# e-book Groups, Rings and Group Rings

Appreciate the significance of unique factorization in rings and integral domains. Apply the theory in the course to solve a variety of problems at an appropriate level of difficulty.

## 4. Basics from Algebra: Groups, Rings, and Fields | SpringerLink

Demonstrate skills in communicating mathematics orally and in writing. Required Resources None. Armstrong, "Groups and Symmetry", Springer, ; covers most of the material about groups in the course, but in addition has many geometric applications and examples. There are many other introductory texts on abstract algebra in the library which students may find useful as references.

Assignments, tutorial exercises, handouts, and course announcements will be posted on MyUni. The lecturer guides the students through the course material in the lectures, working through proofs and examples.

## MA249 Algebra 2: Groups and Rings

In particular students will have opportunity to raise any points of difficulty arising from their own reading of the notes. Whilst they will be recorded, students will not gain the full benefit if not able to attend in person. Fortnightly homework assignments help students strengthen their understanding of the theory and their skills in applying it, and allow them to gauge their progress. Lecture Schedule Week 1 Groups Groups and subgroups.

Week 2 Groups Permutation groups, isomorphisms, cosets and normal subgroups, conjugation. Week 3 Groups Simple groups, homomorphisms and factor groups. Week 7 Groups Groups acting on sets.

### Groups, Rings and Group Rings

Week 8 Groups The Sylow theorems and applications. Week 9 Rings Introduction to rings.

Group Definition (expanded) - Abstract Algebra

Week 10 Rings Integral domains, polynomial rings. Week 11 Rings Factorisation in integral domains, ideals, eudlidean domains. In weeks 2,4,6,8,10 and 12 there will be a tutorial in the Friday class. There will be a mid semester test, most likely in the Wednesday class in week 7 after the mid semester break. There will be one meeting during one of the classes for discussions about the group project. A group project with a written report develops research skills, teamwork skills, and communication skills.

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. 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.