AI 中文总结
介绍群 \(G\) 在集合 \(X\) 上的群作用 - 稳定子图 \(GAS(G)\)，刻画其特殊子图 \(gas(G)\)，确定特定群内群作用数量及 \(GAS(G)=gas(G)\) 的条件，研究有限幂零群 \(G\) 时 \(gas(G)\) 及其补图为导出图的条件。
AI 中文摘要
本文引入了群 \(G\) 在集合 \(X\) 上的群作用 - 稳定子图 \(GAS(G)\)，其顶点集为 \(G\) 在 \(X\) 上所有群作用的集合。当且仅当 \(G\) 的非平凡子群 \(\cap G_x^\phi\) 和 \(\cap G_x^\psi\) 非平凡相交时，两个顶点 \(\phi\) 和 \(\psi\) 相邻，其中 \(G_x^\phi\) 和 \(G_x^\psi\) 分别是关于作用 \(\phi\) 和 \(\psi\) 的 \(x\) 的稳定子。我们刻画了 \(GAS(G)\) 的一个特殊子图 \(gas(G)\)，确定了某些特定群内群作用的数量，并找出 \(GAS(G)\) 等于 \(gas(G)\) 的条件。还研究了对于有限幂零群 \(G\)，\(gas(G)\) 及其补图是导出图的条件。
英文摘要
In this paper we introduce the group action-stabilizer graph $GAS(G)$ of a group $G$ on a set $X$ with vertex set as the collection of all the group actions of $G$ on $X$, and any two vertices $ϕ$ and $ψ$ are adjacent if and only if the non-trivial subgroups $\cap G_x^ϕ$ and $\cap G_x^ψ$ of $G$ intersect non-trivially, where $G_x^ϕ$ and $G_x^ψ$ are two stabilizers of $x$ with respect to the actions $ϕ$ and $ψ$, respectively. We characterize a special subgraph $gas(G)$ of $GAS(G)$ in which the vertex set contains the actions $ϕ$ of $G$ on $X$ such that $\cap G_x^ϕ$'s are distinct. We determine the number of group actions within some specific groups and find certain conditions under which $GAS(G)$ is equal to $gas(G)$. We also examine the conditions under which $gas(G)$ and its complement are derived graph for a finite nilpotent group $G$.
