MAT 310  Abstract Algebra

The course is an intensive introduction to abstract algebraic structures concentrating on groups. Group Theory is the study of symmetry in the abstract. In addition to understanding concrete examples of groups and their applications to the symmetries of physical objects, theorem proving will form an essential part of the course. Students can expect to hone their skills in understanding proofs and how to construct them.

Prerequisite: MAT 250 Linear Algebra or permission of the instructor

Topics:

• Number theory

fundamental properties of the integers: prime numbers, divisibility, common divisors; proofs by mathematical induction

Operations

binary operations, identity, commutative and associative properties, inverses; composition of functions as an operation

Semigroups and Groups

axioms of semigroups and groups; dihedral groups: motions of polygons and the Platonic solids; abelian groups; finite and infinite groups; group tables

Properties of Groups

the cancellation properties; uniqueness of identity and inverses; the order of a group element

Functions and Homomorphisms

injective, surjective, & bijective functions; one-to-one correspondences; the pigeon-hole principle; cardinality; homomorphisms

Subgroups

basic properties of subgroups; methods for checking whether a subset is a subgroup; cyclic subgroups; kernel of a homomorphism

Generating Sets, Cyclic Groups, and Products of Groups

generators; subgroups generated by a subset; direct products of groups

Permutation Groups

definition of a permutation; cycles; cyclic decomposition; transpositions; decomposition of a permutation as a product of cycles and transpositions; even/odd permutations

Cayley's Theorem and the Fundamental Theorem on Finite Abelian Groups

Relations and Cosets

equivalence relations; equivalence classes; partitions; left and right cosets of a subgroup

Lagrange's Theorem

the theorem and its converse; introduction to Sylow subgroups

Normal Subgroups and Quotient Groups

As time permits:

• Rings

fundamental properties

• Subrings and Ideals

left and right ideals; ring homomorphisms

• Polynomial Rings
• Quotient Rings and Field Extensions

The computer software package Exploring Small Groups will also be used during occasional class sessions in the Spidel Computer Lab to reinforce the theory through examples.

Texts: Modern Algebra: A Conceptual Approach, by Franklin D. Pedersen, Wm. C. Brown Publishers, 1993

Laboratory Experiences in Group Theory, by Ellen Maycock Parker, The Mathematical Association of America, 1996

Software: Exploring Small Groups, developed by Ladnor D. Geissinger

Grading: There will be an exam every other week -- on the alternate Fridays. Exams will be non-cumulative: Each will include only material since the previous exam. There will be no other cumulative exams. Problem Sets will be assigned each class and are expected to be completed (or at least seriously attempted) by the next class session. Much of class will be class discussion of problems; these discussions will constitute some of the most valuable in-class learning. Therefore, a class participation grade will also form significant input into your final grade.

Attendance is expected at each class. More than three unexcused absences will begin to affect your grade negatively.

Theorems and their proofs form the language of mathematics. To be articulate in mathematics requires three related abilities: a) to follow proofs of theorems constructed by others; b) to construct your own proofs of theorems; c) to generalize from specific examples to conjectures that might be true in general and then to devise proofs of your conjectures.

Abstract Algebra provides an ideal subject in which to practice these skills -- the ability to understand and devise theorems and proofs -- because the logical structures studied in Abstract Algebra can be reduced to a small number of easily understood and intuitive axioms which provide the foundation for the theorems.

Specific examples of algebraic structures are interesting -- even fascinating -- because of their connections with ideas such as symmetry. We will begin our study of each concept through specific examples. Generalization to theorems, though, constitutes the heart of mathematics; and generalization entails the formulation of theorems and understanding their proofs.

At least half the problems assigned will actually be proofs. There is no standard format for writing up a proof of a theorem, but clarity, coherence, and completeness are essential. These are not easy to attain. The art of writing up a proof will occupy significant amounts of class time. We will analyze the structure of proofs in the text and our own proofs with an eye to understanding them and how they might be improved.

The majority of the questions on the biweekly exams will require you to devise proofs and write them clearly. Clarity, coherence, and completeness will be the standards for grading your proofs. Some of the exams will be take-home to allow the time for you to revise your proof write-ups for optimal clarity, coherence, and completeness.

We will also work on expressing theorems and their relationships in plain English by writing summaries and outlines.