Let G be a transitive subgroup of S_n where n>1. Show that G contains a permutation without fixpoints, i.e., where all cycles has length >1.
Cauchy-Frobenius lemma (wrongly known as Burnside's lemma). let since is transitive, the number of orbits is 1. so we must have
let be the identity permutation. we have if for every then contradiction! so there exists such that Q.E.D.