# MA249 Algebra 2: Groups and Rings

**Lecturer:** Dmitriy Rumynin

**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:** Assignments (15%), two-hour examination (85%)

**Prerequisites:** MA132 Foundations (MA138 Sets and Numbers for non-maths students), MA106 Linear Algebra, and MA251 Algebra I: Advanced Linear Algebra

**Leads To: **The results of this module are used in several modules including: MA377 Rings and Modules, MA3A6 Algebraic Number Theory, MA453 Lie Algebras, MA3G6 Commutative Algebra, MA3D5 Galois Theory, MA3E1 Group and Representations, and MA3J3 Bifurcations Catastrophes and Symmetry, although unfortunately not all of these modules are offered every year.

**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 MA242 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**:

Complete lecture notes for the module will be available from the General Office soon after the beginning of the spring term, and will appear on the module resources page towards the end of term.

One possible book is

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