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On some products of groups

Part of: Other groups of matrices Representation theory of groups

Published online by Cambridge University Press:  02 April 2025

Luigi Iorio  and
Marco Trombetti Open the ORCID record for Marco Trombetti [Opens in a new window]
Luigi Iorio
Affiliation:
Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, Napoli, Campania, Italy
Marco Trombetti*
Affiliation:
Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Università degli Studi di Napoli Federico II, Complesso Universitario Monte S. Angelo, Via Cintia, Napoli, Campania, Italy
*
Corresponding author: Marco Trombetti, email: [email protected]
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Abstract

A classical result of Reinhold Baer states that a group G = XN, which is the product of two normal supersoluble subgroups X and N, is supersoluble if and only if Gʹ is nilpotent. This result has been weakened in [6] for a finite group G: in fact, we do not need that both X and N are normal, but only that N is normal and X permutes with every maximal subgroup of each Sylow subgroup of N.

In our paper, we improve the result mentioned above by showing that we only need X to permute with the maximal subgroups of the non-cyclic Sylow subgroups of N. Also, we extend this result (and several others) to relevant classes of infinite groups.

The central idea behind our results stems from grasping the key aspects of what happens in [6]. It turns out that tensor products play a very crucial role, and it is precisely this shift of perspective that makes it possible not only to improve those results but also extend them to infinite groups.

Keywords

periodic linear groupresidual subgroupsemidirect productsupersoluble group

MSC classification

Primary: 20D10: Solvable groups, theory of formations, Schunck classes, Fitting classes, $pi$-length, ranks
Secondary: 20D20: Sylow subgroups, Sylow properties, $pi$-groups, $pi$-structure 20H20: Other matrix groups over fields
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , First View , pp. 1 - 29
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Copyright
© The Author(s), 2025. Published by Cambridge University Press on Behalf of The Edinburgh Mathematical Society.

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