On characterizing pairs of permutations in determining their generated group

Among the unsolved problems in mathematics listed on Wolfram Mathworld's website is finding a formula for the probability that two permutations chosen at random generate the symmetric group. We show that two permutations of order two cannot generate the symmetric group using the maximum order of permutations. We also define a new notion of distance in a permutation and use a concept of distance preservation to compare the generated subgroup with the symmetric group. We find that if a permutation preserves a distance which is not relatively prime to the lengths of the cycles in another permutation, then the pair of permutations will not generate the symmetric group. All commutative permutation subgroups and some imprimitive subgroups generated by pairs of permuations can be described by distance preservation. Using these results we are able to completely classify pairs of permutations involving a transposition which generate the symmetric group. Lastly, we describe how to form the multiplication table for the symmetric group without performing any multiplications of permutations in the hopes that this description can be utilized in determining generated groups.