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The Examination Of The Arithmetic Surface (3, 5) Over Q, Rachel J. Arguelles

Electronic Theses, Projects, and Dissertations

This thesis is centered around the construction and analysis of the principal arithmetic surface (3, 5) over Q. By adjoining the two symbols i,j, where i² = 3, j² = 5, such that ij = -ji, I can produce a quaternion algebra over Q. I use this quaternion algebra to find a discrete subgroup of SL₂(R), which I identify with isometries of the hyperbolic plane. From this quaternion algebra, I produce a large list of matrices and apply them via Mobius transformations to the point (0, 2), which is the center of my Dirichlet domain. This …

On The Hamiltonicity Of Subgroup Lattices, Nicholas Charles Fleece

MSU Graduate Theses

In this paper we discuss the Hamiltonicity of the subgroup lattices of

different classes of groups. We provide sufficient conditions for the

Hamiltonicity of the subgroup lattices of cube-free abelian groups. We also

prove the non-Hamiltonicity of the subgroup lattices of dihedral and

dicyclic groups. We disprove a conjecture on non-abelian p-groups by

producing an infinite family of non-abelian p-groups with Hamiltonian

subgroup lattices. Finally, we provide a list of the Hamiltonicity of the

subgroup lattices of every finite group up to order 35 barring two groups.

Group Theory Visualized Through The Rubik's Cube, Ashlyn Okamoto

University Honors Theses

In my thesis, I describe the work done to implement several Group Theory concepts in the context of the Rubik’s cube. A simulation of the cube was constructed using Processing-Java and with help from a YouTube series done by TheCodingTrain. I reflect on the struggles and difficulties that came with creating this program along with the inspiration behind the project. The concepts that are currently implemented at this time are: Identity, Associativity, Order, and Inverses. The functionality of the cube is described as it moves like a regular cube but has extra keypresses that demonstrate the concepts listed. Each concept …

Some Model Theory Of Free Groups, Christopher James Natoli

Dissertations, Theses, and Capstone Projects

There are two main sets of results, both pertaining to the model theory of free groups. In the first set of results, we prove that non-abelian free groups of finite rank at least 3 or of countable rank are not A-homogeneous. We then build on the proof of this result to show that two classes of groups, namely finitely generated free groups and finitely generated elementary free groups, fail to form A-Fraisse classes and that the class of non-abelian limit groups fails to form a strong A-Fraisse class.

The second main result is that if a countable group is elementarily …

Cayley Map Embeddings Of Complete Graphs, Miriam Scheinblum

Honors Program Theses

This paper looks at Cayley map embeddings of complete graphs on orientable surfaces. Cayley maps constrain graph embeddings to those with cyclical edge rotations, so optimal embeddings on surfaces with the minimum genus may not always be possible. We explore instances when Cayley maps succeed at optimally embedding complete graphs, and when optimal embeddings are not possible, we determine how close to optimal they can get by finding vertex rotations that result in the smallest possible genus. Many of the complete graphs we consider have prime numbers of vertices, so for each complete graph Kn we focus on mappings with …

Group Theory And Particles, Elizabeth V. Hawkins

Honors College Theses

We begin by a brief overview of the notion of groups and Lie groups. We then explain what group representations are and give their main properties. Finally, we show how group representation form a natural framework to understand the Standard Model of physics.

Symmetric Presentations, Representations, And Related Topics, Adam Manriquez

Electronic Theses, Projects, and Dissertations

The purpose of this thesis is to develop original symmetric presentations of finite non-abelian simple groups, particularly the sporadic simple groups. We have found original symmetric presentations for the Janko group J₁, the Mathieu group M₁₂, the Symplectic groups S(3,4) and S(4,5), a Lie type group Suz(8), and the automorphism group of the Unitary group U(3,5) as homomorphic images of the progenitors 2^*60 : (2 x A₅), 2^{*60 :} A₅, 2^*56 : (2³ : 7), and 2^*28 : (PGL(2,7):2), respectively. We have also discovered the groups 2 …

Homeomorphisms Of The Sierpinski Carpet, Karuna S. Sangam

Senior Projects Spring 2018

The Sierpinski carpet is a fractal formed by dividing the unit square into nine congruent squares, removing the center one, and repeating the process for each of the eight remaining squares, continuing infinitely many times. It is a well-known fractal with many fascinating topological properties that appears in a variety of different contexts, including as rational Julia sets. In this project, we study self-homeomorphisms of the Sierpinski carpet. We investigate the structure of the homeomorphism group, identify its finite subgroups, and attempt to define a transducer homeomorphism of the carpet. In particular, we find that the symmetry groups of platonic …

A Computational Introduction To Elliptic And Hyperelliptic Curve Cryptography, Nicholas Wilcox

