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].

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 SOME SEMI-p-COVER-AVOIDING OR S-QUASINORMALLY EMBEDDED SUBGROUPS

2009 ◽  
Vol 02 (04) ◽  
pp. 667-680 ◽  
Author(s):  
Shouhong Qiao ◽  
Yanming Wang
Keyword(s):  
Finite Group ◽  
Finite Groups ◽  
Prime Power ◽  
Chief Factor ◽  
Prime Power Order ◽  
Quasinormal Subgroup ◽  
A Finite Group

A subgroup H of a group G is called S-quasinormally embedded in G if, for each prime p dividing the order of H, a Sylow p-subgroup of H is a Sylow p-subgroup of an S-quasinormal subgroup of G. H is said to be semi-p-cover-avoiding in G if there is a chief series 1 = G0 < G1 < ⋯ < Gt = G of G such that, for every i = 1, 2, ⋯, t, if Gi/Gi-1 is a p-chief factor, then H either covers or avoids Gi/Gi-1. We give the structure of a finite group G in which some subgroups of G with prime-power order are either semi-p-cover-avoiding or S-quasinormally embedded in G.

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.

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.

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.

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.

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 weakly ℋC-embedded subgroups

2019 ◽  
Vol 19 (05) ◽  
pp. 2050093 ◽  
Author(s):  
M. Ramadan
Keyword(s):  
Finite Group ◽  
Normal Subgroup ◽  
Finite Groups ◽  
Normal Closure ◽  
A Finite Group

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 of [Formula: see text] if [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 [Formula: see text] has a normal subgroup [Formula: see text] such that [Formula: see text] and [Formula: see text] for all [Formula: see text] where [Formula: see text] is the normal closure of [Formula: see text] in [Formula: see text]. For each prime [Formula: see text] dividing the order of [Formula: see text] let [Formula: see text] be a Sylow [Formula: see text]-subgroup of [Formula: see text]. We fix a subgroup of [Formula: see text] of order [Formula: see text] with [Formula: see text] and study the structure of [Formula: see text] under the assumption that every subgroup of [Formula: see text] of order [Formula: see text] [Formula: see text] is weakly [Formula: see text]-embedded in [Formula: see text]. Our results improve and generalize several recent results in the literature.

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