The finite primitive groups with soluble stabilizers, and the edge-primitive s -arc transitive graphs

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Motivated by the study of several problems in algebraic graph theory, we study finite primitive permutation groups whose point stabilizers are soluble. Such primitive permutation groups are divided into three types: affine, almost simple and product action, and the product action type can be reduced to the almost simple type. This paper gives an explicit list of the soluble maximal subgroups of almost simple groups. The classification is then applied to classify edge-primitive s-arc transitive graphs with s ⩾ 4, solving a problem proposed by Richard M. Weiss (1999).