Sets of permutations that generate the symmetric group pairwise

The paper contains proofs of the following results. For all sufficiently large odd integers n, there exists a set of 2 n − 1 permutations that pairwise generate the symmetric group S n . There is no set of 2 n − 1 + 1 permutations having this property. For all sufficiently large integers n with n ≡ 2 mod 4 , there exists a set of 2 n − 2 even permutations that pairwise generate the alternating group A n . There is no set of 2 n − 2 + 1 permutations having this property.