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.

Author Bio

Claudia Raithel is an undergraduate at the University of Michigan- Ann Arbor and is majoring in Honors Mathematics and Physics. She worked on the contents of this paper during the 2009 Louisiana State University REU.