On relations between generalized norms in locally finite groups
In the paper the relations between such generalized norms as the norm of Abelian non-cyclic subgroups and the norm of decomposable subgroups in the class of infinite locally finite groups are studied. The local nilpotency and non-Dedekindness of the norm of Abelian non-cyclic subgroups are considere...
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Lugansk National Taras Shevchenko University
2025
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| Online Zugang: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2347 |
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| Назва журналу: | Algebra and Discrete Mathematics |
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Algebra and Discrete Mathematics| _version_ | 1856543347086196736 |
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| author | Lukashova, Tetiana Drushlyak, Marina |
| author_facet | Lukashova, Tetiana Drushlyak, Marina |
| author_sort | Lukashova, Tetiana |
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| collection | OJS |
| datestamp_date | 2025-01-19T19:44:59Z |
| description | In the paper the relations between such generalized norms as the norm of Abelian non-cyclic subgroups and the norm of decomposable subgroups in the class of infinite locally finite groups are studied. The local nilpotency and non-Dedekindness of the norm of Abelian non-cyclic subgroups are considered as the restrictions. It was proved that any infinite locally finite group with mentioned restrictions on the norm of Abelian non-cyclic subgroups is a finite extension of a quasicyclic \(p\)-subgroup and does not contain Abelian non-cyclic \(p'\)-subgroups. Moreover, in such groups the norm of Abelian non-cyclic subgroups necessarily includes Abelian non-cyclic subgroups and therefore is a non-Hamiltonian \(\overline{HA}\)-group (i.e., a group with the normality condition for Abelian non-cyclic subgroups), whose structure is known. It was shown that for infinite locally finite groups with the non-Dedekind locally nilpotent norm \(N_G^A\) the relation \(N^A_G \supseteq N^d_G\) holds. The inclusion is proper for infinite torsion non-primary locally nilpotent groups with the mentioned restrictions on the norm \(N_G^A\), as well as for infinite locally finite groups in which the norm \(N_G^A\) is a non-Dedekind non-primary locally nilpotent group. |
| first_indexed | 2026-02-08T08:01:47Z |
| format | Article |
| id | admjournalluguniveduua-article-2347 |
| institution | Algebra and Discrete Mathematics |
| language | English |
| last_indexed | 2026-02-08T08:01:47Z |
| publishDate | 2025 |
| publisher | Lugansk National Taras Shevchenko University |
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| spelling | admjournalluguniveduua-article-23472025-01-19T19:44:59Z On relations between generalized norms in locally finite groups Lukashova, Tetiana Drushlyak, Marina norm of a group, generalized norms, norm of Abelian non-cyclic subgroups of a group, norm of decomposable subgroups of a group, non- Dedekindness, locally finite group In the paper the relations between such generalized norms as the norm of Abelian non-cyclic subgroups and the norm of decomposable subgroups in the class of infinite locally finite groups are studied. The local nilpotency and non-Dedekindness of the norm of Abelian non-cyclic subgroups are considered as the restrictions. It was proved that any infinite locally finite group with mentioned restrictions on the norm of Abelian non-cyclic subgroups is a finite extension of a quasicyclic \(p\)-subgroup and does not contain Abelian non-cyclic \(p'\)-subgroups. Moreover, in such groups the norm of Abelian non-cyclic subgroups necessarily includes Abelian non-cyclic subgroups and therefore is a non-Hamiltonian \(\overline{HA}\)-group (i.e., a group with the normality condition for Abelian non-cyclic subgroups), whose structure is known. It was shown that for infinite locally finite groups with the non-Dedekind locally nilpotent norm \(N_G^A\) the relation \(N^A_G \supseteq N^d_G\) holds. The inclusion is proper for infinite torsion non-primary locally nilpotent groups with the mentioned restrictions on the norm \(N_G^A\), as well as for infinite locally finite groups in which the norm \(N_G^A\) is a non-Dedekind non-primary locally nilpotent group. Lugansk National Taras Shevchenko University 2025-01-19 Article Article Peer-reviewed Article application/pdf https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2347 10.12958/adm2347 Algebra and Discrete Mathematics; Vol 38, No 2 (2024): A special issue 2415-721X 1726-3255 en https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2347/pdf Copyright (c) 2025 Algebra and Discrete Mathematics |
| spellingShingle | norm of a group generalized norms norm of Abelian non-cyclic subgroups of a group norm of decomposable subgroups of a group non- Dedekindness locally finite group Lukashova, Tetiana Drushlyak, Marina On relations between generalized norms in locally finite groups |
| title | On relations between generalized norms in locally finite groups |
| title_full | On relations between generalized norms in locally finite groups |
| title_fullStr | On relations between generalized norms in locally finite groups |
| title_full_unstemmed | On relations between generalized norms in locally finite groups |
| title_short | On relations between generalized norms in locally finite groups |
| title_sort | on relations between generalized norms in locally finite groups |
| topic | norm of a group generalized norms norm of Abelian non-cyclic subgroups of a group norm of decomposable subgroups of a group non- Dedekindness locally finite group |
| topic_facet | norm of a group generalized norms norm of Abelian non-cyclic subgroups of a group norm of decomposable subgroups of a group non- Dedekindness locally finite group |
| url | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2347 |
| work_keys_str_mv | AT lukashovatetiana onrelationsbetweengeneralizednormsinlocallyfinitegroups AT drushlyakmarina onrelationsbetweengeneralizednormsinlocallyfinitegroups |