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Abstract

A (v,k,lambda) difference set D in a group G is a subset of G such that every nonidentity element of G is covered exactly lambda times by quotients d1d2-1 where d1 and d2 are in D. In the group ring, this means that D obeys the equation DD(-1) = k1 + lambda(G - 1). An (m,n,k,lambda) relative difference set R is a difference set relative to a normal subgroup N of G satisfying the similar equation RR(-1) = k·1 + lambda(G - N). We will describe various search techniques for relative difference sets (RDS), including the exhaustive search method for small groups using the computer program GAP, as well as the multiplier theorem and group representations methods used for larger groups. We will provide a catalog of RDS found, as well as those eliminated, using these methods. Next, a proof is presented of the non-existence of (2m,2,2m,m) relative difference sets in quaternion groups of order 4m where m is odd. In conclusion, we will state several interesting results found for specific parameters, including (12,2,12,6) and (12,3,12,4).

Author Bio

This research was done during the Summer 2002 REU at Central MichiganUniversity under the direction of Dr. Ken W. Smith. This was just after mysophomore year at Butler University. I enjoy group theory and topology. Whilenot working on math, I can be found practicing violin, which I have playedsince I was three years old.

This research was done after my sophomore year during the Summer 2002 REU at my school, Central Michigan University under the direction of Dr. Ken W. Smith. My primary interest in mathematics is algebra. During my free time I like to read. My favorite authors include Ernest Hemingway and Haruki Murakami.

This research was done during the Summer 2002 REU at Central Michigan University under the direction of Dr. Ken W. Smith. This was after my sophomore year at the University of Missouri - Rolla. My mathematical interests are group, graph, and number theories. Outside of academia, I love to jitterbug and lindy hop and am an active member (currently president) of the Ballroom Dancing Club at my university. In addition, I am slowly but surely learning the art of Tai Chi Chu'an. In what spare time I have left, I am also working on my first reading of the Robert Jordan Wheel of Time series.

This research was done after my sophomore year during the Summer 2002 REU at my school, Central Michigan University under the direction of Dr. Ken W. Smith. I enjoy all types of math and am interested in many research projects. I have lived in Kalamazoo, Michigan all my life and enjoy my time here in Mount Pleasant. I love tennis, softball, reading, biking and Harry Potter.

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