Infinite locally finite groups groups with the given properties of the norm of Abelian non-cyclic subgroups
In the paper, the properties of infinite locally finite groups with non-Dedekind locally nil\-potent norms of Abelian non-cyclic subgroups are studied. It is proved that such groups are finite extensions of a quasicyclic subgroup and contain Abelian non-cyclic $p$-subgroups for a unique prime $p$. I...
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Ivan Franko National University of Lviv
2024-09-01
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| Online Access: | http://matstud.org.ua/ojs/index.php/matstud/article/view/533 |
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| author | T. D. Lukashova M. G. Drushlyak |
| author_facet | T. D. Lukashova M. G. Drushlyak |
| author_sort | T. D. Lukashova |
| collection | DOAJ |
| description | In the paper, the properties of infinite locally finite groups with non-Dedekind locally nil\-potent norms of Abelian non-cyclic subgroups are studied. It is proved that such groups are finite extensions of a quasicyclic subgroup and contain Abelian non-cyclic $p$-subgroups for a unique prime $p$. In particular, in the paper is prove the following assertions: 1) Let $G$ be an infinite locally finite group and contain the locally nilpotent norm $N_{G}^{A}$ with the non-Hamiltonian Sylow $p$-subgroup $(N_{G}^{A})_{p}$. Then $G$ is a finite extension of a quasicyclic $p$-subgroup, all Sylow $p'$-subgroups are finite and do not contain Abelian non-cyclic subgroups. In particular, Sylow $q$-subgroups ($q$ is an odd prime, $q\in \pi(G)$, $q\neq p$) are cyclic, Sylow $2$-subgroups ($p\neq 2$) are either cyclic or finite quaternion $2$-groups (Theorem 1).
2) Let $G$ be a locally finite non-locally nilpotent group with the infinite locally nilpotent non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups. Then $G=G_{p} \leftthreetimes H,$ where $G_{p}$ is an infinite $\overline{HA}_{p}$-group of one of the types (1)--(4) of Proposition~2 in present paper, which coincides with the Sylow $p$-subgroup of the norm $N_{G}^{A}$, $H$ is a finite group, all Abelian subgroups of which are cyclic, and $(|H|,p)=1$. Any element $h\in H$ of odd order that centralizes some Abelian non-cyclic subgroup $M\subset N_{G}^{A}$ is contained in the centralizer of the norm $N_{G}^{A}$. (Theorem 2).
3) Let $G$ be an infinite locally finite non-locally nilpotent group with the finite nilpotent non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups. Then
$G=H\leftthreetimes K,$ where $H$ is a finite group, all Abelian subgroups of which are cyclic,
$\left(\left|H\right|,2\right)=1$, $K$ is an infinite 2-group of one of the types (5)--(6) of Proposition~2 (in present paper). Moreover, the norm $N_{K}^{A}$ of Abelian non-cyclic subgroups of the group $K$ is finite, $K\cap N_{G}^{A}=N_{K}^{A}$ and coincides with the Sylow 2-subgroup $(N_{G}^{A})_2$ of the norm $N_{G}^{A}$ of a group $G$.
