# MA249 Algebra II: Groups and Rings

**Lecturer:** Nicholas Jackson

**Term(s):** Term 2

**Status for Mathematics students:** Core for Year 2 mathematics students. It could be suitable as a usual or unusual option for non-maths students

**Commitment:** 30 lectures

**Assessment:** 85% 2 hour June examination, 15% Assignments

**Formal registration prerequisites: **None

**Assumed knowledge: **

MA132 Foundations or MA138 Sets and Numbers:

- Number theory: congruence modulo-n, prime factorisation, Euclidean algorithm, gcd and lcm
- Sets and functions: basic set theory, injective and surjective functions, relations
- Polynomials: multiplication and division, Euclidean algorithm, Remainder Theorem, algebraic and transcendental numbers

**Useful background: **

MA136 Introduction to Abstract Algebra:

- Group theory: definitions of groups, subgroups, homomorphisms. Key examples (numbers, permutations, dihedral and other symmetry groups), Lagrange's Theorem, Normal subgroups and quotient groups
- Ring theory: definitions of rings, subrings, integral domains, fields. Key examples (Z, Q, R, C, Z_n, matrix rings, polynomial rings). (This material will be covered again, but prior knowledge will be helpful.)

- Classification of finitely generated Abelian groups

**Synergies:** The following modules goes well with Algebra II:

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

- MA3E1 Groups and Representations
- MA3G6 Commutative Algebra
- MA3J9 Historical Challenges in Mathematics
- MA377 Rings and Modules
- MA3F1 Introduction to Topology
- MA3K4 Introduction to Group Theory
- MA3J3 Bifurcations, Catastrophes and Symmetry
- MA3D5 Galois Theory
- MA3H6 Algebraic Topology
- MA3J2 Combinatorics II
- MA3A6 Algebraic Number Theory
- MA4L6 Analytic Number Theory
- MA4H4 Geometric Group Theory
- MA426 Elliptic Curves
- MA473 Reflection Groups
- MA453 Lie Algebras
- MA4J8 Commutative Algebra II

**Content**: This is an introductory abstract algebra module. As the title suggests, the two main objects of study are groups and rings. You already know that a group is a set with one binary operation. Examples include groups of permutations and groups of non-singular matrices. Rings are sets with two binary operations, addition and multiplication. The most notable example is the set of integers with addition and multiplication, but you will also be familiar already with rings of polynomials. We will develop the theories of groups and rings.

Some of the results proved in MA251 Algebra I: Advanced Linear Algebra for abelian groups are true for groups in general. These include Lagrange's Theorem, which says that the order of a subgroup of a finite group divides the order of the group. We defined quotient groups for abelian groups in Algebra I, but for general groups these can only be defined for certain special types of subgroups *H* of *G*, known as normal subgroups. We can then prove the isomorphism theorems for groups in general. An analogous situation occurs in rings. For certain substructures *I* of rings *R*, known as ideals, we can define the quotient ring , and again we get corresponding isomorphism theorems.

Other results to be discussed include the Orbit-Stabiliser Theorem for groups acting as permutations of finite sets, the Chinese Remainder Theorem, and Gauss' theorem on unique factorisation in polynomial rings.

**Aims**: To study abstract algebraic structures, their examples and applications.

**Objectives**: By the end of the module the student should know several fundamental results about groups and rings as well as be able to manipulate with them.

**Books**:

Niels Lauritzen, *Concrete Abstract Algebra*, Cambridge University Press

M A Armstrong,* Groups and Symmetry*, Springer

John B Fraleigh, *A First Course in Abstract Algebra*, Pearson

Joseph Gallian, *Contemporary Abstract Algebra*, Chapman Hall

John M Howie, *Fields and Galois Theory*, Springer

Nicholas Jackson, *A Course in Abstract Algebra*, Oxford University Press (forthcoming, draft sections available on request)