The following statement is equivalent to the one Galois wrote in a paper submitted in 1830. Is this correct?
Let G be a finite solvable group acting faithfully and primitively on a set S. If a, b are distinct elements of S, then the point-wise stabilizer of {a, b} is 1.
I think this is false as the following examples show. Let p be a prime number. Let Z be the ring of rational integers. Let F = Z/pZ be the finite prime field. Let G = AGL(2, F) be the affine general linear group over F. G acts on F^2 faithfully and primitively. The point-stabilizer of {(0, 0), (0, 1)} is not 1. But G is solvable in case p = 2 or 3, as the order of G is (2^3)3 or (2^4)(3^3) respectively.