We study the abstract regular polyhedra with automorphism groups that act faithfully on their vertices, and show that each non-flat abstract regular polyhedron covers a "vertex-faithful" polyhedron with the same number of vertices. We then use this result and earlier work on flat polyhedra to study abstract regular polyhedra based on the size of their vertex set. In particular, we classify all regular polyhedra where the number of vertices is prime or twice a prime. We also construct the smallest regular polyhedra with a prime squared number of vertices.

中文翻译：

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顶点忠实正多面体

我们研究了具有忠实于顶点的自同构群的抽象正则多面体，并表明每个非平面抽象正则多面体都覆盖了一个具有相同顶点数的“忠实顶点”多面体。然后，我们使用这个结果和早期关于平面多面体的工作，根据顶点集的大小来研究抽象的规则多面体。特别是，我们对顶点数为素数或素数两倍的所有正多面体进行分类。我们还用素数平方的顶点构造了最小的正多面体。