论文标题
$ \ mathrm {gl}（n，q）$和$ \ mathrm {pgl}（n，q）$的子组晶格上的关闭操作员
A closure operator on the subgroup lattice of $\mathrm{GL}(n,q)$ and $\mathrm{PGL}(n,q)$ in relation to the zeros of the Möbius function
论文作者
论文摘要
令$ \ mathbb {f} _q $为带有$ q $元素的有限字段，并考虑$ n $ -dimensional $ \ mathbb {f} _q $ -vector-vector space $ v = \ mathbb {f} _q^n \，$，$。在本文中，我们在组$ g = \ mathrm {pgl}（v）$的子组晶格上定义了一个封闭操作员。令$μ$表示此格子的Möbius功能。目的是使用此封闭操作员来表征$ g $的子组$ h $ of $ g $，其中$μ（h，g）\ neq 0 $。此外，我们建立了一个多项式限制的封闭子组的$ c（m）$ h $ in Index $ m $ in $ g $的$ h $，$ h $ invariant子空间$ v $的晶格对链条的产物是同构的。此限制仅取决于$ m $，而不取决于$ n $和$ q $的选择。它是通过考虑$ \ mathrm {gl}（v）$的子组晶格的类似闭合操作员来实现的，并为该组证明了相同的结果。
Let $\mathbb{F}_q$ be the finite field with $q$ elements and consider the $n$-dimensional $\mathbb{F}_q$-vector space $V=\mathbb{F}_q^n\,$. In this paper we define a closure operator on the subgroup lattice of the group $G = \mathrm{PGL}(V)$. Let $μ$ denote the Möbius function of this lattice. The aim is to use this closure operator to characterize subgroups $H$ of $G$ for which $μ(H,G)\neq 0$. Moreover, we establish a polynomial bound on the number $c(m)$ of closed subgroups $H$ of index $m$ in $G$ for which the lattice of $H$-invariant subspaces of $V$ is isomorphic to a product of chains. This bound depends only on $m$ and not on the choice of $n$ and $q$. It is achieved by considering a similar closure operator for the subgroup lattice of $\mathrm{GL}(V)$ and the same results proven for this group.
