The paper determines all permutation groups with a transitive minimal normal subgroup that have no fixed point free elements of prime order. All such groups are primitive and are wreath products in a product action involving $M_{11}$ in its action on 12 points. These groups are not 2-closed and so substantial progress is made towards asserting the truth of the polycirculant conjecture that every 2-closed transitive permutation group has a fixed point free element of prime order. All finite simple groups $T$ with a proper subgroup meeting every ${\rm Aut}(T)$ -conjugacy class of elements of $T$ of prime order are also determined.