#### Event Title

Application of the Ordered Erdos Ginzburg Ziv Theorem to the Non-Abelian Dihedral Group

#### Session Number

O01

#### Advisor(s)

Micah Fogel, Illinois Mathematics and Science Academy

#### Location

B-116

#### Start Date

28-4-2016 8:50 AM

#### End Date

28-4-2016 9:15 AM

#### Abstract

In additive number theory and group theory the Erdos-Ginzburg-Ziv theorem describes the length of the shortest zero-sum subsequence modulo n, given a sequence of length 2n – 1 and has been described well for unordered sequences in non-abelian groups. However there is little to be found on ordered sequences and therefore, the goal of our research is to make progress on this problem by applying an ordered version of the Erdos Ginzburg Ziv theorem to the non-abelian dihedral group. We wrote programs in Python to assist us with the experimentation and data collection, which consisted of non-zero-sum sequences in Dn. We observed patterns in the data, especially looking for something similar to the Erdos-Ginzburg-Ziv Theorem. Our results showed that for odd n ≥ 3, the lower bound on the length of a sequence made from elements in the dihedral group of order 2n that contains a zero – sum sequence is 4n -1. For the even case, the bound is 3n. These results are supported by the results of similar studies on non- abelian groups, and expand on those studies as well. Our study broaches into new areas in group theory and provides a starting point for future analysis

Application of the Ordered Erdos Ginzburg Ziv Theorem to the Non-Abelian Dihedral Group

B-116

In additive number theory and group theory the Erdos-Ginzburg-Ziv theorem describes the length of the shortest zero-sum subsequence modulo n, given a sequence of length 2n – 1 and has been described well for unordered sequences in non-abelian groups. However there is little to be found on ordered sequences and therefore, the goal of our research is to make progress on this problem by applying an ordered version of the Erdos Ginzburg Ziv theorem to the non-abelian dihedral group. We wrote programs in Python to assist us with the experimentation and data collection, which consisted of non-zero-sum sequences in Dn. We observed patterns in the data, especially looking for something similar to the Erdos-Ginzburg-Ziv Theorem. Our results showed that for odd n ≥ 3, the lower bound on the length of a sequence made from elements in the dihedral group of order 2n that contains a zero – sum sequence is 4n -1. For the even case, the bound is 3n. These results are supported by the results of similar studies on non- abelian groups, and expand on those studies as well. Our study broaches into new areas in group theory and provides a starting point for future analysis