Moreover, any element $h\in H$ of the centralizer of some Abelian non-cyclic subgroup $M \subset N_{G}^{A}$ is contained in the centralizer of the norm $N_{G}^{A}$. (Theorem 4). |
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| spelling | doaj-art-d72ce911d7bf4b58bf739de8b9b2c0b52025-08-20T03:33:17ZdeuIvan Franko National University of LvivМатематичні Студії1027-46342411-06202024-09-01621112010.30970/ms.62.1.11-20533Infinite locally finite groups groups with the given properties of the norm of Abelian non-cyclic subgroupsT. D. Lukashova0M. G. Drushlyak1Sumy State Pedagogical University named after A. S. Makarenko Sumy, UkraineSumy State Pedagogical University named after A. S. Makarenko Sumy, UkraineIn the paper, the properties of infinite locally finite groups with non-Dedekind locally nil\-potent norms of Abelian non-cyclic subgroups are studied. It is proved that such groups are finite extensions of a quasicyclic subgroup and contain Abelian non-cyclic $p$-subgroups for a unique prime $p$. In particular, in the paper is prove the following assertions: 1) Let $G$ be an infinite locally finite group and contain the locally nilpotent norm $N_{G}^{A}$ with the non-Hamiltonian Sylow $p$-subgroup $(N_{G}^{A})_{p}$. Then $G$ is a finite extension of a quasicyclic $p$-subgroup, all Sylow $p'$-subgroups are finite and do not contain Abelian non-cyclic subgroups. In particular, Sylow $q$-subgroups ($q$ is an odd prime, $q\in \pi(G)$, $q\neq p$) are cyclic, Sylow $2$-subgroups ($p\neq 2$) are either cyclic or finite quaternion $2$-groups (Theorem 1). 2) Let $G$ be a locally finite non-locally nilpotent group with the infinite locally nilpotent non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups. Then $G=G_{p} \leftthreetimes H,$ where $G_{p}$ is an infinite $\overline{HA}_{p}$-group of one of the types (1)--(4) of Proposition~2 in present paper, which coincides with the Sylow $p$-subgroup of the norm $N_{G}^{A}$, $H$ is a finite group, all Abelian subgroups of which are cyclic, and $(|H|,p)=1$. Any element $h\in H$ of odd order that centralizes some Abelian non-cyclic subgroup $M\subset N_{G}^{A}$ is contained in the centralizer of the norm $N_{G}^{A}$. (Theorem 2). 3) Let $G$ be an infinite locally finite non-locally nilpotent group with the finite nilpotent non-Dedekind norm $N_{G}^{A}$ of Abelian non-cyclic subgroups. Then $G=H\leftthreetimes K,$ where $H$ is a finite group, all Abelian subgroups of which are cyclic, $\left(\left|H\right|,2\right)=1$, $K$ is an infinite 2-group of one of the types (5)--(6) of Proposition~2 (in present paper). Moreover, the norm $N_{K}^{A}$ of Abelian non-cyclic subgroups of the group $K$ is finite, $K\cap N_{G}^{A}=N_{K}^{A}$ and coincides with the Sylow 2-subgroup $(N_{G}^{A})_2$ of the norm $N_{G}^{A}$ of a group $G$. Moreover, any element $h\in H$ of the centralizer of some Abelian non-cyclic subgroup $M \subset N_{G}^{A}$ is contained in the centralizer of the norm $N_{G}^{A}$. (Theorem 4).http://matstud.org.ua/ojs/index.php/matstud/article/view/533infinite locally finite groupgeneralized norms of a groupnorm of abelian non-cyclic subgroups of a grouplocally nilpotent group |
| spellingShingle | T. D. Lukashova M. G. Drushlyak Infinite locally finite groups groups with the given properties of the norm of Abelian non-cyclic subgroups Математичні Студії infinite locally finite group generalized norms of a group norm of abelian non-cyclic subgroups of a group locally nilpotent group |
| title | Infinite locally finite groups groups with the given properties of the norm of Abelian non-cyclic subgroups |
| title_full | Infinite locally finite groups groups with the given properties of the norm of Abelian non-cyclic subgroups |
| title_fullStr | Infinite locally finite groups groups with the given properties of the norm of Abelian non-cyclic subgroups |
| title_full_unstemmed | Infinite locally finite groups groups with the given properties of the norm of Abelian non-cyclic subgroups |
| title_short | Infinite locally finite groups groups with the given properties of the norm of Abelian non-cyclic subgroups |
| title_sort | infinite locally finite groups groups with the given properties of the norm of abelian non cyclic subgroups |
| topic | infinite locally finite group generalized norms of a group norm of abelian non-cyclic subgroups of a group locally nilpotent group |
| url | http://matstud.org.ua/ojs/index.php/matstud/article/view/533 |
| work_keys_str_mv | AT tdlukashova infinitelocallyfinitegroupsgroupswiththegivenpropertiesofthenormofabeliannoncyclicsubgroups AT mgdrushlyak infinitelocallyfinitegroupsgroupswiththegivenpropertiesofthenormofabeliannoncyclicsubgroups |