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On finite products of groups and supersolubility
Abstract
Two subgroups X and Y of a group G are said to be conditionally permutable in G if X permutes with Yg for some element g∈G, i.e., XYg is a subgroup of G. Using this permutability property new criteria for the product of finite supersoluble groups to be supersoluble are obtained and previous results are recovered. Also the behaviour of the supersoluble residual in products of finite groups is studied.
... Using these permutability properties new criteria for a product of finite supersoluble subgroups to be supersoluble are obtained in [21], [31], [32] and by the authors in [1], extending known results. Also in [1] the behaviour of the supersoluble residual in products of finite groups is studied, by considering conditional permutability (not necessarily complete) as mentioned below in this Introduction (Theorem 2). ...
... Using these permutability properties new criteria for a product of finite supersoluble subgroups to be supersoluble are obtained in [21], [31], [32] and by the authors in [1], extending known results. Also in [1] the behaviour of the supersoluble residual in products of finite groups is studied, by considering conditional permutability (not necessarily complete) as mentioned below in this Introduction (Theorem 2). Then, inspired by the previous research on totally permutable products of subgroups, an initial study on conditional permutability in the framework of formation theory has been developed in [3]. ...
... More precisely, a celebrated result by Beidleman and Heineken ([12, Theorem 1]) states that a group G = AB which is the product of totally permutable subgroups A and B is close to be a central product in the sense that the nilpotent residual of each factor centralizes the other factor. Example 3 in [1] (also [3, Example 3.6]) shows that this is not true if every subgroup of A is completely c-permutable with every subgroup of B, also it is not true for the supersoluble residuals. However, under this weaker hypothesis, we prove in this paper that the nilpotent residuals of the factors are normal subgroups in the product (Theorem 3). ...
Two subgroups
A
and
B
of a group
G
are said to be totally completely conditionally permutable (tcc-permutable) if
X
permutes with
for some
, for all
and all
. In this paper, we study finite products of tcc-permutable subgroups, focussing mainly on structural properties of such products. As an application, new achievements in the context of formation theory are obtained.
... The following concept is introduced in [5]. ...
... The totally permutable [6] and totally c-permutable [7] subgroups are NS-permutable [5, Lemma 2]. The supersolubility of a group G = AB which is the NS-permutable product of supersoluble subgroups A and B is obtained in [5]. ...
A subgroup A of a group~G is said to be {\sl NS-supplemented} in G, if there exists a subgroup~B of G such that G=AB and whenever X~is a normal subgroup of~A and , there exists a Sylow p-subgroup~ of~B such that . In this paper, we proved the supersolubility of a group with NS-supplemented non-cyclic Sylow subgroups. The solubility of a group with NS-supplemented maximal subgroups is obtained.
... Let A and B be subgroups of a group G and X a nonempty subset of G. Then A is said to be X -permutable with B if there exists some element x in X such that AB x = B x A (in particular, if X = G, then, in [10], A is said to be conditionally Dapeng Yu yudapeng0@sina.com permutable with B); A is said to be X -semipermutable in G if A is X -permutable with all subgroups of some supplement T of A in G. Based on these generalized permutable subgroups, one has given a series of new and interesting characterizations of the structure of finite groups (see [2,6,[9][10][11][12][13][14][15][16]24]). ...
... Since D 1 is subnormal in D by condition (1), D 1 is normal in D. This means that D ≤ N G (D 1 ). Since G = H N G (D 1 ) by (2), N G (D 1 ) contains a nilpotent Hall πsubgroup of G by the solubility of G, B say. Without loss of generality, we may suppose that D 2 ≤ B. If p ∈ π , then, clearly, P D = D P as P is normal in G. ...
Let A be a subgroup of a group G and X a non-empty subset of G. A is said to be X-s-semipermutable in G if A has a supplement T in G such that A is X-permutable with every Sylow subgroup of T. In this paper, some new criteria for a finite group G to be p-nilpotent or supersoluble in terms of X-s-semipermutable subgroups are obtained. In particular, a characterization of finite groups all of whose subgroups are G-s-semipermutable is presented. © 2015, Malaysian Mathematical Sciences Society and Universiti Sains Malaysia.
... монографию [1]). Результат, связанный с тотальной перестановочностью, получил обобщение в [4][5][6]. ...
A subgroup A is called seminormal in a group G if there exists a subgroup B such that G = AB and AX is a subgroup of G for every subgroup X of B. Studying a group of the form G = AB with seminormal supersoluble subgroups A and B, we prove that . Moreover, if the indices of the subgroups A and B of G are coprime then . Here , , and are the formations of all nilpotent, supersoluble, and metanilpotent groups respectively, while is the -residual of H. We also prove the supersolubility of G = AB when all Sylow subgroups of A and B are seminormal in G.
