Let G be a finite group. A subgroup H of G is called pronormal in G if for every g∈G, H and Hg are conjugate in <H,Hg>. A subgroup H …
Let G be a finite group. A subgroup H of G is called pronormal in G if for every g∈G, H and Hg are conjugate in <H,Hg>. A subgroup H of G is called weakly pronormal in G if there exists a subgroup K of G such that G = HK and H∩K is pronormal in G. In this paper, we investigate the structure of G under the assumption that some subgroups of G are weakly pronormal in G.
After publication, we realized that the proofs of Lemma 2.17 and Theorem 3.4 below are incorrect. We present here the correct proofs of these two results.
After publication, we realized that the proofs of Lemma 2.17 and Theorem 3.4 below are incorrect. We present here the correct proofs of these two results.
Abstract Let G be a finite group and H a subgroup of G. We say that H is an ℌ -subgroup of G if N G ( H ) ∩ …
Abstract Let G be a finite group and H a subgroup of G. We say that H is an ℌ -subgroup of G if N G ( H ) ∩ H g ≤ H for all g ∈ G ; H is called weakly ℌ -embedded in G if G has a normal subgroup K such that H G = HK and H ∩ K is an ℌ -subgroup of G , where H G is the normal clousre of H in G , i. e., H G = 〈 H g | g ∈ G 〉. In this paper, we study the p -nilpotence of a group G under the assumption that every subgroup of order d of a Sylow p -subgroup P of G with 1 < d < | P | is weakly ℌ -embedded in G . Many known results related to p -nilpotence of a group G are generalized.
Let [Formula: see text] be a finite group. If [Formula: see text] is a subgroup of [Formula: see text] and [Formula: see text] a subgroup of [Formula: see text], we …
Let [Formula: see text] be a finite group. If [Formula: see text] is a subgroup of [Formula: see text] and [Formula: see text] a subgroup of [Formula: see text], we say that [Formula: see text] is strongly closed in [Formula: see text] with respect to [Formula: see text] if [Formula: see text] for any [Formula: see text]. We say that a subgroup [Formula: see text] of [Formula: see text] is strongly closed in [Formula: see text] if [Formula: see text] is strongly closed in [Formula: see text] with respect to [Formula: see text]. A subgroup [Formula: see text] of a group [Formula: see text] is said to be weakly [Formula: see text]-supplemented in [Formula: see text] if [Formula: see text] has a subgroup [Formula: see text] such that [Formula: see text] and [Formula: see text] is strongly closed in [Formula: see text]. In this paper, we study the structure of a group [Formula: see text] under the assumption, that some subgroups of prime power order are weakly [Formula: see text]-supplemented in [Formula: see text]. Our results extend and generalize several recent results in the literature.
Let G be a finite group. A subgroup H of G is said to be s -permutable in G if H permutes with all Sylow subgroups of G . Let …
Let G be a finite group. A subgroup H of G is said to be s -permutable in G if H permutes with all Sylow subgroups of G . Let H be a subgroup of G and let H sG be the subgroup of H generated by all those subgroups of H which are s -permutable in G . A subgroup H of G is called n -embedded in G if G has a normal subgroup T such that H G = HT and H ∩ T ≦ H sG , where H G is the normal closure of H in G . We investigate the influence of n -embedded subgroups of the p -nilpotency and p -supersolvability of G .
Let G be a finite group. A subgroup H of G is said to be weakly s-supplemented in G if there exists a subgroup K of G such that G …
Let G be a finite group. A subgroup H of G is said to be weakly s-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ H s G , where H s G is the subgroup of H generated by all those subgroups of H which are s-quasinormal in G. In this article, we investigate the structure of G under the assumption that some families of subgroups of G are weakly s-supplemented in G. Some recent results are generalized.
Abstract. Let G be a finite group. A subgroup H of G is called pronormal in G if for each <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>g</m:mi> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> ${g\in G}$ …
Abstract. Let G be a finite group. A subgroup H of G is called pronormal in G if for each <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>g</m:mi> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> ${g\in G}$ , the subgroups H and H g are conjugate in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>〈</m:mo> <m:mi>H</m:mi> <m:mo>,</m:mo> <m:msup> <m:mi>H</m:mi> <m:mi>g</m:mi> </m:msup> <m:mo>〉</m:mo> </m:mrow> </m:math> ${\langle H, H^g \rangle }$ . A subgroup H of G is called weakly pronormal in G if there exists a subgroup K of G such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>=</m:mo> <m:mi>H</m:mi> <m:mspace width="0.166667em" /> <m:mi>K</m:mi> </m:mrow> </m:math> ${G=H\,K}$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>∩</m:mo> <m:mi>K</m:mi> </m:mrow> </m:math> ${H\cap K}$ is pronormal in G . The main purpose of this paper is to investigate the structure of G under the assumption that some subgroups of G are weakly pronormal in G .
