Let $G$ be a finite primitive permutation group on a set $\Omega$ with nontrivial point stabilizer $G_{\alpha}$. We say that $G$ is extremely primitive if $G_{\alpha}$ acts primitively on each of its orbits in $\Omega \setminus \{\alpha\}$. In earlier work, Mann, Praeger and Seress have proved that every extremely primitive group is either almost simple or of affine type and they have classified the affine groups up to the possibility of at most finitely many exceptions. More recently, the almost simple extremely primitive groups have been completely determined. If one assumes Wall's conjecture on the number of maximal subgroups of almost simple groups, then the results of Mann et al. show that it just remains to eliminate an explicit list of affine groups in order to complete the classification of the extremely primitive groups. Mann et al. have conjectured that none of these affine candidates are extremely primitive and our main result confirms this conjecture.

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关于极原始仿射群的注记

令 $G$ 是集合 $\Omega$ 上的有限原始置换群，具有非平凡点稳定器 $G_{\alpha}$。如果 $G_{\alpha}$ 原始地作用于 $\Omega \setminus \{\alpha\}$ 中的每个轨道，我们就说 $G$ 是极其原始的。在早期的工作中，Mann、Praeger 和 Seress 证明了每个极端原始群要么几乎是简单的，要么是仿射群，并且他们将仿射群分类为最多有有限多个例外的可能性。最近，几乎简单的极端原始群体已经完全确定。如果假设沃尔对几乎单群的最大子群数的猜想，那么 Mann 等人的结果。表明它只需要消除一个明确的仿射群列表，以完成对极端原始群的分类。曼等人。