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.

FINITE GROUPS WITH GIVEN NEARLY S-QUASINORMAL SUBGROUPS

2008 ◽  
Vol 01 (03) ◽  
pp. 369-382
Author(s):  
Nataliya V. Hutsko ◽  
Vladimir O. Lukyanenko ◽  
Alexander N. Skiba
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Cyclic Subgroup ◽  
Saturated Formation ◽  
Supersoluble Groups ◽  
Sylow Subgroups ◽  
Quasinormal Subgroup ◽  
A Finite Group

Let G be a finite group and H a subgroup of G. Then H is said to be S-quasinormal in G if HP = PH for all Sylow subgroups P of G. Let HsG be the subgroup of H generated by all those subgroups of H which are S-quasinormal in G. Then we say that H is nearly S-quasinormal in G if G has an S-quasinormal subgroup T such that HT = G and T ∩ H ≤ HsG. Our main result here is the following theorem. Let [Formula: see text] be a saturated formation containing all supersoluble groups and G a group with a normal subgroup E such that [Formula: see text]. Suppose that every non-cyclic Sylow subgroup P of E has a subgroup D such that 1 < |D| < |P| and all subgroups H of P with order |H| = |D| and every cyclic subgroup of P with order 4 (if |D| = 2 and P is a non-abelian 2-group) having no supersoluble supplement in G are nearly S-quasinormal in G. Then [Formula: see text].

Finite groups with some s-conditionally permutable subgroups

2017 ◽  
Vol 16 (12) ◽  
pp. 1750224
Author(s):  
S. E. Mirdamadi ◽  
G. R. Rezaeezadeh
Keyword(s):  
Finite Group ◽  
Solvable Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
Subnormal Subgroup ◽  
Permutable Subgroups ◽  
A Finite Group

A subgroup [Formula: see text] of a finite group [Formula: see text] is said to be [Formula: see text]-conditionally permutable in [Formula: see text] if for every Sylow subgroup [Formula: see text] of [Formula: see text], there exists an element [Formula: see text] such that [Formula: see text]. In this paper, the structure of solvable group [Formula: see text] in which every [Formula: see text]-subgroup of [Formula: see text] or every subnormal subgroup of [Formula: see text] is [Formula: see text]-conditionally permutable in [Formula: see text] is described. Let [Formula: see text] be a solvable group and [Formula: see text] the largest prime dividing [Formula: see text]. Suppose further that [Formula: see text] is the Sylow [Formula: see text]-subgroup of [Formula: see text] and [Formula: see text]. We are going to show that [Formula: see text] is a PST-group if and only if every subnormal subgroup of [Formula: see text] is [Formula: see text]-conditionally permutable in [Formula: see text].

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 Mutually Permutable Products of Finite Groups

ISRN Algebra ◽  
2011 ◽  
Vol 2011 ◽  
pp. 1-4
Author(s):  
Rola A. Hijazi
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroups ◽  
A Finite Group ◽  
Mutually Permutable Products

Let G be a finite group and G1, G2 are two subgroups of G. We say that G1 and G2 are mutually permutable if G1 is permutable with every subgroup of G2 and G2 is permutable with every subgroup of G1. We prove that if is the product of three supersolvable subgroups G1, G2, and G3, where Gi and Gj are mutually permutable for all i and j with and the Sylow subgroups of G are abelian, then G is supersolvable. As a corollary of this result, we also prove that if G possesses three supersolvable subgroups whose indices are pairwise relatively prime, and Gi and Gj are mutually permutable for all i and j with , then G is supersolvable.

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.

Finite Groups with Normal Normalizers

1968 ◽  
Vol 20 ◽  
pp. 1256-1260 ◽  
Author(s):  
C. Hobby
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Sylow Subgroup ◽  
A Finite Group

We say that a finite group G has property N if the normalizer of every subgroup of G is normal in G. Such groups are nilpotent since every Sylow subgroup is normal (the normalizer of a Sylow subgroup is its own normalizer). Thus it is sufficient to study p-groups which have property N. Note that property N is inherited by subgroups and factor groups.

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.

On weakly 𝓗-permutable subgroups of finite groups

2019 ◽  
Vol 69 (4) ◽  
pp. 763-772
Author(s):  
Chenchen Cao ◽  
Venus Amjid ◽  
Chi Zhang
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
Permutable Subgroups ◽  
A Finite Group ◽  
New Criteria

Abstract Let σ = {σi ∣i ∈ I} be some partition of the set of all primes ℙ, G be a finite group and σ(G) = {σi∣σi ∩ π(G) ≠ ∅}. G is said to be σ-primary if ∣σ(G)∣ ≤ 1. A subgroup H of G is said to be σ-subnormal in G if there exists a subgroup chain H = H0 ≤ H1 ≤ … ≤ Ht = G such that either Hi−1 is normal in Hi or Hi/(Hi−1)Hi is σ-primary for all i = 1, …, t. A set 𝓗 of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of 𝓗 is a Hall σi-subgroup of G for some i and 𝓗 contains exactly one Hall σi-subgroup of G for every σi ∈ σ(G). Let 𝓗 be a complete Hall σ-set of G. A subgroup H of G is said to be 𝓗-permutable if HA = AH for all A ∈ 𝓗. We say that a subgroup H of G is weakly 𝓗-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ H𝓗, where H𝓗 is the subgroup of H generated by all those subgroups of H which are 𝓗-permutable. By using the weakly 𝓗-permutable subgroups, we establish some new criteria for a group G to be σ-soluble and supersoluble, and we also give the conditions under which a normal subgroup of G is hypercyclically embedded.

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.

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