A subgroup H of a group G is said to be c-supplemented inG if there exists a subgroup K of G such thatHK=G and H \cap K is contained in …
A subgroup H of a group G is said to be c-supplemented inG if there exists a subgroup K of G such thatHK=G and H \cap K is contained in Core_G (H). We follow Hall's ideas to characterize the structure of the finite groups in which every subgroup is c-supplemented. Properties of c-supplemented subgroups are also applied to determine the structure of some finite groups.
Let G be a finite group. A subgroup H of G is said to be weakly s-supplemented in G if there exists a subgroup K of G such that G …
Let G be a finite group. A subgroup H of G is said to be weakly s-supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K ≤ H s G , where H s G is the subgroup of H generated by all those subgroups of H which are s-quasinormal in G. In this article, we investigate the structure of G under the assumption that some families of subgroups of G are weakly s-supplemented in G. Some recent results are generalized.
Let G be a finite group. A subgroup A of G is said to be S-quasinormal in G if AP = PA for all Sylow subgroups P of G. The …
Let G be a finite group. A subgroup A of G is said to be S-quasinormal in G if AP = PA for all Sylow subgroups P of G. The symbol HsG denotes the subgroup generated by all those subgroups of H which are S-quasinormal in G. A subgroup H is said to be S-supplemented in G if G has a subgroup T such that T ∩ H ⩽ HsG and HT = G; see [Skiba, J. Algebra 315: 192–209, 2007].
Abstract A subgroup H of a finite group G is said to be c –supplemented in G if there exists a subgroup K of G such that G = HK …
Abstract A subgroup H of a finite group G is said to be c –supplemented in G if there exists a subgroup K of G such that G = HK and H ∩ K is contained in core G (H) . In this paper some results for finite p –nilpotent groups are given based on some subgroups of P c –supplemented in G , where p is a prime factor of the order of G and P is a Sylow p –subgroup of G . We also give some applications of these results.
1. Discussion of results 1·1. Introduction: The classical theorem of Lagrange states that the order of a subgroup of a finite group G divides the order, ( G ), of …
1. Discussion of results 1·1. Introduction: The classical theorem of Lagrange states that the order of a subgroup of a finite group G divides the order, ( G ), of G . More generally, if H and K are subgroups of G , and H ≥ K , then ( G : K ) = ( G : H )( H : K ), where ( G : K ) denotes the index of K in G , etc. We call a number a possible order of a subgroup of G if it is a divisor of ( G ), and a possible order of a subgroup of G containing a subgroup H if it is a divisor of ( G ) and a multiple of ( H ). In this paper we discuss conditions on G for the existence of subgroups of every possible order, the existence of subgroups of every possible order containing arbitrary subgroups, and similar properties.
A subgroup H is said to be an ℋ-subgroup of a group G if Hg ∩ NG(H) ⩽ H for all g ∈ G. In this paper, we investigate the …
A subgroup H is said to be an ℋ-subgroup of a group G if Hg ∩ NG(H) ⩽ H for all g ∈ G. In this paper, we investigate the structure of a group G under the assumption that every subgroup of order pm of a Sylow p-subgroup of G belongs to ℋ(G) for a given positive integer m. Some results related to p-nilpotence and supersolvability of a group G are obtained.
Abstract A subgroup H of G is said to be c-normal in G if there exists a normal subgroup N of G such that HN = G and H ∩ …
Abstract A subgroup H of G is said to be c-normal in G if there exists a normal subgroup N of G such that HN = G and H ∩ N ≤ H G = Core(H). We extend the study on the structure of a finite group under the assumption that all maximal or minimal subgroups of the Sylow subgroups of the generalized Fitting subgroup of some normal subgroup of G are c-normal in G. The main theorems we proved in this paper are: Theorem Let ℱ be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ℱ. If all maximal subgroups of any Sylow subgroup of F*(H) are c-normal in G, then G ∈ ℱ. Theorem Let ℱ be a saturated formation containing 𝒰. Suppose that G is a group with a normal subgroup H such that G/H ∈ ℱ. If all minimal subgroups and all cyclic subgroups of F*(H) are c-normal in G, then G ∈ ℱ.
