Let be a permutation group on letters. So a subgroup of .
There is a theorem of Fein, Kantor and Schacher (1981) that if is transitive, then has a fix point free element of prime order.
The proof is very hard and relies on the classification of finite simple groups. I would like to prove a special case: let , then if the group is transitive there exists an element of order with no fixed points.
I can show that there must be a fixed point:
The transitivity of the group implies that there is only 1 orbit. Burnside's lemma then implies that the average number of fixed points for an element in the group is 1. Since the identity fixes everything, there is a fixed point free element.
I can show that there exists an element of order :
Again we use that the group is transitive and that there is therefore only one orbit. Since there is only one orbit, the order of the orbit of is . By the orbit-stabilizer theorem, the order of the stabilizer of some element is equal to the order of the group divided by . Since the order of a stabilizer is an integer, divides the order of the group. By Cauchy's theorem, there is an element of order in the group.
I cannot, however, show that there is an element of order with no fixed points. Does anyone have any hints?