Sharp permutation groups whose point stabilizers are Frobenius groups with cyclic Frobenius kernel

Abstract

Let $(G,X)$ be a transitive non-geometric sharp permutation group of type $\{0,k\}$ and let $x\in X$. We prove that if the point stabilizer $G_x$ is a Frobenius group with cyclic Frobenius kernel, then $G_x\cong \text{AGL}(1,p)$ for some odd prime p and $(G,X)$ is permutation isomorphic to a certain subgroup of $\text{AGL}(2,p)$ acting on its natural module.

KEYWORDS:

• Finite permutation group
• frobenius group
• geometric permutation group
• imprimitive permutation group
• sharp permutation group

2020 MATHEMATICS SUBJECT CLASSIFICATION:

• 20B05
• 20B10