## Author ORCID Identifier

## Date Available

4-13-2021

## Year of Publication

2021

## Degree Name

Doctor of Philosophy (PhD)

## Document Type

Doctoral Dissertation

## College

Arts and Sciences

## Department/School/Program

Mathematics

## First Advisor

Dr. David Leep

## Abstract

Let G be a finite abelian group, written additively. The Davenport constant, *D(G)*, is the smallest positive number *s* such that any subset of the group *G*, with cardinality at least *s*, contains a non-trivial zero-subsum. We focus on a variation of the Davenport constant where we allow addition and subtraction in the non-trivial zero-subsum. This constant is called the plus-minus Davenport constant, *D*_{±}(*G*). In the early 2000’s, Marchan, Ordaz, and Schmid proved that if the cardinality of *G* is less than or equal to 100, then the *D*_{±}(*G*) is the floor of log_{2} *n* + 1, the basic upper bound, with few exceptions. The value of *D*_{±}(*G*) is primarily known when the rank of G at most two and the cardinality of G is less than or equal to 100. In most cases, when *D*_{±}(*G*) is known, *D*_{±}(*G*)= floor(log_{2} |*G*|) + 1, with the exceptions of when *G* is a 3-group or a 5-group. We have studied a class of groups where the cardinality of G is a product of two prime powers. We look more closely to when the primes are 2 and 3, since the plus-minus Davenport constant of a 2-group attains the basic upper bound and while the plus-minus Davenport constant of a 3-group does not. To help us compute *D*_{±}(*G*), we define the even plus-minus Davenport constant, *De*_{±}(*G*), that guarantees a pm zero-subsum of even length.

Let *C _{n}* be a cyclic group of order n. Then

*D*(

*C*) =

_{n}*n*and

*D*

_{±}(

*C*) =floor( log

_{n}_{2}

*n*)+1. We have shown that

*De*

_{±}(

*C*) depends on whether

_{n}*n*is even or odd. When

*n*is even and not a power of 2, then

*De*

_{±}(

*C*) = floor(log

_{n}_{2}

*n*) + 2. When

*n*= 2

*, then*

^{k}*De*

_{±}(

*C*) = floor(log

_{n}_{2}

*n*) + 1. The case when

*n*is odd,

*De*

_{±}(

*C*) varies depending on how close

_{n}*n*is to a power of 2. We have also shown that a subset containing the Jacobsthal numbers provides a subset of

*C*that does not contain an even pm zero-subsum for certain values of n.

_{n}When *G* is a finite abelian group, we provide bounds for *De*_{±}(*G*). If *D*_{±}(*G*) is known, then we given an improvement to the lower bound of *De*_{±}(*G*). Additional improvements are shown when *G* is a direct sum an elementary abelian p-groups where p is prime. Then we compute the values of *De*_{±}(*C _{r}^{3}* ) when 2 ≤ r ≤ 9 and provide an optimal lower bound for larger r. For the group

*C*⊕

_{2}*C*,

_{r}^{3}*D*

_{±}(

*C*⊕

_{2}*C*) =

_{r}^{3}*De*

_{±}(

*C*). When r < 10,

_{r}^{3}*D*

_{±}(

*C*⊕

_{2}*C*) does not attain the basic upper bound. We conjecture that as r increases,

_{r}^{3}*D*

_{±}(

*C*⊕

_{2}*C*) will not attain the basic upper bound. Now, let

_{r}^{3}*G*=

*C*⊕

_{q}^{2}*C*. We compute the values of

_{r}^{3}*D*

_{±}(

*G*) for general q and small r. In this case, we show that if

*D*

_{±}(

*G*) attains the basic upper bound then so does

*De*

_{±}(

*G*). We then look at the case when the cardinality of G is a product of two prime powers and show improvements on the lower bound by using the fractional part of log

_{2}

*p*of each prime. Furthermore, we compute the values of

*D*

_{±}(

*G*) when 100 < |

*G*| ≤ 200, with some exceptions.

## Digital Object Identifier (DOI)

https://doi.org/10.13023/etd.2021.059

## Funding Information

Department of Defense, Science, Mathematics, and Research for Transformation (SMART) Fellowship; August 2018 - May 2021

## Recommended Citation

Perez-Lavin, Darleen S., "The Plus-Minus Davenport Constant of Finite Abelian Groups" (2021). *Theses and Dissertations--Mathematics*. 79.

https://uknowledge.uky.edu/math_etds/79