scholarly journals ON CERTAIN PRODUCTS OF PERMUTABLE SUBGROUPS

2021 ◽  
pp. 1-8
Author(s):  
A. BALLESTER-BOLINCHES ◽  
S. Y. MADANHA ◽  
T. M. MUDZIIRI SHUMBA ◽  
M. C. PEDRAZA-AGUILERA
Keyword(s):  
Finite Groups ◽  
Subnormal Subgroup ◽  
Permutable Subgroups

Abstract In this paper, we study the structure of finite groups $G=AB$ which are a weakly mutually $sn$ -permutable product of the subgroups A and B, that is, A permutes with every subnormal subgroup of B containing $A \cap B$ and B permutes with every subnormal subgroup of A containing $A \cap B$ . We obtain generalisations of known results on mutually $sn$ -permutable products.

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 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.

scholarly journals ON GENERALISED PRONORMAL SUBGROUPS OF FINITE GROUPS

2014 ◽  
Vol 56 (3) ◽  
pp. 691-703 ◽  
Author(s):  
A. BALLESTER-BOLINCHES ◽  
J. C. BEIDLEMAN ◽  
A. D. FELDMAN ◽  
M. F. RAGLAND
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
Normal Subgroups ◽  
Transitive Relation ◽  
Formula Group ◽  
Permutable Subgroups ◽  
Sylow Permutable ◽  
A Finite Group ◽  
Subnormal Subgroups

AbstractFor a formation $\mathfrak F$, a subgroup M of a finite group G is said to be $\mathfrak F$-pronormal in G if for each g ∈ G, there exists x ∈ 〈U,Ug〉$\mathfrak F$ such that Ux = Ug. Let f be a subgroup embedding functor such that f(G) contains the set of normal subgroups of G and is contained in the set of Sylow-permutable subgroups of G for every finite group G. Given such an f, let fT denote the class of finite groups in which f(G) is the set of subnormal subgroups of G; this is the class of all finite groups G in which to be in f(G) is a transitive relation in G. A subgroup M of a finite group G is said to be $\mathfrak F$-normal in G if G/CoreG(M) belongs to $\mathfrak F$. A subgroup U of a finite group G is called K-$\mathfrak F$-subnormal in G if either U = G or there exist subgroups U = U0 ≤ U1 ≤ . . . ≤ Un = G such that Ui–1 is either normal or $\mathfrak F$-normal in Ui, for i = 1,2, …, n. We call a finite group G an $fT_{\mathfrak F}$-group if every K-$\mathfrak F$-subnormal subgroup of G is in f(G). In this paper, we analyse for certain formations $\mathfrak F$ the structure of $fT_{\mathfrak F}$-groups. We pay special attention to the $\mathfrak F$-pronormal subgroups in this analysis.

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 weakly m-σ-permutable subgroups

2020 ◽  
pp. 2150056
Author(s):  
Muhammad Tanveer Hussain ◽  
Venus Amjid
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Subnormal Subgroup ◽  
Permutable Subgroup ◽  
Modular Subgroup ◽  
Permutable Subgroups ◽  
A Finite Group

Let [Formula: see text] be a finite group, [Formula: see text] be a partition of the set of all primes [Formula: see text] and [Formula: see text]. A set [Formula: see text] of subgroups of [Formula: see text] is said to be a complete Hall[Formula: see text]-set of [Formula: see text] if every non-identity member of [Formula: see text] is a Hall [Formula: see text]-subgroup of [Formula: see text] and [Formula: see text] contains exactly one Hall [Formula: see text]-subgroup of [Formula: see text] for every [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-permutable in [Formula: see text] if [Formula: see text] possesses a complete Hall [Formula: see text]-set [Formula: see text] such that [Formula: see text] for all [Formula: see text] and all [Formula: see text]. Let [Formula: see text] be a subgroup of [Formula: see text]. [Formula: see text] is: [Formula: see text]-[Formula: see text]-permutable in [Formula: see text] if [Formula: see text] for some modular subgroup [Formula: see text] and [Formula: see text]-permutable subgroup [Formula: see text] of [Formula: see text]; weakly[Formula: see text]-[Formula: see text]-permutable in [Formula: see text] if there are an [Formula: see text]-[Formula: see text]-permutable subgroup [Formula: see text] and a [Formula: see text]-subnormal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text]. In this paper, we investigate the influence of weakly [Formula: see text]-[Formula: see text]-permutable subgroups on the structure of finite groups.

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.

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.

THE INFLUENCE OF Φ-S-SUPPLEMENTED SUBGROUPS ON THE STRUCTURE OF FINITE GROUPS

2012 ◽  
Vol 11 (04) ◽  
pp. 1250064
Author(s):  
CHANGWEN LI
Keyword(s):  
Finite Groups ◽  
Subnormal Subgroup ◽  
Frattini Subgroup

A subgroup H of a group G is called Φ-s-supplemented in G if there exists a subnormal subgroup K of G such that G = HK and H ∩ K ≤ Φ (H), where Φ(H) is the Frattini subgroup of H. We investigate the influence of Φ-s-supplemented subgroups on the p-nilpotency, p-supersolvability and supersolvability of finite groups.

On X-Permutable Subgroups

2012 ◽  
Vol 19 (04) ◽  
pp. 699-706
Author(s):  
Baojun Li ◽  
Zhirang Zhang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Hall Subgroups ◽  
Permutable Subgroups

A subgroup A of a group G is said to be X-permutable with another subgroup B in G, where ∅ ≠ X ⊆ G, if there exists some element x ∈ X such that ABx=BxA. In this paper, the solubility and supersolubility of finite groups are described by X-permutability of the Hall subgroups and their subgroups, in addition, the well known theorem of Schur-Zassenhaus in finite group is generalized.

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