Mathematics, Statistics & Computer Science
University of Wisconsin-Stout
Jarvis Hall Science Wing 231
Menomonie, WI 54751-0790
COURSE NUMBER/TITLE: MATH-370 [355-370] MODERN ALGEBRA I
CREDITS:
3
COURSE DESCRIPTION:
A first course in abstract algebra. Basic number theory, groups,
subgroups, cyclic groups, permutation groups, factor and product groups,
homomorphisms, isomorphisms, and an introduction to rings.
Prerequisites: MATH-154 Calculus II or MATH-157 Calculus and
Analytic Geometry II, MATH-180 Foundations of the Language of
Mathematics or MATH-270 Discrete Mathematics, and MATH-275
Linear Algebra.
TEXTBOOK: Abstract Algebra, 8th Ed., by Gallian (adopted F13)
Previous:
Abstract Algebra, 6th Ed., by Gallian (adopted F06)
Abstract Algebra, 5th Ed., by Gallian (adopted F03)
Modern Algebra, 8th Ed., by Gallian (adopted F00)
Abstract Algebra, 2nd Ed., by Hungerford (adopted S00)
Abstract Algebra, 1st Ed., by Hungerford (Prior to F97)
COURSE OBJECTIVES:
Upon successful completion of the course, the student will be able to
1. Formulate a rigorous mathematical proof.
2. Demonstrate understanding of basic number theory concepts, make computations with modular
arithmetic, and perform the Euclidean algorithm.
3. Demonstrate understanding of fundamental properties of groups and make computations within
groups.
4. Demonstrate understanding of common families of groups including modular groups, unit groups,
dihedral groups, and permutation groups.
5. Determine whether a subset of a group is or is not a (normal) subgroup.
6. Determine the subgroup structure and order of elements in a cyclic group.
7. Carry out operations with homomorphisms.
8. Determine whether two groups are isomorphic or not.
9. Apply the Fundamental Theorem of Finite Abelian Groups.
10. Demonstrate understanding of fundamental properties of rings and make computations within rings.
11. Apply abstract algebra to computer science, geometry, or physical science.
COURSE OUTLINE:
1. Number Theory (Objectives 1, 2)
a. Integers and Primes
b. Division Algorithm
c. Greatest Common Divisor and Euclidean Algorithm
d. Fundamental Theorem of Arithmetic
e. Modular Arithmetic
2.
Groups (Objectives 1, 3, 4)
a. Definition and Examples
b. Basic Properties
c. Subgroups
d. Subgroup Tests
e. Special Subgroups: center, centralizer, cyclic
3. Cyclic Groups (Objectives 1, 3, 6)
a. Definition and Examples
b. Classification of Cyclic Groups
4. Permutation Groups (Objectives 1, 3, 4)
a. Definition
b. Matrix Notation and Operations
c. Cycle Notation and Operations
d. Even and Odd Permutations
e. Alternating Subgroup
5. Homomorphisms (Objectives 1, 7, 8)
a. Definition and Examples
b. Properties
c. Isomorphisms
d. Automorphisms
e. Kernels
6. Cosets and Factor Groups (Objectives 1, 3, 5)
a. Cosets
b. Lagrange’s Theorem and Consequences
c. Normal Subgroups
d. Factor Groups
e. First Isomorphism Theorem
7. Direct Products (Objectives 1, 3)
a. Definition and Examples
b. Properties
8. The Fundamental Theorem of Finite Abelian Groups (Objectives 1, 9)
a. The Fundamental Theorem
b. Isomorphism Classes
c. Identifying Groups up to Isomorphism
9. Rings (Objectives 1, 10)
a. Definition and Examples
b. Subrings
c. Integral Domains and Fields
10. Applications (at the discretion of the instructor) (Objective 11)
a. Check Digits
b. RSA Algorithm
c. Symmetry Groups and Geometry
d. Other
Updated 10/2015
Revised 2/2015
Revised 4/2003
9/1967