Honors Papers

At its core, cryptography relies on problems that are simple to construct but difficult to solve unless certain information (the “key”) is known. Many of these problems come from number theory and group theory. One method of obtaining groups from which to build cryptosystems is to define algebraic curves over finite fields and then derive a group structure from the set of points on those curves. This thesis serves as an exposition of Elliptic Curve Cryptography (ECC), preceded by a discussion of some basic cryptographic concepts and followed by a glance into one generalization of ECC: cryptosystems based on hyperelliptic …

Cayley Graphs Of Groups And Their Applications, Anna Tripi

MSU Graduate Theses

Cayley graphs are graphs associated to a group and a set of generators for that group (there is also an associated directed graph). The purpose of this study was to examine multiple examples of Cayley graphs through group theory, graph theory, and applications. We gave background material on groups and graphs and gave numerous examples of Cayley graphs and digraphs. This helped investigate the conjecture that the Cayley graph of any group (except Z_2) is hamiltonian. We found the conjecture to still be open. We found Cayley graphs and hamiltonian cycles could be applied to campanology (in particular, to the …

Solving Algorithmic Problems In Finitely Presented Groups Via Machine Learning, Jonathan Gryak

Dissertations, Theses, and Capstone Projects

Machine learning and pattern recognition techniques have been successfully applied to algorithmic problems in free groups. In this dissertation, we seek to extend these techniques to finitely presented non-free groups, in particular to polycyclic and metabelian groups that are of interest to non-commutative cryptography.

As a prototypical example, we utilize supervised learning methods to construct classifiers that can solve the conjugacy decision problem, i.e., determine whether or not a pair of elements from a specified group are conjugate. The accuracies of classifiers created using decision trees, random forests, and N-tuple neural network models are evaluated for several non-free groups. …

Normal Subgroups Of Wreath Product 3-Groups, Ryan Gopp

Williams Honors College, Honors Research Projects

Consider the regular wreath product group P of Z₉ with (Z₃ x Z₃). The problem of determining all normal subgroups of P that are contained in its base subgroup is equivalent to determining the subgroups of a certain matrix group M that are invariant under two particular endomorphisms of M. This thesis is a partial solution to the latter. We use concepts from linear algebra and group theory to find and count so-called doubly-invariant subgroups of M.

On The Number Of Distinct Cyclic Subgroups Of A Given Finite Group, Joseph Dillstrom

MSU Graduate Theses

In the study of finite groups, it is a natural question to consider the number of distinct cyclic subgroups of a given finite group. Following an article by M. Tarnauceanu in the American Mathematical Monthly, we consider arithmetic relations between the order of a finite group and the number of its cyclic subgroups. We classify several infinite families of finite groups in this fashion and expand upon an open problem posed in the article.

Monomial Characters Of Finite Groups, John Mchugh

Graduate College Dissertations and Theses

An abundance of information regarding the structure of a finite group can be obtained by studying its irreducible characters. Of particular interest are monomial characters – those induced from a linear character of some subgroup – since Brauer has shown that any irreducible character of a group can be written as an integral linear combination of monomial characters. Our primary focus is the class of M-groups, those groups all of whose irreducible characters are monomial. A classical theorem of Taketa asserts that an M-group is necessarily solvable, and Dade proved that every solvable group can be embedded as a subgroup …

The Automorphism Group Of The Halved Cube, Benjamin B. Mackinnon

Theses and Dissertations

An n-dimensional halved cube is a graph whose vertices are the binary strings of length n, where two vertices are adjacent if and only if they differ in exactly two positions. It can be regarded as the graph whose vertex set is one partite set of the n-dimensional hypercube, with an edge joining vertices at hamming distance two. In this thesis we compute the automorphism groups of the halved cubes by embedding them in R n and realizing the automorphism group as a subgroup of GLn(R). As an application we show that a halved cube is a circulant graph if …

Constructions And Isomorphism Types Of Images, Jessica Luna Ramirez

Electronic Theses, Projects, and Dissertations

In this thesis, we have presented our discovery of true finite homomorphic images of various permutation and monomial progenitors, such as 2^*7: D₁₄, 2^*7 : (7 : 2), 2^*6 : S₃ x 2, 2^*8: S₄, 2*⁷²: (3²:(2S₄)), and 11^*2 :_m D₁₀. We have given delightful symmetric presentations and very nice permutation representations of these images which include, the Mathieu groups M₁₁, M₁₂, the 4-fold cover of the Mathieu group M₂₂, 2 x …

Exploring Platform (Semi)Groups For Non-Commutative Key-Exchange Protocols, Ha Lam

Dissertations, Theses, and Capstone Projects

In this work, my advisor Delaram Kahrobaei, our collaborator David Garber, and I explore polycyclic groups generated from number fields as platform for the AAG key-exchange protocol. This is done by implementing four different variations of the length-based attack, one of the major attacks for AAG, and submitting polycyclic groups to all four variations with a variety of tests. We note that this is the first time all four variations of the length-based attack are compared side by side. We conclude that high Hirsch length polycyclic groups generated from number fields are suitable for the AAG key-exchange protocol.

