In 2011, Fang et al. in [9] posed the following problem: Classify non-normal locally primitive Cayley graphs of finite simple groups of valency d, where either $d\le 20$ or d is a prime number. The only case for which the complete solution of this problem is known is of $d=3$. Except this, a lot of efforts have been made to attack this problem by considering the following problem: Characterize finite nonabelian simple groups which admit non-normal locally primitive Cayley graphs of certain valency $d\ge 4$. Even for this problem, it was only solved for the cases when either $d\le 5$ or $d=7$ and the vertex stabilizer is solvable. In this paper, we make crucial progress towards the above problems by completely solving the second problem for the case when $d\ge 11$ is a prime and the vertex stabilizer is solvable.

2011年，Fang等人。在[9]中提出了以下问题：对化合价为d的有限简单组的非正规局部原始Cay​​ley图进行分类，其中 $d\le 20$ 或d是质数。唯一知道此问题的完整解决方案的情况是$d=3$。除此之外，通过考虑以下问题，人们为解决该问题做出了许多努力：表征有限的非阿贝尔简单群，这些非阿贝尔简单群允许某些价态的非正规局部本机Cayley图 $d\ge 4$。即使是这个问题，也只能在以下情况下解决：$d\le 5$ 要么 $d=7$顶点稳定剂是可解决的。在本文中，我们通过针对以下情况完全解决第二个问题，在解决上述问题方面取得了重要进展：$d\ge 11$ 是素数，顶点稳定剂是可解的。