Overview

Description

CALENDAR DESCRIPTION:

Finite groups and subgroups. Cyclic groups and permutation groups. Cosets, normal subgroups and factor groups. Homomorphisms and isomorphisms. Fundamental theorem of finite abelian groups. Sylow theorems. Students with credit for MATH 339 may not take this course for further credit.

COURSE DETAILS:

Course Details:


Groups:

•     Definition and examples of Groups

•     Elementary Properties of Groups

Finite Groups: Subgroups:

•     Terminology and Notation

•     Subgroup Tests

•     Examples of Subgroups

Cyclic Groups:

•     Properties of Cyclic Groups

•     Classification of Subgroups of Cyclic Groups

Permutation Groups:

•     Definition and Notation

•     Cycle Notation

•     Properties of Permutations

  
Isomorphisms:

•     Motivation

•     Definition and Examples

•     Cayley's Theorem

•     Properties of Isomorphisms

•     Automorphisms

Cosets and Lagranges Theorem:

•     Properties of Cosets

•     Lagranges Theorem and Consequences

•     An Application of Cosets to Permutation Groups [Orbit-Stabilizer Theorem]

•     The Rotation Group of a Cube

Normal Subgroups and Factor Group:

•     Normal Subgroups

•     Factor Groups

•     Applications of Factor Groups [including Cauchy's Theorem]

    

Group Homomorphisms:

•     Definition and Examples

•     Properties of Homomorphisms

•     The First Isomorphism Theorem

Sylow Theorems:

•     Conjugacy Classes

•     The Class Equation

•     The Sylow Theorems

•     Applications of Sylow theorems

Other topics:

•     The Fundamental Theorem of Finite Abelian Groups

•     Simple Groups

•     Composition Series

•     Solvable Groups

Grading

Materials

REQUIRED READING:

RECOMMENDED READING: