Over the non-existence of sharply 3-transitive permutation sets containing sharply 2-transitive permutation subsets

The following results are proved: Let E be a finite set, ¦E¦>4, and let G be a sharply 3-transitive permutation set on E. Then G contains no subset which is a sharply 2-transitive permutation set on E (Theorem 1). In the case when G is a sharply 3-transitive permutation group which is also planar, the finiteness condition on E can be dropped (Theorem 2).Dedicated to G. Zappa on his 70th birthdayResearch done within the activity of GNSAGA of CNR, supported by the 40% grants of MPI.