On weakly σ-semipermutable subgroups of finite groups

2021 ◽  
Author(s):  
Venus Amjid ◽  
Muhammad Tanveer Hussain ◽  
Zhenfeng Wu
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Permutable Subgroup ◽  
A Finite Group

Let [Formula: see text] be some partition of the set of all primes [Formula: see text], [Formula: see text] be a finite group 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] for some [Formula: see text] and [Formula: see text] contains exactly one Hall [Formula: see text]-subgroup of [Formula: see text] for every [Formula: see text]. Let [Formula: see text] be a complete Hall [Formula: see text]-set of [Formula: see text]. A subgroup [Formula: see text] of [Formula: see text] is said to be [Formula: see text]-semipermutable with respect to [Formula: see text] if [Formula: see text] for all [Formula: see text] and all [Formula: see text] such that [Formula: see text]; [Formula: see text]-semipermutablein [Formula: see text] if [Formula: see text] is [Formula: see text]-semipermutable in [Formula: see text] with respect to some complete Hall [Formula: see text]-set of [Formula: see text]. We say that a subgroup [Formula: see text] of [Formula: see text] is weakly[Formula: see text]-semipermutable in [Formula: see text] if there exists a [Formula: see text]-permutable subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] is [Formula: see text]-permutable in [Formula: see text] and [Formula: see text], where [Formula: see text] is the subgroup of [Formula: see text] generated by all those subgroups of [Formula: see text] which are [Formula: see text]-semipermutable in [Formula: see text]. In this paper, we study the structure of [Formula: see text] under the condition that some subgroups of [Formula: see text] are weakly [Formula: see text]-semipermutable in [Formula: see text].

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.

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 Nearly σ-Embedded Subgroups of Finite Groups

2019 ◽  
Vol 26 (04) ◽  
pp. 677-688
Author(s):  
Muhammad Tanveer Hussain ◽  
Chenchen Cao ◽  
Li Zhang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Permutable Subgroup ◽  
A Finite Group

Let G be a finite group, σ={σi|i∈I} be some partition of the set of all primes and σ(G)={σi|σi∩π(G)≠∅}. We say that a subgroup H of G is nearly σ-embedded in G if there exists a σ-permutable subgroup T of G such that HT is a σ-permutable subgroup of G and H∩T≤HσeG, where HσeG is the subgroup of H generated by all those subgroups of H which are σ-permutably embedded in G. In this paper, we study the structure of G under the condition that some given subgroups of G are nearly σ-embedded in G. Some known results are generalized.

scholarly journals The Influence of C- Z-permutable Subgroups on the Structure of Finite Groups

2018 ◽  
Vol 11 (1) ◽  
pp. 160
Author(s):  
Mohammed Mosa Al-shomrani ◽  
Abdlruhman A. Heliel
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Nonempty Subset ◽  
Permutable Subgroup ◽  
Sylow Subgroups ◽  
Permutable Subgroups ◽  
Complete Set ◽  
A Finite Group ◽  
Power Orders

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, say Gp. Let C be a nonempty subset of G. A subgroup H of G is said to be C-Z-permutable (conjugateZ-permutable) subgroup of G if there exists some x ∈ C such that HxGp = GpHx, for all Gp ∈ Z. We investigate the structure of the finite group G under the assumption that certain subgroups of prime power orders of G are C-Z-permutable subgroups of G.

scholarly journals Cyclic permutable subgroups of finite groups

2001 ◽  
Vol 71 (2) ◽  
pp. 169-176 ◽  
Author(s):  
John Cossey ◽  
Stewart E. Stonehewer
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Normal Closure ◽  
Permutable Subgroup ◽  
Odd Order ◽  
Permutable Subgroups ◽  
A Finite Group

AbstractThe authors describe the structure of the normal closure of a cyclic permutable subgroup of odd order in a finite group.

Finite groups with given nearly SΦ-embedded subgroups

2020 ◽  
pp. 2150135
Author(s):  
Venus Amjid ◽  
Chenchen Cao ◽  
Yuemei Mao
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Permutable Subgroup ◽  
A Finite Group

Let [Formula: see text] be a finite group and [Formula: see text] a subgroup of [Formula: see text]. Then [Formula: see text] is said to be [Formula: see text]-permutably embedded in [Formula: see text] if a Sylow [Formula: see text]-subgroup of [Formula: see text] is also a Sylow [Formula: see text]-subgroup of some [Formula: see text]-permutable subgroup of [Formula: see text] for every prime dividing the order of [Formula: see text]. We say that [Formula: see text] is nearly[Formula: see text]-embedded in [Formula: see text] if [Formula: see text] has a normal subgroup [Formula: see text] such that [Formula: see text] is [Formula: see text]-permutable in [Formula: see text] and [Formula: see text], where [Formula: see text] is the subgroup of [Formula: see text] generated by all those subgroups of [Formula: see text] which are [Formula: see text]-permutably embedded in [Formula: see text]. In this paper, we study the properties of the nearly [Formula: see text]-embedded subgroups and use them to determine the structure of finite groups. Some known results are generalized.

ON CONJUGATE-ℨ-PERMUTABLE SUBGROUPS OF FINITE GROUPS

2013 ◽  
Vol 12 (08) ◽  
pp. 1350060 ◽  
Author(s):  
A. A. HELIEL ◽  
T.M. Al-GAFRI
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Nonempty Subset ◽  
Permutable Subgroup ◽  
Sylow Subgroups ◽  
Permutable Subgroups ◽  
Complete Set ◽  
A Finite Group ◽  
Power Orders

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, say Gp. Let C be a nonempty subset of G. A subgroup H of G is said to be C-ℨ-permutable (conjugate-ℨ-permutable) subgroup of G if there exists some x ∈ C such that HxGp = GpHx, for all Gp ∈ ℨ. We investigate the structure of the finite group G under the assumption that certain subgroups of prime power orders of G are C-ℨ-permutable (conjugate-ℨ-permutable) subgroups of G. Our results improve and generalize several results in the literature.

scholarly journals Finite groups with some weakly pronormal subgroups

2020 ◽  
Vol 18 (1) ◽  
pp. 1742-1747
Author(s):  
Jianjun Liu ◽  
Mengling Jiang ◽  
Guiyun Chen
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
A Finite Group

Abstract A subgroup H of a finite group G is called weakly pronormal in G if there exists a subgroup K of G such that G = H K G=HK and H ∩ K H\cap K is pronormal in G. In this paper, we investigate the structure of the finite groups in which some subgroups are weakly pronormal. Our results improve and generalize many known results.

scholarly journals A on simplicity criterion for finite groups

1969 ◽  
Vol 10 (3-4) ◽  
pp. 359-362
Author(s):  
Nita Bryce
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Odd Order ◽  
A Finite Group

M. Suzuki [3] has proved the following theorem. Let G be a finite group which has an involution t such that C = CG(t) ≅ SL(2, q) and q odd. Then G has an abelian odd order normal subgroup A such that G = CA and C ∩ A = 〈1〉.

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.

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