Abstract
We prove that a transitive permutation group of degree n with a cyclic point stabilizer and whose order is n(n-1) is isomorphic to the affine group of degree 1 over a field with n elements. More generally we show that if a finite group G has an abelian and core-free Hall subgroup Q, then either Q has a small order (2|Q|2 < |G|) or G is a direct product of 2-transitive Frobenius groups.
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Lucchini, A., Mainardis, M. & Stellmacher, B. Transitive Permutation Groups with Cyclic Point Stabilizers of Maximum Order. Geometriae Dedicata 100, 117–121 (2003). https://doi.org/10.1023/A:1025886014568
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DOI: https://doi.org/10.1023/A:1025886014568