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On conditional permutability and saturated formations
Abstract
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.
... Focusing in products of groups, along the last decades, some relations of permutability between the factors have been considered by many authors, as, for instance, total permutability, mutual permutability (see [4]) and tcc-permutability (see [1,2]). These last permutability relations are inherited by quotients, and they ensure the existence of a minimal normal subgroup contained in one of the factors. ...
... Example 1 Let 1 = G = AB be the product of the subgroups A and B, and let assume that A and B satisfy one of the following permutability properties: Applying [4,Theorem 4.3.11] in (i) and [2,Lemma 2.5] in (ii), it can be seen that A G B G = 1. Also, the above permutability properties are clearly inherited by quotients. ...
... Proof If either S is a group of Lie type or p ≥ 5, then [14, Proposition 2.1] applies and S has an irreducible character of p-defect zero (note that this case includes the groups A 5 ∼ = P SL (2,5) and A 6 ∼ = P SL (2,9)). Hence, it remains to consider sporadic simple groups and alternating groups, and p ∈ {2, 3}. ...
An element g of a finite group G is said to be vanishing inG if there exists an irreducible character (Formula presented.) of G such that (Formula presented.); in this case, g is also called a zero of G. The aim of this paper is to obtain structural properties of a factorised group (Formula presented.) when we impose some conditions on prime power order elements (Formula presented.) which are (non-)vanishing in G. © 2018 Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag GmbH Germany, part of Springer Nature
... 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]. A compilation of recent results can be found in[2]. ...
... More exactly, the supersoluble residuals of the factors centralize each other in a product of tcc-permutable subgroups. Moreover, Corollary 5 allows us to avoid restrictions to soluble groups in[3]and to extend the research in this reference to the universe of all finite groups (see Section 3). ...
... To be more specific, we recall first that a formation is a class F of groups closed under homomorphic images, such that G/M ∩N ∈ F whenever G is a group and M, N are normal subgroups of G with G/M ∈ F and G/N ∈ F. In this case the F-residual G F of G is the smallest normal subgroup of G such that G/G F ∈ F. The formation F is saturated if G ∈ F whenever G/Φ(G) ∈ F, where Φ(G) denotes the Frattini subgroup of G. U denotes the class of all finite supersoluble groups. Now we can state the main result in[3], which is the following: ...
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.
An element g of a finite group G is said to be vanishing in G if there exists an irreducible character of G such that ; in this case, g is also called a zero of G. The aim of this paper is to obtain structural properties of a factorised group G=AB when we impose some conditions on prime power order elements which are (non-)vanishing 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 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.
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.
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.
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 the structure of finite groups which are the product of two totally permutable subgroups is studied. In fact we can obtain the -residual, where is a formation, -projectors and -normalisers, where is a saturated formation, of the group from the corresponding subgroups of the factor subgroups.
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.
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 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.
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 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.
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.
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.
Theorems of Kegel–Wielandt type, in Groups St Andrews 1997 in Bath I
- A Carocca
- R Maier
A. Carocca and R. Maier, Theorems of Kegel–Wielandt type, in Groups St Andrews 1997 in Bath I, in London Mathematical Society Lecture Note Series, Volume 260, pp. 195– 201 (Cambridge University Press, 1999).