MTH 3340 - Abstract Algebra with Applications

3 lecture hours 0 lab hours 3 credits

Course Description
This course covers methods of mathematical reasoning and proof. It also covers elementary number theory including divisibility, primes, congruence and modular arithmetic, and Euler's formula. It includes introduction to group theory including subgroups, symmetry and permutation groups, homomorphism and isomorphism, direct products, structure of cyclic and abelian groups, and group actions. Applications to cryptography are addressed. Other possible topics include rings and fields, solvability of polynomials, impossibility of compass and straight edge constructions, or primality testing.
Prereq: MTH 2340 or MTH 2140 or MTH 2310 (quarter system prereq: MA 383 or MA 2310)
Note: None
This course meets the following Raider Core CLO Requirement: Think Critically

Course Learning Outcomes
Upon successful completion of this course, the student will be able to:

Prove elementary facts in number theory
Solve linear congruence equations
Determine if a given set with a binary operation is a group
Determine if a given subset of a group is a subgroup
Determine the possible orders of elements and subgroups of a given group
Determine the cycle structure and orders of elements in a permutation group
Determine if two groups are isomorphic; recognize standard classes of groups
Determine the kernel and image of a group homomorphism; the cosets of a normal subgroup and the associated quotient group
Describe the structure of a finite abelian group as a direct sum
Use the RSA algorithm to encrypt and decrypt messages

Prerequisites by Topic

Basic set theory
Functions and the algebra of functions including composition

Course Topics

Direct and indirect proof methods
Divisibility and prime numbers
Modular arithmetic and linear congruence
Euler's formula and applications including k-th roots
RSA public key cryptosystem
Groups and subgroups, order of a group, cyclic groups, and Lagrange's theorem
Examples including symmetry groups, permutation groups, matrix groups
Permutations, cycles, parity of permutations, alternating group
Homomorphisms, normal subgroups, quotient groups
Isomorphism and the isomorphism theorems
Direct Products and structure theorems for cyclic and finitely generated abelian groups
Group actions, orbits and stabilizers
Other topics at instructor discretion: rings and fields, solvability of polynomials, impossibility of compass and straight edge constructions, or primality testing

Coordinator
Mathematics Staff

Add to Portfolio.

Close Window