... In the papers [9][10][11] (see also the literature in [11]), we can see that the supersolubility of a group can also be obtained for other generalizations of totally permutable product. So, for example, the product G = AB is said to be tcc-permutable [11], if for any X ≤ A and for any Y ≤ B, there exists an element u ∈ X, Y such that XY u ≤ G. ...
A subgroup A of a group G is called tcc-subgroup in G, if there is a subgroup T of G such that G = AT and for any X ≤ A and for any Y ≤ T, there exists an element u ∈〈X,Y 〉 such that XYu ≤ G. The notation H ≤ G means that H is a subgroup of a group G. In this paper, we proved the supersolubility of a group G = AB in the following cases: A and B are supersoluble tcc-subgroups in G; all Sylow subgroups of A and of B are tcc-subgroups in G; all maximal subgroups of A and of B are tcc-subgroups in G. Besides, the supersolubility of a group G is obtained in each of the following cases: all maximal subgroups of every Sylow subgroup of G are tcc-subgroups in G; every subgroup of prime order or 4 is a tcc-subgroup in G; all 2-maximal subgroups of G are tcc-subgroups in G.
... The totally permutable [4] and totally c-permutable [7] subgroups are NS-permutable [1, Lemma 2]. The supersolubility of a group G = AB such that it is the NS-permutable product of supersoluble subgroups A and B was obtained in [1]. ...
A subgroup A of a finite group G is said to be NS-supplemented in G, if there exists a subgroup B of G such that G=AB and whenever X is a normal subgroup of A and , there exists a Sylow p-subgroup Bp of B such that . In this paper, we prove the supersolubility of a group G in the following cases: every non-cyclic Sylow subgroup of G is NS-supplemented in G; G is soluble and all maximal subgroups of every non-cylic Sylow subgroup of G are NS-supplemented in G. The solubility of a group with NS-supplemented maximal subgroups is obtained.
... The importance of the concept of -permutability is connected first of all to the observation that many important for applications classes of groups can be described in terms of -permutable subgroups [24,38,27,25,29,1]. The concept of -permutability was first introduced and applied in [24] as the form of -permutability. ...
In this paper, some new ideas, recent results and some open questions in the theory of finite partially soluble groups will be given and discussed.
Let G be a finite group and G be a split extension of A by B, that is, G is a semidirect product: , where A and B are subgroups of G. Under the condition that B permutes with every maximal subgroup of Sylow subgroups of A, every maximal subgroup of A or every nontrivial normal subgroup of A, we prove that the supersolvable residual of G is the product of the supersolvable residuals of A and B.
In this paper, we will continue to discuss the topic on NS-supplemented subgroups proposed by Monakhov and Trofimuk. In detail, we obtained the classification of nonabelian pd-composition factors by using the NS-supplemented Sylow p-subgroup (or NS-supplemented sylowizer for every maximal subgroup of a Sylow p-subgroup) and we also investigated p-supersolvable hypercentre ZpU(G) of a group G.
A subgroup A of a group G is called tcc-subgroup in G, if there is a subgroup T of G such that G=AT and for any X≤A and Y≤T there exists an element u∈⟨X,Y⟩ such that XYu≤G. The notation H≤G means that H is a subgroup of a group G. In this paper we consider a group G=AB such that A and B are tcc-subgroups in G. We prove that G belongs to F, when A and B belong to F and F is a saturated formation of soluble groups such that U⊆F. Here U is the formation of all supersoluble groups.
A subgroup A of a group G is called seminormal in G if there exists a subgroup B such that {G=AB} and AX is a subgroup of G for every subgroup X of B . We introduce the new concept that unites subnormality and seminormality. A subgroup A of a group G is called semisubnormal in G if A is subnormal in G or seminormal in G . A group {G=AB} with semisubnormal supersoluble subgroups A and B is studied. The equality {G^{\mathfrak{U}}=(G^{\prime})^{\mathfrak{N}}} is established; moreover, if the indices of subgroups A and B in G are relatively prime, then {G^{\mathfrak{U}}=G^{\mathfrak{N}^{2}}} . Here {\mathfrak{N}} , {\mathfrak{U}} and {\mathfrak{N}^{2}} are the formations of all nilpotent, supersoluble and metanilpotent groups, respectively; {H^{\mathfrak{X}}} is the {\mathfrak{X}} -residual of H . Also we prove the supersolubility of {G=AB} when all Sylow subgroups of A and of B are semisubnormal in G .
