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Articles 1 - 18 of 18
Full-Text Articles in Algebra
Cryptography Through The Lens Of Group Theory, Dawson M. Shores
Cryptography Through The Lens Of Group Theory, Dawson M. Shores
Electronic Theses and Dissertations
Cryptography has been around for many years, and mathematics has been around even longer. When the two subjects were combined, however, both the improvements and attacks on cryptography were prevalent. This paper introduces and performs a comparative analysis of two versions of the ElGamal cryptosystem, both of which use the specific field of mathematics known as group theory.
Computable Model Theory On Loops, Josiah Schmidt
Computable Model Theory On Loops, Josiah Schmidt
All NMU Master's Theses
We give an introduction to the problem of computable algebras. Specifically, the algebras of loops and groups. We start by defining a loop and group, then give some of their properties. We then give an overview of comptability theory, and apply it to loops and groups. We conclude by showing that a finitely presented residually finite algebra has a solvable word problem.
About Automorphisms Of Some Finite Groups, Sarah Emery
About Automorphisms Of Some Finite Groups, Sarah Emery
West Chester University Master’s Theses
This thesis gives an introduction to some topics from group theory, with a focus on automorphism groups of finite groups. Chapter one introduces the basic definitions and properties of groups and subgroups. In chapter two, the different classifications of functions between groups are defined and some properties thereof are given. Here we define automorphisms which are the focus of the paper. Chapters three and four deal with permutation groups and Sylow theorems respectively, and are discussions of some important groups, subgroups, and theorems pertaining thereto. The topics of these chapters help with our discussion of automorphism groups in the final …
The Last Digits Of Infinity (On Tetrations Under Modular Rings), William Stowe
The Last Digits Of Infinity (On Tetrations Under Modular Rings), William Stowe
Celebration of Learning
A tetration is defined as repeated exponentiation. As an example, 2 tetrated 4 times is 2^(2^(2^2)) = 2^16. Tetrated numbers grow rapidly; however, we will see that when tetrating where computations are performed mod n for some positive integer n, there is convergent behavior. We will show that, in general, this convergent behavior will always show up.
Simple Groups, Progenitors, And Related Topics, Angelica Baccari
Simple Groups, Progenitors, And Related Topics, Angelica Baccari
Electronic Theses, Projects, and Dissertations
The foundation of the work of this thesis is based around the involutory progenitor and the finite homomorphic images found therein. This process is developed by Robert T. Curtis and he defines it as 2^{*n} :N {pi w | pi in N, w} where 2^{*n} denotes a free product of n copies of the cyclic group of order 2 generated by involutions. We repeat this process with different control groups and a different array of possible relations to discover interesting groups, such as sporadic, linear, or unitary groups, to name a few. Predominantly this work was produced from transitive groups …
An Implementation Of The Solution To The Conjugacy Problem On Thompson's Group V, Rachel K. Nalecz
An Implementation Of The Solution To The Conjugacy Problem On Thompson's Group V, Rachel K. Nalecz
Senior Projects Spring 2018
We describe an implementation of the solution to the conjugacy problem in Thompson's group V as presented by James Belk and Francesco Matucci in 2013. Thompson's group V is an infinite finitely presented group whose elements are complete binary prefix replacement maps. From these we can construct closed abstract strand diagrams, which are certain directed graphs with a rotation system and an associated cohomology class. The algorithm checks for conjugacy by constructing and comparing these graphs together with their cohomology classes. We provide a complete outline of our solution algorithm, as well as a description of the data structures which …
Investigation Of Finite Groups Through Progenitors, Charles Baccari
Investigation Of Finite Groups Through Progenitors, Charles Baccari
Electronic Theses, Projects, and Dissertations
The goal of this presentation is to find original symmetric presentations of finite groups. It is frequently the case, that progenitors factored by appropriate relations produce simple and even sporadic groups as homomorphic images. We have discovered two of the twenty-six sporadic simple groups namely, M12, J1 and the Lie type group Suz(8). In addition several linear and classical groups will also be presented. We will present several progenitors including: 2*12: 22 x (3 : 2), 2*11: PSL2(11), 2*5: (5 : 4) which have produced the homomorphic images: …
Mckay Graphs And Modular Representation Theory, Polina Aleksandrovna Vulakh
Mckay Graphs And Modular Representation Theory, Polina Aleksandrovna Vulakh
Senior Projects Spring 2016
Ordinary representation theory has been widely researched to the extent that there is a well-understood method for constructing the ordinary irreducible characters of a finite group. In parallel, John McKay showed how to associate to a finite group a graph constructed from the group's irreducible representations. In this project, we prove a structure theorem for the McKay graphs of products of groups as well as develop formulas for the graphs of two infinite families of groups. We then study the modular representations of these families and give conjectures for a modular version of the McKay graphs.
