Group Theory

The details
Mathematical Sciences
Colchester Campus
Postgraduate: Level 7
Monday 13 January 2020
Friday 20 March 2020
01 October 2019


Requisites for this module



Key module for


Module description

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.

Module aims

The module 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.

Module 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

Module information

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.

Learning and teaching methods

Lectures and classes plus revision lectures.


  • (2014-09-01) Representation Theory: Springer.
  • Fraleigh, John B. (©2014) A first course in abstract algebra: Pearson.
  • Joseph A. Gallian. (2017) Contemporary Abstract Algebra, Boston, MA: Cengage Learning.

The above list is indicative of the essential reading for the course. The library makes provision for all reading list items, with digital provision where possible, and these resources are shared between students. Further reading can be obtained from this module's reading list.

Assessment items, weightings and deadlines

Coursework / exam Description Deadline Weighting
Coursework Problem set 1 13/02/2020
Coursework Problem set 2 19/03/2020
Exam 120 minutes during Summer (Main Period) (Main)

Overall assessment

Coursework Exam
20% 80%


Coursework Exam
0% 100%
Module supervisor and teaching staff
Dr Jesus Martinez-Garcia, e-mail
Dr Jesus Martinez-Garcia (



External examiner

No external examiner information available for this module.
Available via Moodle
Of 38 hours, 33 (86.8%) hours available to students:
5 hours not recorded due to service coverage or fault;
0 hours not recorded due to opt-out by lecturer(s).


Further information
Mathematical Sciences

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