The purpose of this paper is to study a class of finite groups whose subgroups of prime power order are all pro-normal. Following P. Hall, we say that a subgroup …
The purpose of this paper is to study a class of finite groups whose subgroups of prime power order are all pro-normal. Following P. Hall, we say that a subgroup H of a group G is pro-normal in G if and only if, for all x in G, H and Hz are conjugate in (H, HX), the subgroup generated by H and HX. We shall determine the structure of all finite groups whose subgroups of prime power order are pro-normal. It turns out that these groups are in fact soluble t-groups. A t-group is a group G whose subnormal subgroups are all normal in G. The structure of finite soluble t-groups has been determined by Gaschuitz [1]. Are soluble t-groups precisely those, groups whose subgroups of prime power order are all pro-normal? The answer is yes. Thus our study furnishes another characterization of soluble t-groups. Except for the definitions given above, our terminology and notation are standard (see, for example, M. Hall [2 ]). We first prove two general facts about pro-normal subgroups.
ABSTRACT A subgroup K of a finite group G is called an ℋ-subgroup of G if the following condition is satisfied: The set of all ℋ-subgroups of a finite group …
ABSTRACT A subgroup K of a finite group G is called an ℋ-subgroup of G if the following condition is satisfied: The set of all ℋ-subgroups of a finite group G will be denoted by ℋ(G). In this paper, we investigate the structure of a finite group G under the assumption that certain subgroups of prime power orders belong to ℋ(G). Communicated by M. Dixon
Notation and definitions 384 3. Statement of main theorem and corollaries 388 4. Proofs of corollaries 389 5. Preliminary lemmas 389 5.1.Inequalities and modules 389 5.2.7r-reducibility and fl,(®) 395 5.3.Groups …
Notation and definitions 384 3. Statement of main theorem and corollaries 388 4. Proofs of corollaries 389 5. Preliminary lemmas 389 5.1.Inequalities and modules 389 5.2.7r-reducibility and fl,(®) 395 5.3.Groups of symplectic type 397 5.4.^-groups, ^-solvability and F((&) 400 5.5.Groups of low order 404 5.6.2-groups, involutions and 2-length 409 5.7.Factorizations 423 5.8.Miscellaneous 426 6.A transitivity theorem 428 i 9 68]
We introduce a new subgroup embedding property in a finite group called c*-normality. We investigate the influence of c*-normal maximal subgroups on the p-nilpotency, p-supersolvability and supersolvability of finite groups. …
We introduce a new subgroup embedding property in a finite group called c*-normality. We investigate the influence of c*-normal maximal subgroups on the p-nilpotency, p-supersolvability and supersolvability of finite groups. Our results unify and generalize some earlier results.
Let G be a group and H be a subgroup of G. We say that H is weakly Φ-supplemented in G if G has a subgroup T such that HT …
Let G be a group and H be a subgroup of G. We say that H is weakly Φ-supplemented in G if G has a subgroup T such that HT = G and , where denotes the Frattini subgroup of H. In this paper, properties of this new kind of inequalities of subgroups are investigated and new characterizations of nilpotency and supersolubility of finite groups in terms of the new inequalities are obtained. MSC:20D10, 20D15, 20D20.
Suppose G is a finite group and H is subgroup of G. H is said to be s-permutable in G if HG p = G p H for any Sylow …
Suppose G is a finite group and H is subgroup of G. H is said to be s-permutable in G if HG p = G p H for any Sylow p-subgroup G p of G; H is called weakly s-supplemented subgroup of G if there is a subgroup T of G such that G = HT and H ∩ T ≤ H sG , where H sG is the subgroup of H generated by all those subgroups of H which are s-permutable in G. We investigate the influence of minimal weakly s-supplemented subgroups on the structure of finite groups and generalize some recent results. Furthermore, we give a positive answer in the minimal subgroup case for Skiba's Open Questions in [On weakly s-permutable subgroups of finite groups, J. Algebra315 (2007) 192–209].
