Products of $\scr N$-connected groups

Hauck, Peter; Martínez-Pastor, A.; Pérez‐Ramos, M. D.

doi:10.1215/ijm/1258138089

Cited by 10 publications

(19 citation statements)

References 12 publications

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“…(2) (Ballester-Bolinches and Pedreza-Aguilera, 1999;Ballester-Bolinches et al, 2003;Hauck, 2003). Let F be a saturated formation containing N. Then G 2 F if and only if G i 2 F for all i 2 f1; 2; .…”

Section: _lagb32_12_r3_101604mentioning

confidence: 99%

“…Lemma 1 (Hauck et al, 2003). Let the group G ¼ G 1 G 2 Á Á Á G r be the product of the pairwise permutable and N-connected subgroups G 1 ; G 2 ; .…”

Section: Preliminary Lemmasmentioning

confidence: 99%

“…The study of N-connected products was taken further by Ballester-Bolinches and Pedraza-Aguilera (1999) where they considered products of two N-connected groups within the framework of formation theory of soluble groups. They prove, for example, that if G ¼ HK is an Nconnected product, F is a saturated formation of soluble groups containing N, and if A is an F-projector of H and B is an F-projector of K, then A permutes with B and AB is an F-projector of G. More recently Hauck et al (2003) investigated N-connected products within the entire finite universe and the products of finitely many factors. Hauck et al (2003) studied the behaviour of projectors and residuals associated with saturated formations and radicals and injectors associated with Fitting classes.…”

Section: Introduction and Statement Of Theoremsmentioning

confidence: 99%

“…They prove, for example, that if G ¼ HK is an Nconnected product, F is a saturated formation of soluble groups containing N, and if A is an F-projector of H and B is an F-projector of K, then A permutes with B and AB is an F-projector of G. More recently Hauck et al (2003) investigated N-connected products within the entire finite universe and the products of finitely many factors. Hauck et al (2003) studied the behaviour of projectors and residuals associated with saturated formations and radicals and injectors associated with Fitting classes. Their results on Fischer classes are surprising.…”

Section: Introduction and Statement Of Theoremsmentioning

confidence: 99%

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Pairwise -connected Products of Certain Classes of Finite Groups

Beidleman

Heineken

2004

Communications in Algebra

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Subgroups A and B of a finite group are said to be N-connected if the subgroup generated by elements x and y is a nilpotent group, for every pair of elements x in A and y in B. The behaviour of finite pairwise permutable and N-connected products are studied with respect to certain classes of groups including those groups where all the subnormal subgroups permute with all the maximal subgroups, the so-called SM-groups, and also the class of soluble groups where all the subnormal subgroups permute with all the Carter subgroups, the so-called C-groups.

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Section: _lagb32_12_r3_101604mentioning

confidence: 99%

“…Lemma 1 (Hauck et al, 2003). Let the group G ¼ G 1 G 2 Á Á Á G r be the product of the pairwise permutable and N-connected subgroups G 1 ; G 2 ; .…”

Section: Preliminary Lemmasmentioning

confidence: 99%

Section: Introduction and Statement Of Theoremsmentioning

confidence: 99%

Section: Introduction and Statement Of Theoremsmentioning

confidence: 99%

See 2 more Smart Citations

Pairwise -connected Products of Certain Classes of Finite Groups

Beidleman

Heineken

2004

Communications in Algebra

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“…Ballester-Bolinches and Pedraza-Aguilera [2] and Hauck, Martínez-Pastor and Pérez-Ramos [13]); for instance, G = A B is an N -connected product of A and B if and only if G modulo its hypercenter is a direct product of the images of A and B. For N 2 , the class of metanilpotent groups, and for saturated formations F ⊆ N A, the class of nilpotent-by-abelian groups, there are also very satisfactory characterizations of finite (soluble) groups which are products of F -connected subgroups (cf.…”

Section: Introduction and Notationmentioning

confidence: 98%

Saturated formations and products of connected subgroups

2011

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For a non-empty class of groups C, two subgroups A and B of a group G are said to be C-connected if a, b ∈ C for all a ∈ A and b ∈ B. Given two sets π and ρ of primes, S π S ρ denotes the class of all finite soluble groups that are extensions of a normal π -subgroup by a ρ-group.It is shown that in a finite group G = A B, with A and B soluble subgroups, then A and B are S π S ρ -connected if and only if O ρ (B) centralizes A O π (G)/O π (G), O ρ (A) centralizes B O π (G)/O π (G)and G ∈ S π ∪ρ . Moreover, if in this situation A and B are in S π S ρ , then G is in S π S ρ . This result is then extended to a large family of saturated formations F , the so-called nilpotent-like Fitting formations of soluble groups, and to finite groups that are products of arbitrarily many pairwise permutable F -connected F -subgroups.

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Thompson-like characterization of solubility for products of finite groups

Hauck

Kazarin

Martínez-Pastor

et al. 2020

Annali di Matematica

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A remarkable result of Thompson states that a finite group is soluble if and only if all its two-generated subgroups are soluble. This result has been generalized in numerous ways, and it is in the core of a wide area of research in the theory of groups, aiming for global properties of groups from local properties of two-generated (or more generally, n-generated) subgroups. We contribute an extension of Thompson's theorem from the perspective of factorized groups. More precisely, we study finite groups G = AB with subgroups A, B such that a, b is soluble for all a ∈ A and b ∈ B. In this case, the group G is said to be an S-connected product of the subgroups A and B for the class S of all finite soluble groups. Our main theorem states that G = AB is S-connected if and only if [A, B] is soluble. In the course of the proof we derive a result about independent primes regarding the soluble graph of almost simple groups that might be interesting in its own right.

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Products of $\scr N$-connected groups

Cited by 10 publications

References 12 publications

Pairwise -connected Products of Certain Classes of Finite Groups

Pairwise -connected Products of Certain Classes of Finite Groups

Saturated formations and products of connected subgroups

Thompson-like characterization of solubility for products of finite groups

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