MA136 Introduction to Abstract Algebra

Lecturer: Richard Lissaman

Term(s): Term 1

Status for Mathematics students: Core for Maths

Commitment: 15 one hour lectures

Assessment: Assignments (15%), Written exam (85%)

Formal registration prerequisites: None

Assumed knowledge: A-level Mathematics and Further Mathematics

Useful background: Some elementary knowledge of matrices, functions, modular arithmetic

Synergies:

Leads To: The following modules have this module listed as assumed knowledge or useful background:

Content:

Section 1 Group Theory:

Motivating examples: numbers, symmetry groups
Definitions, elementary properties
Subgroups, including subgroups of $Z$
Arithmetic modulo n and the group $Z_n$
Lagrange's Theorem
Permutation groups, odd and even permutations (proof optional)
Normal subgroups and quotient groups

Section 2 Ring Theory:

Definitions: Commutative and non-commutative rings, integral domains, fields
Examples: $Z, Q, R, C, Z_n$, matrices, polynomials, Gaussian integers

Aims:

To introduce First Year Mathematics students to abstract Algebra, covering Group Theory and Ring Theory.

Objectives:

By the end of the module students should be able to understand:

the abstract definition of a group, and be familiar with the basic types of examples, including numbers, symmetry groups and groups of permutations and matrices.
what subgroups are, and be familiar with the proof of Lagrange’s Theorem.
the definition of various types of ring, and be familiar with a number of examples, including numbers, polynomials, and matrices.
unit groups of rings, and be able to calculate the unit groups of the integers modulo n.

Books:

Any library book with Abstract Algebra in the title would be useful.

Additional Resources

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