# 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:**

- MA249 Algebra II
- MA251 Algebra I: Advanced Linear Algebra
- MA257 Introduction to Number Theory
- MA3F1 Introduction to Topology
- MA3E1 Groups and Representations
- MA4H4 Geometric Group Theory

**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.