The Plus-Minus Davenport Constant of Finite Abelian Groups

Author

Darleen S. Perez-Lavin, University of KentuckyFollow

Author ORCID Identifier

https://orcid.org/0000-0003-4300-379X

Date Available

4-13-2021

Year of Publication

2021

Document Type

Doctoral Dissertation

Degree Name

Doctor of Philosophy (PhD)

College

Arts and Sciences

Department/School/Program

Mathematics

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 log₂ 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(log₂ |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_n) =floor( log₂ n)+1. We have shown that De_±(C_n) depends on whether n is even or odd. When n is even and not a power of 2, then De_±(C_n) = floor(log₂ n) + 2. When n = 2^k , then De_±(C_n) = floor(log₂ n) + 1. The case when n is odd, De_±(C_n) varies depending on how close n is to a power of 2. We have also shown that a subset containing the Jacobsthal numbers provides a subset of C_n that does not contain an even pm zero-subsum for certain values of 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_r³ ) when 2 ≤ r ≤ 9 and provide an optimal lower bound for larger r. For the group C₂ ⊕ C_r³ , D_±(C₂ ⊕ C_r³ ) = De_±(C_r³ ). When r < 10, D_±(C₂ ⊕ C_r³ ) does not attain the basic upper bound. We conjecture that as r increases, D_±(C₂ ⊕ C_r³ ) will not attain the basic upper bound. Now, let G = C_q² ⊕ C_r³ . We compute the values of 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 log₂ 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