标题： 确定4度半弧传递图的顶点稳定子 显示英文标题

标题： Determining the vertex stabilizers of 4-valent half-arc-transitive graphs

摘要： 我们说一个群是$4$-HAT-stabilizer，如果它是某个连通$4$-valent 半弧传递图的顶点稳定子群。 2001年，Marušič和Nedela证明了每个$4$-HAT-stabilizer 必须是一个同心群。 然而，在过去的二十年中，只有非常小比例的同心群被证明是$4$-HAT-stabilizers。 本文发展了一种理论，该理论提供了一个通用框架，用于确定一个同心群是否是$4$-HAT-stabilizer。 通过这种方法，我们显著扩展了已知的$4$-HAT-stabilizers 列表。 作为推论，我们确认$\mathcal{H}_7\times C_2^{m-7}$是$4$-HAT-稳定器，对于$m\geq 7$，实现了 Spiga 和 Xia 提出的猜想的目标。 显示英文摘要

摘要： We say that a group is a $4$-HAT-stabilizer if it is the vertex stabilizer of some connected $4$-valent half-arc-transitive graph. In 2001, Maru\v{s}i\v{c} and Nedela proved that every $4$-HAT-stabilizer must be a concentric group. However, over the past two decades, only a very small proportion of concentric groups have been shown to be $4$-HAT-stabilizers. This paper develops a theory that provides a general framework for determining whether a concentric group is a $4$-HAT-stabilizer. With this approach, we significantly extend the known list of $4$-HAT-stabilizers. As a corollary, we confirm that $\mathcal{H}_7\times C_2^{m-7}$ are $4$-HAT-stabilizers for $m\geq 7$, achieving the goal of a conjecture posed by Spiga and Xia.