◽  
Frattini Subgroup ◽  
Sylow Subgroups ◽  
The Family ◽  
A Finite Group ◽  
Main Definition

Given a finite group G, the Frattini subgroup of G, Φ(G) is defined to be the intersection of all the maximal subgroups of G. Of late there have been several attempts to consider generalizations of Φ(G). For example, Gaschutz [7] and Rose [13] have investigated the intersection of all non-normal, maximal subgroups of a finite group. Deskins [6] has discussed the intersection of the family of maximal subgroups of a finite group whose indices are co-prime to a given prime. In [4-5, 12] we have considered the investigation of the family of all maximal subgroups of a finite group whose indices are composite and co-prime to a given prime. We have obtained several results about the family . In this paper which is a sequel to [4] we prove some further results about this family indicating the interesting role it plays especially when G is solvable or p-solvable. First we recall the main definition from [4].

scholarly journals A generalized Frattini subgroup of a finite group

1989 ◽  
Vol 12 (2) ◽  
pp. 263-266
Author(s):  
Prabir Bhattacharya ◽  
N. P. Mukherjee
Keyword(s):  
Finite Group ◽  
Maximal Subgroups ◽  
Frattini Subgroup ◽  
Definition Of ◽  
A Finite Group

For a finite group G and an arbitrary prime p, letSP(G)denote the intersection of all maximal subgroups M of G such that [G:M] is both composite and not divisible by p; if no such M exists we setSP(G)= G. Some properties of G are considered involvingSP(G). In particular, we obtain a characterization of G when each M in the definition ofSP(G)is nilpotent.

On S-Quasinormal and C-Normal Subgroups of Prime Power Order in Finite Groups

2011 ◽  
Vol 18 (04) ◽  
pp. 685-692
Author(s):  
Xuanli He ◽  
Shirong Li ◽  
Xiaochun Liu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Maximal Subgroups ◽  
Prime Power Order ◽  
Normal Subgroups ◽  
A Finite Group

Let G be a finite group, p the smallest prime dividing the order of G, and P a Sylow p-subgroup of G with the smallest generator number d. Consider a set [Formula: see text] of maximal subgroups of P such that [Formula: see text]. It is shown that if every member [Formula: see text] of is either S-quasinormally embedded or C-normal in G, then G is p-nilpotent. As its applications, some further results are obtained.

Finite Groups with Some Subgroups of Sylow Subgroups s∗-Semipermutable

2021 ◽  
Vol 58 (2) ◽  
pp. 147-156
Author(s):  
Qingjun Kong ◽  
Xiuyun Guo
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Subnormal Subgroup ◽  
Sylow Subgroups ◽  
A Finite Group

We introduce a new subgroup embedding property in a finite group called s∗-semipermutability. Suppose that G is a finite group and H is a subgroup of G. H is said to be s∗-semipermutable in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K is s-semipermutable in G. We fix in every non-cyclic Sylow subgroup P of G some subgroup D satisfying 1 < |D| < |P | and study the structure of G under the assumption that every subgroup H of P with |H | = |D| is s∗-semipermutable in G. Some recent results are generalized and unified.

scholarly journals A remark on the C-normality of maximal subgroups of finite groups

1997 ◽  
Vol 40 (2) ◽  
pp. 243-246
Author(s):  
Yanming Wang
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Maximal Subgroups ◽  
Primitive Groups ◽  
A Finite Group

A subgroup H is called c-normal in a group G if there exists a normal subgroup N of G such that HN = G and H∩N ≤ HG, where HG =: Core(H) = ∩g∈GHg is the maximal normal subgroup of G which is contained in H. We use a result on primitive groups and the c-normality of maximal subgroups of a finite group G to obtain results about the influence of the set of maximal subgroups on the structure of G.

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