Triality Groups

Dr. Stephen Gagola

Department of Mathematics
University of Arizona

Friday, 13 March 2009
219 Shelby Center
3:00 (Refreshements at 2:30)

Abstract

A Moufang loop L is the generalization of a group that arises when the associative law is replaced by any one of the Moufang identities:

(xy)(zx) = (x((yz)x)

((xy)x)z) = x(y(xz))

((zx)y)x) = z(x(yx))

for all x, y, z in L. Such identities first arose among Zorn vector matrices in octonion algebras. In this talk we will define what it means for a group to admit triality. From there we will focus on Latin Square Designsto see the connections between Moufang loops and groups with triality. By seeing the correspondence between triality groups and Moufang loops one can get a better understanding of why Moufang loops behave like groups. In this talk we will also expand upon some of the recent results (Lagrange's Theorem & Sylow's Theorems) that have been achieved using triality groups.