The $k$-orbit theory and Fein-Kantor-Schacher Theorem
  By the investigation of $k$-orbits symmetry properties it is obtained a simple proof of the B. Fein, W. M. Kantor and M. Schacher Theorem: any transitive permutation group contains a non-trivial fixed-point-free prime-power element.   Key words: $k$-orbits, partitions, permutations, symmetry, groups 
Aleksandr Golubchik