论文标题
有限的完全$ k $ cluped的群体
Finite totally $k$-closed groups
论文作者
论文摘要
对于一个正整数$ k $，如果在每个忠实的排列表示中，则$ g $完全限制为$ k $，例如在$ω$中，$ g $是$ \ operatatorname {sym}（sym}（sym}（ω）$）的最大亚组，它在$ g $ -orbits中都在$ g $ -orbits tere pocused time time time time time time time time k $ k $ k $ k $ k \ ocy of form.我们证明，每个Abelian Group $ g $都是完全$（n（g）+1）$ - 关闭，但并不完全是$ n（g）$ - 关闭，其中$ n（g）$是$ g $不变因子分解中不变因素的数量。特别是，我们证明，对于每个$ k \ geq2 $和每个prime $ p $，都有无限的有限的Abelian $ p $ groups，它们完全是$ k $ c的，但并非完全$ $（K-1）$ - 关闭。在特殊情况下，此结果$ k = 2 $归因于Abdollahi和Arezoomand。我们提出了几个关于$ k $ clubersus的公开问题。
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 $Ω$, $G$ is the largest subgroup of $\operatorname{Sym}(Ω)$ which leaves invariant each of the $G$-orbits in the induced action on $Ω\times\dots\times Ω=Ω^k$. 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.
