Non-Abelian Groups with Many Abelian Subgroups

Marco Trombetti

doi:10.6093/978-88-6887-331-8

Non-Abelian Groups with Many Abelian Subgroups

Authors

Marco Trombetti

University of Naples Federico II

https://orcid.org/0000-0003-4532-3690

DOI: https://doi.org/10.6093/978-88-6887-331-8

Keywords:

Group theory, Non-abelian groups, Abelian subgroups, Metahamiltonian groups, Metabelian groups

Synopsis

Publisher: FedOA - Federico II University Press

Series: ATPAM-DLS Advanced Topics in Pure and Applied Mathematics: Doctoral Lecture Series

Pages: 56

Language: English

Abstract: The present volume is largely based on the lectures from a Ph.D. course I taught in early 2024 at the Dipartimento di Matematica e Applicazioni "Renato Caccioppoli" of the Università degli Studi di Napoli Federico II. The course was titled "Groups with Many Abelian Subgroups." In these notes, we explore the structure of (soluble) non-abelian groups whose proper subgroups are all abelian, commonly referred to as minimal non-abelian groups. Additionally, we examine some properties of groups in which every subgroup is either normal or abelian, known as metahamiltonian groups. The final chapter focuses on metabelian groups with a trivial center, whose proper quotients are abelian, showing that even in this context, there are numerous abelian subgroups.

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Author Biography

Marco Trombetti, University of Naples Federico II

Marco Trombetti is Associate Professor of Algebra at the Dipartimento di Matematica e Applicazioni "Renato Caccioppoli" of the Università degli Studi di Napoli Federico II. He is vice-president of the no-profit association "AGTA - Advances in Group Theory and Applications" and Editor-in-Chief of the journal "Advances in Group Theory and Applications". On January 30, 2020, he was awarded (ex aequo with E. Giannelli) the Premio Nazionale "Mario Curzio" for the best young algebra researcher (non-professor) in group theory by the Accademia Pontaniana.