σ-Subnormality in locally finite groups

Ferrara, Maria; Trombetti, Marco

doi:10.1016/j.jalgebra.2022.10.013

Journal of Algebra

2023

DOI: 10.1016/j.jalgebra.2022.10.013

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σ-Subnormality in locally finite groups

Maria Ferrara

Marco Trombetti

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Cited by 8 publications

References 25 publications

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On locally finite groups whose derived subgroup is locally nilpotent

Trombetti

2024

Mathematische Nachrichten

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A celebrated theorem of Helmut Wielandt shows that the nilpotent residual of the subgroup generated by two subnormal subgroups of a finite group is the subgroup generated by the nilpotent residuals of the subgroups. This result has been extended to saturated formations in Ballester‐Bolinches, Ezquerro, and Pedreza‐Aguilera [Math. Nachr. 239–240 (2002), 5–10]. Although Wielandt's result is not true in arbitrary locally finite groups, we are able to extend it (even in a stronger form) to homomorphic images of periodic linear groups. Also, all results in Ballester‐Bolinches, Ezquerro, and Pedreza‐Aguilera [Math. Nachr. 239–240 (2002), 5–10] are extended to locally finite groups, so it is possible to characterize the class of locally finite groups with a locally nilpotent derived subgroup as the largest subgroup‐closed saturated formation such that, for all ‐closed saturated formations , the ‐residual of an ‐group generated by ‐subnormal subgroups is the subgroup generated by their ‐residuals. Our proofs are based on a reduction theorem that is of an independent interest. Furthermore, we provide strengthened versions of Wielandt's result for other relevant classes of groups, among which we mention the class of paranilpotent groups. A brief discussion on the permutability of the residuals is given at the end of the paper.

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On locally finite groups whose derived subgroup is locally nilpotent

Trombetti

2024

Mathematische Nachrichten

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Periodic linear groups in which permutability is a transitive relation

Ferrara,

Trombetti

2023

Annali di Matematica

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A PT-group is a group in which the relation of being a permutable subgroup is transitive. The main aim of this paper is to show that a (homomorphic image of a) periodic linear group is a soluble PT-group if and only if each subgroup of a Sylow subgroup is permutable in the corresponding Sylow normalizer (see Theorem 4.7); for a fixed prime p, the latter condition is denoted by $$\mathfrak {X}_p$$ X p . In order to prove our main theorem, we need (i) to characterize (homomorphic images of) periodic linear groups that are PT-groups (see Sect. 2), (ii) to develop a fusion theory for locally finite groups (see Sect. 3), (iii) to carefully study (homomorphic images of) periodic linear groups with the property $$\mathfrak {X}_p$$ X p for a fixed prime p (see for instance Theorem 4.6). As a by-product we obtain (among other results) a characterization of (homomorphic images of) periodic linear $$\mathfrak {X}_p$$ X p -groups in terms of pronormality (see Theorem 4.11) that will allow us to show that, on some occasions, the property $$\mathfrak {X}_p$$ X p is inherited by subgroups.

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On a question of T. Hawkes

Murashka

2023

Ricerche mat

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σ-Subnormality in locally finite groups

Cited by 8 publications

References 25 publications

On locally finite groups whose derived subgroup is locally nilpotent

On locally finite groups whose derived subgroup is locally nilpotent

Periodic linear groups in which permutability is a transitive relation

On a question of T. Hawkes

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