COURSE NUMBER AND TITLE: MATH 4211 Modern Abstract Algebra I

CREDIT HOURS: 3

CATALOG DESCRIPTION: A study of abstract algebraic structure. Topics include groups, subgroups, permutation groups, homomorphisms, and quotient groups.

PREREQUISITE(S): MATH 2030 or permission of instructor.

SUGGESTED TEXT(S):
A First Course in Abstract Algebra by John Fraleigh
Contemporary Abstract Algebra by Joseph Gallian
A Book of Abstract Algebra by Charles Pinter

COURSE OUTLINE:

1. Fundamental concepts and groups
   • Binary operations on a set
   • The group axioms and examples of groups
   • The concept of isomorphism in algebra
   • Subgroups of a group
   • Cyclic groups
   • Abelian groups
2. More advanced group theory
   • Permutation groups and Cayley's theorem
   • Alternating groups
   • Cosets and Lagrange's theorem
   • Direct products of groups
   • The fundamental theorem of finitely generated abelian groups
   • Group homomorphisms
   • Normal subgroups and quotient groups
   • The First Isomorphism Theorem
3. Other topics (as time permits)
   • Other isomorphism theorems
   • The center of a group and the commutator subgroup
   • p-groups and Cauchy's Theorem
   • The class equation and the Sylow theorems
   • The action of a group on a set
   • Burnside's Lemma and applications to counting