c * -Supplemented subgroups and p -nilpotency of finite groups
A subgroup $H$ of a finite group $G$ is said to be $c^{*}$-supplemented in $G$ if there exists a subgroup $K$ such that $G = HK$ and $H ⋂ K$ is permutable in $G$. It is proved that a finite group $G$ that is $S_4$-free is $p$-nilpotent if $N_G (P)$ is $p$-nilpotent and, for all $x ∈ G \backslash N_G...
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| Дата: | 2007 |
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| Автори: | , , , |
| Формат: | Стаття |
| Мова: | Англійська |
| Опубліковано: |
Institute of Mathematics, NAS of Ukraine
2007
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| Онлайн доступ: | https://umj.imath.kiev.ua/index.php/umj/article/view/3364 |
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| Назва журналу: | Ukrains’kyi Matematychnyi Zhurnal |
| Завантажити файл: |  |
Репозитарії
Ukrains’kyi Matematychnyi Zhurnal| Резюме: | A subgroup $H$ of a finite group $G$ is said to be $c^{*}$-supplemented in $G$ if there exists a subgroup $K$ such that $G = HK$ and $H ⋂ K$ is permutable in $G$. It is proved that a finite group $G$ that is $S_4$-free is $p$-nilpotent if $N_G (P)$ is $p$-nilpotent and, for all $x ∈ G \backslash N_G (P)$, every minimal subgroup of $P ∩ P^x ∩ G^{N_p}$ is $c^{*}$-supplemented in $P$ and (if $p = 2$) one of the following conditions is satisfied:
(a) every cyclic subgroup of $P ∩ P^x ∩ G^{N_p}$ of order 4 is $c^{*}$-supplemented in $P$,
(b) $[Ω2(P ∩ P^x ∩ G^{N_p}),P] ⩽ Z(P ∩ G^{N_p})$,
(c) $P$ is quaternion-free, where $P$ a Sylow $p$-subgroup of $G$ and $G^{N_p}$ is the $p$-nilpotent residual of $G$.
This extends and improves some known results. |
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