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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.

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    Let f(X) be a separable irreducible polynomial over a field K. If he meant the decomposition of f(X) took place in an algebraic extention of K, I think his definition is equivalent to that the Galois group of f(X) acts non-primitively on the set of its roots.2012-04-06

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