A special case of the Yelton-Gaines Conjecture on Isomorphic Dessins

Authors

Claudia Raithel, University of Michigan- Ann ArborFollow

Abstract

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.

Faculty Sponsor

Robert Perlis

Recommended Citation

Raithel, Claudia (2009) "A special case of the Yelton-Gaines Conjecture on Isomorphic Dessins," Rose-Hulman Undergraduate Mathematics Journal: Vol. 10: Iss. 2, Article 14.
Available at: https://scholar.rose-hulman.edu/rhumj/vol10/iss2/14