Let P1 be the class of all finite groups that are products of two normal supersoluble subgroups. Let P be the class of all nonsupersoluble P1-groups G such that all proper P1-subgroups of G and nontrivial factor groups of G are supersoluble. We classify the P-groups.
Two subgroups A and B of a finite group G are said to be tcc-permutable if X permutes with for some , for all and all . Some aspects about the normal structure of a product of two tcc-permutable subgroups are analyzed. The obtained results allow to study the behaviour of such products in relation with certain classes of groups, namely the class of T-groups and some generalizations.
This book offers a systematic introduction to recent achievements and development in research on the structure of finite non-simple groups, the theory of classes of groups and their applications. In particular, the related systematic theories are considered and some new approaches and research methods are described - e.g., the F-hypercenter of groups, X-permutable subgroups, subgroup functors, generalized supplementary subgroups, quasi-F-group, and F-cohypercenter for Fitting classes. At the end of each chapter, we provide relevant supplementary information and introduce readers to selected open problems.
Let H be a subgroup of a finite group G. H is said to be λ-supplemented in G if G has a subgroup T such that G = HT and H ∩ T ≤ HSE, where HSE denotes the subgroup of H generated by all those subgroups of H, which are S-quasinormally embedded in G. In this article, some results about the λ-supplemented subgroups are obtained, by which we determine the structure of some classes of finite groups. In particular, some new characterizations of p-supersolubility of finite groups are given under the assumption that some primary subgroups are λ-supplemented. As applications, a number of previous known results are generalized.
Two subgroups A and B of a group G are said to be totally completely conditionally permutable (tcc-permutable) in G if X permutes with Yg for some g ∊ 〈X, Y〉, for all X ≤ A and Y ≤ B. We study the belonging of a finite product of tcc-permutable subgroups to a saturated formation of soluble groups containing all finite supersoluble groups.
Let H and T be subgroups of a group G. Then we call H conditionally permutable (or in brevity, c-permutable) with T in G if there exists an element x ∈ G such that HT x = T x H. If H is c-permutable with T in H, T , then we call H completely c-permutable with T in G. By using the above concepts , we will give some new criterions for the supersolubility of a finite group G = AB, where A and B are both supersoluble groups. In particular, we prove that a finite group G is supersoluble if and only if G = AB, where both A, B are nilpotent subgroups of the group G and B is completely c-permutable in G with every term in some chief series of A. We will also give some applications of our new criterions.
In this paper finite groups factorized as products of pairwise totally permutable subgroups are studied in the framework of Fitting classes.
AMS 2000 Mathematics subject classification: Primary 20D10; 20D40
In this paper, we research p-supersolubility of finite groups. We determine the structure of some groups by using the conditionally permutable subgroups. We obtain some sufficient or necessary and sufficient conditions of a finite group is p-supersolvabe.
Let
\frak Z \frak Z be a complete set of Sylow subgroups of a finite group G, that is, for each prime p dividing the order of G,
\frak Z \frak Z contains exactly one and only one Sylow p-subgroup of G. A subgroup H of a finite group G is said to be
\frak Z \frak Z -permutable if H permutes with every member of
\frak Z \frak Z . The purpose here is to study the influence of
\frak Z \frak Z -permutability of some subgroups on the structure of finite groups. Some recent results are generalized.
Let H and T be subgroups of a finite group G. We say that H is completely c-permutable with T in G if there exists an element x ∈ H, T such that HT x = T x H. In this paper, we shall use the concept of completely c-permutable subgroups to determine the supersolubility of the product groups which are expressed by G = AB, where A and B are supersoluble subgroups of G. Some criteria of supersolubility of the above product groups are obtained and some known results are generalized.
Let H and T be subgroups of a finite group G. We say that H is completely c-permutable with T in G if there exists an element x ∈ 〈H,T〉 such that HTx = TxH. In this paper, we use this concept to determine the supersolubility of a group G = AB, where A and B are supersoluble subgroups of G. Some criterions of supersolubility of such groups are obtained and some known results are generalized. © 2009 Academy of Mathematics and Systems Science, Chinese Academy of Sciences, and Suzhou University.
Let p be a class of finite soluble groups which is closed under epimorphic images and let g be a saturated formation. Then if G is a group of minimal order belonging to p but not to g, F(G), the Fitting subgroup of G, is the unique minimal normal subgroup of G. It is to groups with this property that the following proposition is applicable.
For a non-empty class of groups ℱ, two subgroups A and B of a finite group G are said to be ℱ-connected if 〈a, b〉 ∈ ℱ for all a ∈ A and b ∈ B. This paper is a study of ℱ-connection for saturated formations ℱ ⊆ (where denotes the class of all finite groups with nilpotent commutator subgroup). The class of all finite supersoluble groups constitutes an example of such a saturated formation. It is shown for example that in a finite soluble group G = AB the subgroups A and B are -connected if and only if [A, B] ⩽ F(G), where F(G) denotes the Fitting subgroup of G. Also ℱ-connected finite soluble products for any saturated formation ℱ with ℱ ⊆ are characterized.
