On weakly ℋ-subgroups and p-nilpotency of finite groups

2017 ◽  
Vol 16 (03) ◽  
pp. 1750042 ◽  
Author(s):  
Changwen Li ◽  
Shouhong Qiao
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Prime Power ◽  
Sufficient Conditions ◽  
A Finite Group ◽  
Power Orders

Let [Formula: see text] be a finite group and [Formula: see text] a subgroup of [Formula: see text]. We say that [Formula: see text] is an [Formula: see text]-subgroup in [Formula: see text] if [Formula: see text] for all [Formula: see text]; [Formula: see text] is called weakly [Formula: see text]-subgroup in [Formula: see text] if it has a normal subgroup [Formula: see text] such that [Formula: see text] and [Formula: see text] is an [Formula: see text]-subgroup in [Formula: see text]. In this paper, we present some sufficient conditions for a group to be [Formula: see text]-nilpotent under the assumption that certain subgroups of fixed prime power orders are weakly [Formula: see text]-subgroups in [Formula: see text]. The main results improve and extend new and recent results in the literature.

A note on CAP-subgroups in finite groups

2020 ◽  
pp. 2150171
Author(s):  
Dengfeng Liang ◽  
Guohua Qian
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Prime Power ◽  
Chief Factor ◽  
A Finite Group

A subgroup [Formula: see text] of a finite group [Formula: see text] is called a CAP-subgroup if [Formula: see text] covers or avoids every chief factor of [Formula: see text]. Let [Formula: see text] be a normal subgroup of a finite group [Formula: see text] and [Formula: see text] be a prime power such that [Formula: see text] or [Formula: see text]. In this note, we show that if all subgroups of order [Formula: see text], and all subgroups of order 4 when [Formula: see text] and [Formula: see text] has a nonabelian Sylow [Formula: see text]-subgroup, of [Formula: see text] are CAP-subgroups of [Formula: see text], then every [Formula: see text]-chief factor of [Formula: see text] is either a [Formula: see text]-group or of order [Formula: see text].

On c#-normal subgroups of finite groups

2016 ◽  
Vol 16 (08) ◽  
pp. 1750160
Author(s):  
Guo Zhong ◽  
Shi-Xun Lin
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Normal Subgroups ◽  
A Finite Group ◽  
Necessary And Sufficient

Let [Formula: see text] be a subgroup of a finite group [Formula: see text]. We say that [Formula: see text] is a [Formula: see text]-normal subgroup of [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] is a [Formula: see text]-subgroup of [Formula: see text]. In the present paper, we use [Formula: see text]-normality of subgroups to characterize the structure of finite groups, and establish some necessary and sufficient conditions for a finite group to be [Formula: see text]-supersolvable, [Formula: see text]-nilpotent and solvable. Our results extend and improve some recent ones.

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.

On weakly ℋC-embedded subgroups of finite groups

2016 ◽  
Vol 15 (05) ◽  
pp. 1650077 ◽  
Author(s):  
M. Asaad ◽  
M. Ramadan
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Prime Power ◽  
Prime Power Order ◽  
Group A ◽  
A Finite Group

Let [Formula: see text] be a finite group. A subgroup [Formula: see text] of [Formula: see text] is said to be an [Formula: see text]-subgroup of [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text] We say that [Formula: see text] is weakly [Formula: see text]-embedded in [Formula: see text] if there exists a normal subgroup [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text]. In this paper, we investigate the structure of the finite group [Formula: see text] under the assumption that some subgroups of prime power order are weakly [Formula: see text]-embedded in [Formula: see text]. Our results improve and generalize several recent results in the literature.

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 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.

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 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.

scholarly journals Automorphism orbits of finite groups

1986 ◽  
Vol 40 (2) ◽  
pp. 253-260 ◽  
Author(s):  
Thomas J. Laffey ◽  
Desmond MacHale
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Finite Groups ◽  
Prime Power ◽  
Structure Theorem ◽  
Equivalence Classes ◽  
Prime Power Order ◽  
A Finite Group

AbstractLet G be a finite group and let Aut(G) be its automorphism group. Then G is called a k-orbit group if G has k orbits (equivalence classes) under the action of Aut(G). (For g, hG, we have g ~ h if ga = h for some Aut(G).) It is shown that if G is a k-orbit group, then kGp + 1, where p is the least prime dividing the order of G. The 3-orbit groups which are not of prime-power order are classified. It is shown that A5 is the only insoluble 4-orbit group, and a structure theorem is proved about soluble 4-orbit groups.

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