Finite groups with σ-Frobenius condition for non-normal σ-primary subgroups

2019 ◽  
Vol 19 (03) ◽  
pp. 2050047
Author(s):  
Bin Hu ◽  
Jianhong Huang ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Primary Subgroup ◽  
Group A ◽  
A Finite Group ◽  
Primary Subgroups ◽  
Normal Formula

Let [Formula: see text] be a partition of the set [Formula: see text] of all primes and [Formula: see text] a finite group. A group is said to be [Formula: see text]-primary if it is a finite [Formula: see text]-group for some [Formula: see text]. We say that a [Formula: see text]-subgroup [Formula: see text] of [Formula: see text] satisfies the [Formula: see text]-Frobenius condition in [Formula: see text] if [Formula: see text] is a [Formula: see text]-group. In this paper, we determine the structure of finite groups in which every non-normal [Formula: see text]-primary subgroup satisfies the [Formula: see text]-Frobenius condition.

SOME NEW CHARACTERISATIONS OF FINITE -SUPERSOLUBLE GROUPS

2013 ◽  
Vol 89 (3) ◽  
pp. 514-521 ◽  
Author(s):  
CHANGWEN LI ◽  
NANYING YANG ◽  
NA TANG
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Supersoluble Groups ◽  
Group A ◽  
A Finite Group ◽  
Primary Subgroups

AbstractLet $G$ be a finite group. A subgroup $H$ of $G$ is said to be $E$-supplemented in $G$ if there is a subgroup $T$ of $G$ such that $G= HT$ and $H\cap T\leq {H}_{eG} $, where ${H}_{eG} $ denotes the subgroup of $H$ generated by all those subgroups of $H$ which are $S$-quasinormally embedded in $G$. In this paper, some new characterisations of $p$-supersolubility of finite groups are given under the assumption that some primary subgroups are $E$-supplemented.

scholarly journals Finite Groups with Certain Permutability Criteria

2019 ◽  
Vol 12 (2) ◽  
pp. 571-576 ◽  
Author(s):  
Rola A. Hijazi ◽  
Fatme M. Charaf
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroups ◽  
Group A ◽  
A Finite Group

Let G be a finite group. A subgroup H of G is said to be S-permutable in G if itpermutes with all Sylow subgroups of G. In this note we prove that if P, the Sylowp-subgroup of G (p > 2), has a subgroup D such that 1 <|D|<|P| and all subgroups H of P with |H| = |D| are S-permutable in G, then G′ is p-nilpotent.

On SS-supplemented subgroups of some sylow subgroups of finite groups

2021 ◽  
Author(s):  
Xuanli He ◽  
Qinghong Guo ◽  
Muhong Huang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Necessary Conditions ◽  
Saturated Formation ◽  
Sylow Subgroups ◽  
Group A ◽  
Sufficient And Necessary Conditions ◽  
A Finite Group

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is called to be [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] permutes with all Sylow subgroups of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-supplemented in [Formula: see text] if there exists a subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is [Formula: see text]-permutable in [Formula: see text]. In this paper, we investigate [Formula: see text]-nilpotency of a finite group. As applications, we give some sufficient and necessary conditions for a finite group belongs to a saturated formation.

The Influence of Certain Abelian Subgroups on the Structure of Finite Groups

2008 ◽  
Vol 15 (03) ◽  
pp. 479-484 ◽  
Author(s):  
M. Ramadan
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Power Order ◽  
Group A ◽  
Abelian Subgroups ◽  
A Finite Group

Let G be a finite group. A subgroup K of a group G is called an [Formula: see text]-subgroup of G if NG(K) ∩ Kx ≤ K for all x ∈ G. The set of all [Formula: see text]-subgroups of G is denoted by [Formula: see text]. In this paper, we investigate the structure of a group G under the assumption that certain abelian subgroups of prime power order belong to [Formula: see text].

scholarly journals Finite Groups with Minimal CSS-subgroups

2021 ◽  
Vol 14 (3) ◽  
pp. 1002-1014
Author(s):  
A. A. Heliel ◽  
R. A. Hijazi ◽  
S. M. Al-Shammari
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Group A ◽  
A Finite Group

Let G be a finite group. A subgroup H of G is called SS-quasinormal in G if there is a supplement B of H to G such that H permutes with every Sylow subgroup of B. A subgroup H of G is called CSS-subgroup in G if there exists a normal subgroup K of G such that G = HK and H ∩K is SS-quasinormal in G. In this paper, we investigate the influence of minimal CSS-subgroups of G on its structure. Our results improve and generalize several recent results in the literature.

scholarly journals Finite groups with given systems of generalised σ-permutable subgroups

