A partial linear space is a pair $(\mathcal{P},\mathcal{L})$ where $\mathcal{P}$ is a non-empty set of points and $\mathcal{L}$ is a collection of subsets of $\mathcal{P}$ called lines such that any two distinct points are contained in at most one line, and every line contains at least two points. A partial linear space is proper when it is not a linear space or a graph. A group of automorphisms $G$ of a proper partial linear space acts transitively on ordered pairs of distinct collinear points and ordered pairs of distinct non-collinear points precisely when $G$ is transitive of rank 3 on points. In this paper, we classify the finite proper partial linear spaces that admit rank 3 affine primitive automorphism groups, except for certain families of small groups, including subgroups of $\mathrm{A}\mathrm{\Gamma }{\mathrm{L}}_1(q)$. Up to these exceptions, this completes the classification of the finite proper partial linear spaces admitting rank 3 primitive automorphism groups. We also provide a more detailed version of the classification of the rank 3 affine primitive permutation groups, which may be of independent interest.

中文翻译：

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具有自同构的 3 阶仿射本原群的部分线性空间

部分线性空间是一对 $(\mathcal{磷},\mathcal{升})$ 在哪里 $\mathcal{磷}$ 是一组非空的点，并且 $\mathcal{升}$ 是一个子集的集合 $\mathcal{磷}$任何两个不同的点最多包含在一条线上，并且每条线至少包含两个点。当它不是线性空间或图时，部分线性空间是合适的。一组自同构$G$ 一个适当的部分线性空间传递作用于有序的不同共线点对和有序的不同非共线点对，当 $G$在点上是等级 3 的传递。在本文中，我们对允许 3 阶仿射本原自同构群的有限真偏线性空间进行分类，除了某些小群族，包括$\mathrm{一种}\mathrm{\Gamma }{\mathrm{升}}_1(q)$. 直到这些例外，这完成了接纳秩 3 本原自同构群的有限真局部线性空间的分类。我们还提供了一个更详细的 3 级仿射原始置换组的分类版本，这可能是独立的兴趣。