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| ON INNER$\[{\pi ^'}\]$-CLOSED GROUPS AND NORMAL$\[{\pi}\]$-COMPLEMENTS |
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Citation： |
Wang Xiaofong.ON INNER$\[{\pi ^'}\]$-CLOSED GROUPS AND NORMAL$\[{\pi}\]$-COMPLEMENTS[J].Chinese Annals of Mathematics B,1989,10(3):323~331 |
| Page view： 851
Net amount： 886 |
Authors： |
Wang Xiaofong; |
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| Abstract： |
In this paper, the author classifies the finite inner $\[{\pi ^'}\]$-closed groups, and proves the
following results
1.If each proper subgroup $K$ of a group $G$ is weak $\[{\pi }\]$-homogeneous and weak $\[{\pi ^'}\]$-homogeneous, then $G$ is a Schmidt group, or a direct product of two Hall cubgroups.
2.If $G$ is a weak $\[{\pi }\]$-homogeneous group, then $G$ is $\[{\pi }\]$-closed if one of the following statements is true: (l)Each $\[{\pi }\]$-subgroup of $G$ is 2-closed. (2)Eaqh $\[{\pi }\]$-subgroup of $G$ is 2'-closed.
3. Let $G$ be a group and $\[{\pi }\]$ be a set of odd primes. If $\[{N_G}(Z(J(P)))\]$ has a normal $\[{\pi }\]$-completement for a 16w Sylow p-subgroup of G with prime p in $\[{\pi }\]$, then so does $G$. |
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