Let (r0, r1) and (r0¢, r1¢) 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:< r0,r1> ® < r0¢, r1¢> between the two subgroups of Sn generated by the elements in each ordered pair, then there is a fixed permutation t in Sn that simultaneously conjugates ri to ri¢ 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.
Robert Perlis, Department of Mathematics, Louisiana State University firstname.lastname@example.org
"A special case of the Yelton-Gaines Conjecture on Isomorphic Dessins,"
Rose-Hulman Undergraduate Mathematics Journal: Vol. 10
, Article 14.
Available at: https://scholar.rose-hulman.edu/rhumj/vol10/iss2/14