有限群的交图的控制集
Dominating Sets in Intersection Graphs of Finite Groups
摘要：
设$G$是一个群。$G$的交图$Γ(G)$是一个无环无重边无向图，定义如下：顶点集是$G$的所有真非平凡子群的集合，两个不同顶点$H$和$K$之间存在一条边当且仅当$H\cap K \neq 1$其中$1$表示$G$的平凡子群.本文研究了有限群的交图的控制集。它证明了顶点集的一个子集是一个控制集当且仅当相应子群的并包含所有极小子群的并。我们将阿贝尔群按其控制数进行分类，并找到了某些特定类群的上界。子群交与Burnside环有关。引入了$G$ -集的交图的概念（在一定程度上推广了群的交图的一般定义），并利用Burnside环中满足某种性质的子群建立了$Γ(G)$的控制数的一般上界. $G$的交图是单纯复形的$1$ -骨架，其面是非平凡相交的真子群的集合。我们称之为$G$的单纯复交复形，并证明了它与$G$的真非平凡子群的序复形具有相同的同伦类型。证明了如果$Γ(G)$的控制数是$1$，则$G$的交复形是可收缩的.
英文摘要：
Let $G$ be a group. The intersection graph $Γ(G)$ of $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H$ and $K$ if and only if $H\cap K \neq 1$ where $1$ denotes the trivial subgroup of $G$. In this paper we studied the dominating sets in intersection graphs of finite groups. It turns out a subset of the vertex set is a dominating set if and only if the union of the corresponding subgroups contains the union of all minimal subgroups. We classified abelian groups by their domination number and find upper bounds for some specific classes of groups. Subgroup intersection is related with Burnside rings. We introduce the notion of intersection graph of a $G$-set (somewhat generalizing the ordinary definition of intersection graph of a group) and establish a general upper bound for the domination number of $Γ(G)$ in terms of subgroups satisfying a certain property in Burnside ring. Intersection graph of $G$ is the $1$-skeleton of the simplicial complex whose faces are the sets of proper subgroups which intersect non-trivially. We call this simplicial complex intersection complex of $G$ and show that it shares the same homotopy type with the order complex of proper non-trivial subgroups of $G$. We also prove that if domination number of $Γ(G)$ is $1$, then intersection complex of $G$ is contractible.
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