Delaram Kahrobaei …

Convexity Properties Of The Diestel-Leader Group Γ_3(2), Peter J. Davids

Honors Projects

The Diestel-Leader groups are a family of groups first introduced in 2001 by Diestel and Leader in [7]. In this paper, we demonstrate that the Diestel-Leader group Γ₃(2) is not almost convex with respect to a particular generating set S. Almost convexity is a geometric property that has been shown by Cannon [3] to guarantee a solvable word problem (that is, in any almost convex group there is a finite-step algorithm to determine if two strings of generators, or “words”, represent the same group element). Our proof relies on the word length formula given by Stein and Taback …

Permutation Groups And Puzzle Tile Configurations Of Instant Insanity Ii, Amanda N. Justus

Electronic Theses and Dissertations

The manufacturer claims that there is only one solution to the puzzle Instant Insanity II. However, a recent paper shows that there are two solutions. Our goal is to find ways in which we only have one solution. We examine the permutation groups of the puzzle and use modern algebra to attempt to fix the puzzle. First, we find the permutation group for the case when there is only one empty slot at the top. We then examine the scenario when we add an extra column or an extra row to make the game a 4 × 5 puzzle or …

A Study Of Finite Symmetrical Groups, May Majid

Theses Digitization Project

This study investigated finite homomorphic images of several progenitors, including 2*⁵ : S₅, 2*⁶ : A₆, and 3*⁵ : C₅ The technique of manual of double coset enumeration is used to construct several groups by hand and computer-based proofs are given for the isomorphism types of the groups that are not constructed.

A Study Of Finite Symmetrical Groups, Patrick Kevin Martinez

Theses Digitization Project

This study discovered several important groups that involve the classical and sporadic groups. These groups appeared as finite homomorphic images of the progenitors 3*8 : PGL₂(7), 2*¹⁴ : L₃ (2), 5*³ : S₃ and 7*2 : m S₃.

Enumeration And Symmetric Presentations Of Groups, With Music Theory Applications, Jesse Graham Train

Theses Digitization Project

The purpose of this project is to construct groups as finite homomorphic images of infinite semi-direct products. In particular, we will construct certain classical groups and subgroups of sporadic groups, as well groups with applications to the field of music theory.

Monomial And Permutation Representation Of Groups, Rebeca Maria Blanquet

Theses Digitization Project

The purpose of this project is to introduce another method of working with groups, that is more efficient when the groups we wish to work with are of a significantly large finite order. When we wish to work with small finite groups, we use permutations and matrices. Although these two methods are the general methods of working with groups, they are not always efficient.

Symmetric Generation, Lisa Sanchez

Theses Digitization Project

The purpose of this project is to conduct a systematic search for finite homomorphic images of infinite semi-direct products mn : N, where m = 2,3,5,7 and N <̲ Sn and construct by hand some of the important homomorphic images that emerge from the search.

Symmetric Generation Of M₂₂, Bronson Cade Lim

Theses Digitization Project

This study will prove the Mathieu group M₂₂ contains two symmetric generating sets with control grougp L₃ (2). The first generating set consists of order 3 elements while the second consists of involutions.

Homomorphic Images Of Progenitors Of Order Three, Mark Gutierrez

Theses Digitization Project

The main purpose of this thesis is to construct finite groups as homomorphic images of infinite semi-direct products, 2*n : N, 3*n : N, and 3*n :m N, where 2*n and 3*n are free products of n copies of the cyclic group C₂ extended by N, a group of permutations on n letters.

Symmetric Generators Of Order 3, Stewart Contreras

Theses Digitization Project

The main purpose of this project is to construct finite homomorphic images of infinite semi-direct products.

Symmetric Generation, Dung Hoang Tri

Theses Digitization Project

In this thesis we construct finite homorphic images of infinite semi-direct products, 2*n : N, where 2*n is a free product of n copies the cyclic group of permutations on n letter.

An Investigation Of Kurosh's Theorem, Keith Anthony Earl

Theses Digitization Project

The purpose of this project will be an exposition of the Kurosh Theorem and the necessary and suffcient condition that A must be algebraic and satisfy a P.I. to be locally finite.

On A Symmetric Presentation Of The Double Cover Of M₂₂: 2, Gabriela Laura Maerean

Theses Digitization Project

The purpose of this project is to construct finite homomorphic images of infinite semi-direct products. We will construct two finite homomorphic images, L₂ (8) and PGL₂ (9) of the infinite semi-direct product 2*³ : S₃. The main part of this project is to construct the double cover 2 - M₂₂ : 2 and the automorphism group M₂₂ : 2 of the Matheiu sporadic group M₂₂ as a homomorphic image of the progenitor 2*⁷ : L₃ (2).