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Conditional permutability of subgroups and certain classes of groups
Abstract
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.
... In the articles [9]- [13] (see also the references from [13]) we can see that the supersolubility of a group can also be obtained for other generalizations of totally permutable product. ...
... In the articles [9]- [13] (see also the references from [13]) we can see that the supersolubility of a group can also be obtained for other generalizations of totally permutable product. ...
... The notation H G means that H is a subgroup of a group G. So, for example, the product G = AB is said to be tcc-permutable [13], if for any X A and Y B there exists an element u ∈ X, Y such that XY u G. The subgroups A and B in this product are called tcc-permutable. ...
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.
... 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. ...
... 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. ...
... 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. The subgroups A and B in this product are called tcc-permutable. ...
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.
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 𝑋𝑌^𝑢 ≤ 𝐺. The notation 𝐻 ⩽ 𝐺 means that 𝐻 is a subgroup of a group 𝐺. In this paper we proved that the class of all SM-groups is closed under the product of tcc- subgroups. Here an SM-group is a group where each subnormal subgroup permutes with every maximal subgroup.
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.
We obtain estimations for the p -length of a group G=AB in terms of some invariant parameters of p-soluble conditionally permutable factors A and B.
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 XY u ≤ G. The notation H ≤ G means that H is a subgroup of a group G. In this paper, we obtained the estimations of numerical invariants(the derived length, the nilpotent length, the π-length, the nilpotent π-length) of G = AB in terms of invariants of tcc-subgroups A and B.
The structure of finite groups in which permutability is transitive (PT-groups) is studied in detail. In particular a finite PT-group has simple chief factors and the p-chief factors fall into at most two isomorphism classes. The structure of finite T-groups, that is, groups in which normality is transitive, is also discussed, as is that of groups generated by subnormal or normal PT-subgroups.
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.
In this paper a structural theorem about mutually permutable products of finite groups is obtained. This result is used to derive some results on mutually permutable products of groups whose chief factors are simple. Some earlier results on mutually permutable products of supersoluble groups appear as particular cases.
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
Finite groups which are products of pairwise totally permutable subgroups are studied in this paper. The -residual, -projectors and -normalizers in such groups are obtained from the corresponding subgroups of the factor subgroups under suitable hypotheses.
Finite groups G=AB factorized by two subgroups A and B such that every subgroup of A permutes with every subgroup of B are studied in this paper. The behaviour of such products with respect to the class of finite groups in which Sylow-permutability is transitive is analyzed.
A subgroup of a group G is called quasinormal in G if it permutes with every subgroup of G. For a finite group G it is well known that quasinormal subgroups are not far from being normal: It6 and Sz6p [3] have shown that a quasinormal subgroup of G containing no nontrivial normal subgroup of G is always nilpotent. In this paper we shall prove a stronger result, namely: Theorem. If Q is a quasinormal subgroup of the finite group G, then Q~/QG is contained in the hypercentre Zoo (G/Q~) of G/Q~.
The study of products of groups whose factors are linked by certain permutability conditions has been the subject of fruitful investigations by a good number of authors. A particular starting point was the interest in providing criteria for products of supersoluble groups to be supersoluble. We take further previous research on total and mutual permutability by considering significant weaker permutability hypotheses. The aim of this note is to report about new progress on structural properties of factorized groups within the considered topic. As a consequence, we discuss new attainments in the framework of formation theory.
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.
We describe the finite groups satisfying one of the following conditions: all maximal subgroups permute with all subnormal subgroups, (2) all maximal subgroups and all Sylow p-subgroups for p<7 permute with all subnormal subgroups.
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 are closed under the product of subgroups which are connected by certain permutability properties.
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.
Let G=AB be the mutually permutable product of the nontrivial subgroups A and B of the group G. Then A or B contains a nontrivial normal subgroup of G. It is also established that S(G)∩A=S(A), where S(U) is the solvable radical of U. These facts lead to some generalized results about mutually permutable products of SC-groups. It is also shown that if G is a PTS-group, then A and B are PST-groups.
We investigate the structure of finite groups that are the mutually permutable product of two supersoluble groups. We show that the supersoluble residual is nilpotent and the Fitting quotient group is metabelian. These results are consequences of our main theorem, which states that such a product is supersoluble when the intersection of the two factors is core-free in the group.
Let the finite group G=AB be the mutually permutable product of the subgroups A and B and let F be a Fitting class. Then the F-radicals AF and BF of the factors A and B are mutually permutable. Using this, we also prove the inclusion G′∩AFBF⩽GF, which generalizes the fact that A∈F and B∈F implies G′∈F.
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.
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.
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.
Subgroups A and B of a finite group are said to be totally permutable if every subgroup of A permutes with every subgroup of B. The behaviour of finite pairwise totally permutable products are studied with respect to certain classes of groups including the class of groups where all the subnormal subgroups permute with all the maximal subgroups, the so-called SM-groups, and also the class of groups where all the subnormal subgroups are permutable, the so-called PT-groups.
Let G = AB be the mutually permutable product of the subgroups A and B. It is shown that if A and B are contained in a Fitting class
F\mathcal {F}
, then the commutator subgroup G′ of G is also contained in
F\mathcal {F}
.
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.
We consider the product AB of two finite mutually permutable subgroups A, B and find some subnormal subgroups of the product. This leads to local and otherwise generalized statements about products of supersolvable groups.