有限完全k闭群

D. Churikov, C. Praeger
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引用次数: 5

摘要

对于一个正整数$k$,如果在它的每一个忠实的排列表示中,比如在一个集合$\Omega$上,$G$是$\operatorname{Sym}(\Omega)$的最大的子群,并且在$\Omega\times\dots\times \Omega=\Omega^k$的诱导作用中每个$G$轨道都保持不变,则称群$G$是完全$k$闭的。证明了每个阿贝尔群$G$是完全$(n(G)+1)$ -闭的,但不是完全$n(G)$ -闭的,其中$n(G)$为$G$不变因子分解中不变因子的个数。特别地,我们证明了对于每一个$k\geq2$和每一个质数$p$,存在无穷多个完全$k$ -闭但不完全$(k-1)$ -闭的有限阿贝尔$p$ -群。这种特殊情况$k=2$的结果是由于阿卜杜拉希和阿雷佐曼。我们提出了几个关于$k$ -关闭的开放性问题。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Finite totally k-closed groups
For a positive integer $k$, a group $G$ is said to be totally $k$-closed if in each of its faithful permutation representations, say on a set $\Omega$, $G$ is the largest subgroup of $\operatorname{Sym}(\Omega)$ which leaves invariant each of the $G$-orbits in the induced action on $\Omega\times\dots\times \Omega=\Omega^k$. We prove that every abelian group $G$ is totally $(n(G)+1)$-closed, but is not totally $n(G)$-closed, where $n(G)$ is the number of invariant factors in the invariant factor decomposition of $G$. In particular, we prove that for each $k\geq2$ and each prime $p$, there are infinitely many finite abelian $p$-groups which are totally $k$-closed but not totally $(k-1)$-closed. This result in the special case $k=2$ is due to Abdollahi and Arezoomand. We pose several open questions about total $k$-closure.
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