We collect facts about pairs of torsion subgroups with the property that all subgroups of one of them are permutable with all subgroups of the other. Results are obtained on the structure of the product, the existence of normal subgroups, the embedding of the intersection, and the commutator subgroup of the two factors.
Saturated formations which contain the class of supersoluble groups are closed under the product of two subgroups if every subgroup of one factor is permutable with every subgroup of the other.
LetGbe a finite group and π a set of primes. In this paper, some new criteria for π-separable groups and π-solvable groups in terms of Hall subgroups are proved. The main results are the following:Theorem.G is aπ-separable group if and only if(1)G satisfies Eπand Eπ′;(2)G satisfies Eπ∪{q}and Eπ′∪{p}for all p∈π,q∈π′.Theorem.G is aπ-separable group if and only if(1)G satisfies Eπand Eπ′;(2)G satisfies Ep,qfor all p∈πand q∈π′.
Two subgroups H and K of a group G are said to be totally permutable if every subgroup of H permutes with every subgroup of K. In this paper the behaviour of radicals and injectors associated to Fitting classes in a product of pairwise totally permutable finite groups is studied.
In this paper we deduce some structural properties of a finite group which is a mutually M-permutable product of some subgroups with these properties. We prove the solubility of a finite group G which is a mutually M-permutable product of two soluble subgroups. If moreover the factors are supersoluble subgroups of coprime order, then G is supersoluble. Finally, we characterize what kind of linear orderings ≺ , on the set ℙ of all primes, are such that the mutually M-permutable product of subgroups with Sylow tower of type ≺ is a group with a Sylow tower of the same type.
Two subgroups A and B of a group G are said to be totally completely conditionally permutable (tcc-permutable) in G if X permutes with Yg for some g ∊ 〈X, Y〉, for all X ≤ A and Y ≤ B. We study the belonging of a finite product of tcc-permutable subgroups to a saturated formation of soluble groups containing all finite supersoluble groups.
Using classification theorems of
simple groups, we give a proof of a conjecture on factorized finite groups which is an extension of a
well known theorem due to P. Hall.
Let H and T be subgroups of a finite group G. H is said to be permutable with T in G if HT = TH. In this paper, we use the concept of permutable subgroups to give two new criterions of supersolubility of the product G = AB of finite supersoluble groups A and B.
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In order to understand the effect of different amounts of powder-activated carbon (PAC) injection and bio-solution (NOE-7F) addition on the removal efficiencies of polychlorinated dibenzo-p-dioxins and dibenzofurans (PCDD/Fs) in a fly ash treatment plant with Waelz rotary kiln process, the PCDD/F concentrations in the stack flue gasses were measured and discussed. In the amount of 20, 40 and 50 kg/h PAC injection, the removal efficiencies of PCDD/Fs in the stack flue gas were 86, 96 and 97%, respectively. While adding more amounts of PAC did enhance the removal efficiencies, the reduction fractions of low chlorinated PCDD/F congeners were much higher than those of highly chlorinated PCDD/F congeners. Particularly, a lower amount of PAC injection (20 kg/h), not only cannot remove highly chlorinated PCDD/Fs, but also the carbon surface of the PAC can act as a precursor for the formation promotion of highly chlorinated PCDD/F congeners. The addition of NOE-7F in the raw materials had the dechlorination effect on the PCDD/F removal and mainly inhibited highly chlorinated PCDD/F formation. The combination of both PAC injection and NOE-7F addition has a high potential for practical application.
Mart´ ınez-Pastor and M. D. P´ erez-Ramos, Products of pairwise totally permutable groups
- P Hauck
P. Hauck, A. Mart´ ınez-Pastor and M. D. P´ erez-Ramos, Products of pairwise totally permutable groups, Proc. Edinb. Math. Soc. 46 (2003) 147–157.
- A Carocca
- R Maier
A. Carocca, R. Maier, Theorems of Kegel-Wielandt type, in: Groups St. Andrews 1997 in Bath I, in: London Math. Soc.
Lecture Note Ser., vol. 260, Cambridge University Press, Cambridge, 1999, pp. 195-201.
- P Hauck
- A Martínez-Pastor
- M D Pérez-Ramos
P. Hauck, A. Martínez-Pastor, M.D. Pérez-Ramos, Injectors and radicals in products of totally permutable groups, Comm.
Algebra 31 (12) (2003) 6135-6147.