Module Directory

Groups are essential to abstract algebra. Indeed many algebraic structures (including rings, fields, modules of rings and vector spaces) are groups with extra operations.

Groups are not only key to understand other areas of mathematics such as geometry and representation theory, they have also influenced our understanding of other sciences including physics, chemistry, material science or cryptography.

The aims of this module are:

• To introduce students to basic definitions of group theory


• To develop students’ critical understanding of some main results in group theory


• To develop understanding of how to apply such results in particular problems.

By the end of the module, students will be expected to be able to:

1. Formulate and evaluate statements involving groups and their properties.

2. Recognise and work with various common examples of groups, such as cyclic, dihedral, symmetric and alternating groups and matrix groups.

3. Select and apply relevant theorems to examples.

4. Prove basic results involving abstract groups and their properties, and construct counterexamples to statements.

5. Understand standard constructions in group theory, such as group actions, group presentations, conjugacy, homomorphisms, isomorphisms, quotients and products.

Group theory is the branch of abstract algebra that formalises the rules obeyed by collections of symmetries, and other sets with rules for combining elements, such as various types of numbers with addition or multiplication operations. From a few basic axioms, an extraordinarily rich structure theory emerges, with applications across all of mathematics and the sciences. Common basic questions then include: Can we classify certain types of groups? And what mathematical results apply to all groups?

Indicative syllabus

• Homomorphisms, isomorphisms, automorphisms; kernels and images


• Cosets, normal subgroups and quotient groups. First isomorphism theorem.


• Conjugacy classes and centralisers.


• Symmetric and alternating groups; cycle structure; Cayley's theorem.


• Direct and semi-direct products.


• Classification of finite abelian groups.


• Free groups and group presentations (definition and examples).


• Group actions and orbit-stabiliser theorem


• Abelianisation and derived subgroup.


• Cayley graphs.

Teaching in the School will be delivered using a range of face-to-face lectures, classes, and lab sessions as appropriate for each module. Modules may also include online only sessions where it is advantageous, for example for pedagogical reasons, to do so.