2021 ◽  
pp. 25-33
Author(s):  
Viktoria S. Zakrevskaya
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
Chief Factor ◽  
Permutable Subgroup ◽  
Permutable Subgroups ◽  
Group A ◽  
A Finite Group

Let σ = {σi|i ∈ I } be a partition of the set of all primes ℙ and G be a finite group. A set ℋ  of subgroups of G is said to be a complete Hall σ-set of G if every member ≠1 of ℋ  is a Hall σi-subgroup of G for some i ∈ I and ℋ contains exactly one Hall σi-subgroup of G for every i such that σi ⌒ π(G)  ≠ ∅.  A group is said to be σ-primary if it is a finite σi-group for some i. A subgroup A of G is said to be: σ-permutable in G if G possesses a complete Hall σ-set ℋ  such that AH x = H  xA for all H ∈ ℋ  and all x ∈ G; σ-subnormal in G if there is a subgroup chain A = A0 ≤ A1 ≤ … ≤ At = G such that either Ai − 1 ⊴ Ai or Ai /(Ai − 1)Ai is σ-primary for all i = 1, …, t; 𝔄-normal in G if every chief factor of G between AG and AG is cyclic. We say that a subgroup H of G is: (i) partially σ-permutable in G if there are a 𝔄-normal subgroup A and a σ-permutable subgroup B of G such that H = < A, B >; (ii) (𝔄, σ)-embedded in G if there are a partially σ-permutable subgroup S and a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ S ≤ H. We study G assuming that some subgroups of G are partially σ-permutable or (𝔄, σ)-embedded in G. Some known results are generalised.

On p-supersolvability of finite groups

2015 ◽  
Vol 52 (4) ◽  
pp. 504-510
Author(s):  
Mohamed Asaad
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Normal Closure ◽  
Sylow Subgroups ◽  
Group A ◽  
A Finite Group

Let G be a finite group. A subgroup H of G is said to be s-permutable in G if H permutes with all Sylow subgroups of G. Let H be a subgroup of G and let HsG be the subgroup of H generated by all those subgroups of H which are s-permutable in G. A subgroup H of G is called n-embedded in G if G has a normal subgroup T such that HG = HT and H ∩ T ≦ HsG, where HG is the normal closure of H in G. We investigate the influence of n-embedded subgroups of the p-nilpotency and p-supersolvability of G.

scholarly journals Finite Groups with Some -Supplemented Subgroups

Algebra ◽  
2013 ◽  
Vol 2013 ◽  
pp. 1-5
Author(s):  
Guo Zhong ◽  
Liying Yang ◽  
Huaquan Wei ◽  
Xuanlong Ma ◽  
Jiayi Xia
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Maximal Subgroups ◽  
Group A ◽  
A Finite Group

Let be a subgroup of a finite group , a prime dividing the order of , and a Sylow -subgroup of for prime We say that is -supplemented in if there is a subgroup of such that and where denotes the subgroup of generated by all those subgroups of which are -quasinormally embedded in In this paper, we characterize -nilpotency and supersolvability of under the assumption that all maximal subgroups of are -supplemented in .

scholarly journals Finite Groups Whose Certain Subgroups of Prime Power Order Are -Semipermutable

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-8
Author(s):  
Mustafa Obaid
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Prime Power ◽  
Prime Power Order ◽  
Group A ◽  
Abelian Subgroups ◽  
A Finite Group

Let be a finite group. A subgroup of is said to be S-semipermutable in if permutes with every Sylow -subgroup of with . In this paper, we study the influence of S-permutability property of certain abelian subgroups of prime power order of a finite group on its structure.

On semi-σ-nilpotent finite groups

2019 ◽  
Vol 18 (10) ◽  
pp. 1950200
Author(s):  
Chi Zhang ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Semidirect Product ◽  
Finite Groups ◽  
Nilpotent Groups ◽  
Product Formula ◽  
Nilpotent Subgroup ◽  
Chief Factor ◽  
Group A ◽  
A Finite Group

Let [Formula: see text] be a partition of the set [Formula: see text] of all primes and [Formula: see text] a finite group. A chief factor [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-central if the semidirect product [Formula: see text] is a [Formula: see text]-group for some [Formula: see text]. [Formula: see text] is called [Formula: see text]-nilpotent if every chief factor of [Formula: see text] is [Formula: see text]-central. We say that [Formula: see text] is semi-[Formula: see text]-nilpotent (respectively, weakly semi-[Formula: see text]-nilpotent) if the normalizer [Formula: see text] of every non-normal (respectively, every non-subnormal) [Formula: see text]-nilpotent subgroup [Formula: see text] of [Formula: see text] is [Formula: see text]-nilpotent. In this paper we determine the structure of finite semi-[Formula: see text]-nilpotent and weakly semi-[Formula: see text]-nilpotent groups.

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