Slicing A Puzzle And Finding The Hidden Pieces, Martha Arntson
Slicing A Puzzle And Finding The Hidden Pieces, Martha Arntson
Honors Program Projects
The research conducted was to investigate the potential connections between group theory and a puzzle set up by color cubes. The goal of the research was to investigate different sized puzzles and discover any relationships between solutions of the same sized puzzles. In this research, first, there was an extensive look into the background of Abstract Algebra and group theory, which is briefly covered in the introduction. Then, each puzzle of various sizes was explored to find all possible color combinations of the solutions. Specifically, the 2x2x2, 3x3x3, and 4x4x4 puzzles were examined to find that the 2x2x2 has 24 …
On T-Pure And Almost Pure Exact Sequences Of Lca Groups, Peter Loth
On T-Pure And Almost Pure Exact Sequences Of Lca Groups, Peter Loth
Mathematics Faculty Publications
A proper short exact sequence in the category of locally compact abelian groups is said to be t-pure if φ(A) is a topologically pure subgroup of B, that is, if for all positive integers n. We establish conditions under which t-pure exact sequences split and determine those locally compact abelian groups K ⊕ D (where K is compactly generated and D is discrete) which are t-pure injective or t-pure projective. Calling the extension (*) almost pure if for all positive integers n, we obtain a complete description of the almost pure injectives and almost pure projectives in the category of …
Generalized Conjugacy Classes, Pramod N. Achar
Generalized Conjugacy Classes, Pramod N. Achar
Mathematical Sciences Technical Reports (MSTR)
Generalized conjugation is the action of a group on its underlying set given by (g,x) -> p(g)xg-1, where p is some fixed endomorphism of G. Here we study combinatorial properties of the sizes of the orbits of the preceding action. In particular, we reduce the problem to a simpler case if p has nontrivial kernel, or if it is an inner automorphism, and we give a construction that allows a partial analysis in the general case.
Rectangular Groups, Nick Fiala, Crystal Hanscom, Patrick Keenan, Tung Tran
Rectangular Groups, Nick Fiala, Crystal Hanscom, Patrick Keenan, Tung Tran
Mathematical Sciences Technical Reports (MSTR)
We provide an overview of results and conjectures relating to rectangular groups.
Bounds On Squares Of Two-Sets, Dan Slilaty, Jeff Vanderkam
Bounds On Squares Of Two-Sets, Dan Slilaty, Jeff Vanderkam
Mathematical Sciences Technical Reports (MSTR)
For a finite group G, let pi(G) denote the proportion of (x,y) in GxG for which the set {x2,xy,yx,y2} has cardinality i. In this paper we develop estimates on the pi(G) for various i.
When Is The Number Of P-Subgroups Of A Group Satisfying A Property Congruent To 1 (Mod P)?, Jason Fulman, Jeff Vanderkam
When Is The Number Of P-Subgroups Of A Group Satisfying A Property Congruent To 1 (Mod P)?, Jason Fulman, Jeff Vanderkam
Mathematical Sciences Technical Reports (MSTR)
Let T be a property which holds for a group independent of whether or not this group is embedded in a group G or in a p-Sylow subgroup of G. Using a generalization of Sylow's second Theorem, we prove that if for any p-group P the number of subgroups of P satisfying T is congruent to 1 (mod p), then for any group G, the number of p-subgroups satisfying T is also congruent to 1 (mod p). As an application, we give simple proofs of several theorems, including the well-known Frobenius theorem.
More Upper Bounds On The 3-Rewriteability Of Non-3-Rewriteable Groups, Eric Wepsic
More Upper Bounds On The 3-Rewriteability Of Non-3-Rewriteable Groups, Eric Wepsic
Mathematical Sciences Technical Reports (MSTR)
We find an upper bound on the probability that a randomly selected triple in a group is 3-rewriteable, and a bound for the core set rewriteability.
Cyclicizers, Centralizers, And Normalizers, David Patrick, Eric Wepsic
Cyclicizers, Centralizers, And Normalizers, David Patrick, Eric Wepsic
Mathematical Sciences Technical Reports (MSTR)
Our goal is to define the cyclicizer, which is analogous to the centralizer and normalizer, and to examine groups in which these subsets have certain special properties.
An Upper Bound For 3-Rewriteability In Finite Groups, Jordan Ellenberg
An Upper Bound For 3-Rewriteability In Finite Groups, Jordan Ellenberg
Mathematical Sciences Technical Reports (MSTR)
An ordered triple of group elements (x,y,z) is said to be rewriteable if the product xyz is equal to one of the products xzy, yxz, yzx, zxy, zyx. In the present paper, we shall ask the following question: how rewriteable can a finite group be if its derived group has order greater than 2?
Finite Abelian Groups In Which The Probability Of An Automorphism Fixing An Element Is Large, Gary J. Sherman
Finite Abelian Groups In Which The Probability Of An Automorphism Fixing An Element Is Large, Gary J. Sherman
Mathematical Sciences Technical Reports (MSTR)
Let G be a finite group and let A be its automorphism group. We obtain various results on the probability that a random element of A fixes a random element of G.