The module continues the study of abstract algebra by further developing the theory of groups. The theory will be illustrated through examples in settings that students will already have encountered.

Aims

To continue the study of groups and to teach how an extensive and important theory can be developed by logical deductions from a small number of axioms.

Learning Outcomes

On completion of the course, students should:

1. have a systemic understanding of key definitions in the theory of groups and critical awareness of how they interact and support each other

2. select and apply relevant theorems to examples

3. construct arguments to prove properties of groups

4. solve problems involving homomorphisms between pairs of groups

5. formulate counterexamples to statements

6. understand the concept of a group presentation

7. recognise and work with cyclic, dihedral, Fibonacci, and triangle groups,

8. deploy methods learned to distinguish pairs of groups defined by presentations or to prove they are isomorphic

9. apply geometric techniques to obtain and illustrate algebraic properties of particular groups

10. understand the definition and importance of p-groups and recognise them

11. apply Sylow theory to obtain the subgroup structure of groups

Syllabus

Homomorphisms, isomorphisms, automorphisms. Cosets, normal subgroups, quotient groups, abelianization and derived subgroup. Lagrange's theorem, isomorphism theorems. Free groups, group presentations (definition and examples), Tietze transformations, Cayley Diagrams, Van Kampen diagrams. Sylow theorems, p-groups,conjugacy, special linear groups.

- Module Supervisor: Jesus Martinez-Garcia