Minimal normal subgroups of transitive permutation groups of square-free degree

It is shown that a minimal normal subgroup of a transitive permutation group of square-free degree in its induced action is simple and quasiprimitive, with three exceptions related to A5A5, A7A7, and PSL(2,29)PSL(2,29). Moreover, it is shown that a minimal normal subgroup of a 2-closed permutation group of square-free degree in its induced action is simple. As an almost immediate consequence, it follows that a 2-closed transitive permutation group of square-free degree contains a semiregular element of prime order, thus giving a partial affirmative answer to the conjecture that all 2-closed transitive permutation groups contain such an element (see [D. Marušič, On vertex symmetric digraphs, Discrete Math. 36 (1981) 69–81; P.J. Cameron (Ed.), Problems from the fifteenth British combinatorial conference, Discrete Math. 167/168 (1997) 605–615]).