The distribution of the invariant factors of multiplicative groups
department or school: Science, Faculty of
department or school: Mathematics, Department of
data provider: DSpace
degree campus: UBCV
creator: Simpson, Reginald M.
issued: 2025-05-29T18:00:54Z
issued: 2025
related degree: Doctor of Philosophy - PhD
degree grantor: University of British Columbia
description: For any cyclic group ℤ, this work studies the number of copies that appear in the invariant factor decomposition of the multiplicative group (ℤ/mℤ)⁢ˣ, denoted ℑ(m; d). The work proves the existence of a sequence of singleton and doubleton sets {fᵢ}ᵢ∞₌₁, a “universal profile” such that the natural density of numbers m where ℑ(m; d) > 0 is 1 for d in the singleton sets, 1/2 for d in the doubleton sets, and 0 for all d not found in the sets. The “universal profile” of sets is fully characterized, and can be explicitly computed based on the description given in this work, and has the property that for d₁ ∈ f ᵢ and d₂ ∈ fⱼ with i < j, d₁ divides d₂. For all d ∈ ℕ, we further determine Erdős-Kac type laws for the limiting distributions of ℑ(m; d) under appropriate scaling. These laws do not always follow the standard Erdős-Kac pattern of having normal limiting distributions, and mathematical descriptions of these non-normal distributions is given. Asymptotic formulas for the expectations of ℑ(m; d) are determined using these laws.
aggregated CHO: https://circle.library.ubc.ca/rest/handle/2429/91229
note: THE DISTRIBUTION OF THE INVARIANT FACTORSOF MULTIPLICATIVE GROUPSbyReginald M. SimpsonA THESIS SUBMITTED IN PARTIAL FULFILLMENT OF THEREQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHYinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)May 2025© Reginald M. Simpson, 2025The following individuals certify that they have read, and recommend to the Faculty of Graduateand Postdoctoral Studies for acceptance, the dissertation entitled:The distribution of the invariant factors of multiplicative groupssubmitted by Reginald M. Simpson in partial fulfilment of the requirements forthe degree of Doctor of Philosophyin MathematicsExamining Committee:Greg Martin, Professor, Mathematics, UBCSupervisorJulia Gordon, Professor, Mathematics, UBCSupervisory Committee MemberMichael Bennett, Professor, Mathematics, UBCSupervisory Committee MemberNicholas Harvey, Professor, Computer Science, UBCUniversity ExaminerBrian Wetton, Professor, Mathematics, UBCUniversity ExaminerPaul Pollack, Professor, Mathematics, University of GeorgiaExternal ExamineriiAbstractFor any cyclic group Zd, this work studies the number of copies that appear in the invariantfactor decomposition of the multiplicative group (Z/mZ)×, denoted I(m; d). The work proves theexistence of a sequence of singleton and doubleton sets {fi}∞i=1, a “universal profile” such that thenatural density of numbers m where I(m; d) > 0 is 1 for d in the singleton sets, 1/2 for d inthe doubleton sets, and 0 for all d not found in the sets. The “universal profile” of sets is fullycharacterized, and can be explicitly computed based on the description given in this work, and hasthe property that for d1 ∈ fi and d2 ∈ fj with i < j, d1 divides d2.For all d ∈ N, we further determine Erdo˝s-Kac type laws for the limiting distributions of I(m; d)under appropriate scaling. These laws do not always follow the standard Erdo˝s-Kac pattern of hav-ing normal limiting distributions, and mathematical d…
has type: Thesis/Dissertation
date issued: 2025-11
is shown at: 10.14288/1.0448994
language: eng
degree discipline: Mathematics
provider: Vancouver : University of British Columbia Library
publisher: University of British Columbia
rights: Attribution-NonCommercial-NoDerivatives 4.0 International
rights URI: http://creativecommons.org/licenses/by-nc-nd/4.0/
scholar level: Graduate
contributor: Martin, Gregory George
type: Text
identifier URI: http://hdl.handle.net/2429/91229