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.
Transitive permutation groups with cyclic point stabilizers of maximum order
LUCCHINI, ANDREA;
2003
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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