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...
Збережено в:
| Дата: | 2025 |
|---|---|
| Автори: | , |
| Формат: | Стаття |
| Мова: | English |
| Опубліковано: |
Lugansk National Taras Shevchenko University
2025
|
| Теми: | |
| Онлайн доступ: | https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2347 |
| Теги: |
Додати тег
Немає тегів, Будьте першим, хто поставить тег для цього запису!
|
| Назва журналу: | Algebra and Discrete Mathematics |
Репозитарії
Algebra and Discrete Mathematics| id |
oai:ojs.admjournal.luguniv.edu.ua:article-2347 |
|---|---|
| record_format |
ojs |
| spelling |
oai:ojs.admjournal.luguniv.edu.ua: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 |
| institution |
Algebra and Discrete Mathematics |
| baseUrl_str |
|
| datestamp_date |
2025-01-19T19:44:59Z |
| collection |
OJS |
| language |
English |
| 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 |
| 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 |
| 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 |
| format |
Article |
| author |
Lukashova, Tetiana Drushlyak, Marina |
| author_facet |
Lukashova, Tetiana Drushlyak, Marina |
| author_sort |
Lukashova, Tetiana |
| title |
On relations between generalized norms in locally finite groups |
| title_short |
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_sort |
on relations between generalized norms in locally finite groups |
| 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. |
| publisher |
Lugansk National Taras Shevchenko University |
| publishDate |
2025 |
| url |
https://admjournal.luguniv.edu.ua/index.php/adm/article/view/2347 |
| work_keys_str_mv |
AT lukashovatetiana onrelationsbetweengeneralizednormsinlocallyfinitegroups AT drushlyakmarina onrelationsbetweengeneralizednormsinlocallyfinitegroups |
| first_indexed |
2025-07-17T10:36:20Z |
| last_indexed |
2025-07-17T10:36:20Z |
| _version_ |
1838376098041167872 |