A special case of the Yelton - Gaines Conjecture on Isomorphic

Let (ρ0, ρ1) and (ρ ′ 0, ρ ′ 1) be two ordered pairs of permutations in Sn and let t be a divisor of n. The Yelton-Gaines conjecture states that if at least one of these four permutations is a product of n/t disjoint t-cycles, and if there is a strong isomorphism (definition below) φ : 〈ρ0, ρ1〉 → 〈ρ0, ρ1〉 between the two subgroups of Sn generated by the elements in each ordered pair, then there is a fixed permutation τ in Sn that simultaneously conjugates ρi to ρ ′ i for i = 0, 1. The conclusion of this conjecture can be restated to say that the two dessins d’enfants corresponding to the two ordered pairs are isomorphic. In this paper a proof of this conjecture is given in the case in which all of the initial four permutations are fixed-point-free involutions.