In this article, we investigate the structure of a finite group \g under the assumption that some subgroups of \g are c-normal in $G$. The main theorem is as follows: …
In this article, we investigate the structure of a finite group \g under the assumption that some subgroups of \g are c-normal in $G$. The main theorem is as follows: Let \e be a normal finite group of $G$. If all subgroups of \ep with order \dpp and 2\dpp (if $p=2$ and $E_{p}$ is not an abelian nor quaternion free 2-group) are c-normal in $G$, then \e is \phe $G$. We give some applications of the theorem and generalize some known results.
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 …
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.
Abstract. Let G be a finite group. A subgroup H of G is called pronormal in G if for each <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>g</m:mi> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> ${g\in G}$ …
Abstract. Let G be a finite group. A subgroup H of G is called pronormal in G if for each <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>g</m:mi> <m:mo>∈</m:mo> <m:mi>G</m:mi> </m:mrow> </m:math> ${g\in G}$ , the subgroups H and H g are conjugate in <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>〈</m:mo> <m:mi>H</m:mi> <m:mo>,</m:mo> <m:msup> <m:mi>H</m:mi> <m:mi>g</m:mi> </m:msup> <m:mo>〉</m:mo> </m:mrow> </m:math> ${\langle H, H^g \rangle }$ . A subgroup H of G is called weakly pronormal in G if there exists a subgroup K of G such that <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>G</m:mi> <m:mo>=</m:mo> <m:mi>H</m:mi> <m:mspace width="0.166667em" /> <m:mi>K</m:mi> </m:mrow> </m:math> ${G=H\,K}$ and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>H</m:mi> <m:mo>∩</m:mo> <m:mi>K</m:mi> </m:mrow> </m:math> ${H\cap K}$ is pronormal in G . The main purpose of this paper is to investigate the structure of G under the assumption that some subgroups of G are weakly pronormal in G .
Let G be a finite group and H a subgroup of G. We say that H is an ℋ-subgroup in G if NG(H) ∩ Hg ≤ H for all g …
Let G be a finite group and H a subgroup of G. We say that H is an ℋ-subgroup in G if NG(H) ∩ Hg ≤ H for all g ∈ G; H is called weakly ℋ-subgroup in G if G has a normal subgroup K such that G = HK and H ∩ K is an ℋ-subgroup in G. We say that H is weakly ℋ -embedded in G if G has a normal subgroup K such that HG = HK and H ∩ K is an ℋ-subgroup in G. In this paper, we investigate the structure of the finite group G under the assumption that some subgroups of prime power order are weakly ℋ-embedded in G. Our results improve and generalize several recent results in the literature.
Publisher Summary This chapter describes and presents a theorem of strongly closed 2-subgroups of finite group. S is a strongly closed subgroup of G if NSg (S) ⊆ S for …
Publisher Summary This chapter describes and presents a theorem of strongly closed 2-subgroups of finite group. S is a strongly closed subgroup of G if NSg (S) ⊆ S for all g ∈ G. If S is a p-group, the condition is equivalent to Sg ⋂ P ⊆ S for all g ∈ G, where S ⊆ P ∈ Sylp (G). The following theorem can be obtained. Suppose the finite group G contains a direct product of two strongly closed 2-subgroups S1 × S2, then
A subgroup [Formula: see text] of a finite group [Formula: see text] is called pronormal in [Formula: see text] if for each [Formula: see text], the subgroups [Formula: see text] …
A subgroup [Formula: see text] of a finite group [Formula: see text] is called pronormal in [Formula: see text] if for each [Formula: see text], the subgroups [Formula: see text] and [Formula: see text] are conjugate in [Formula: see text]. In this paper, we analyze how certain properties of pronormal subgroups influence the structure of groups.
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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>G</m:mi><m:mo>=</m:mo><m:mi>H</m:mi><m:mi>K</m:mi></m:math> G=HK and <m:math …
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 <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>G</m:mi><m:mo>=</m:mo><m:mi>H</m:mi><m:mi>K</m:mi></m:math> G=HK and <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"><m:mi>H</m:mi><m:mo>∩</m:mo><m:mi>K</m:mi></m:math> 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.
The major aim of the note is to strengthen the results of Asaad in Communications in Algebra, 42 (2014), 2319–2330.
The major aim of the note is to strengthen the results of Asaad in Communications in Algebra, 42 (2014), 2319–2330.