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    • Student Forum

      Forum Description: This forum is available for everyone to post messages to. Students can raise questions or discuss issues related to the module. Students are encouraged to post to this forum and it will be checked daily by the module leaders. Students should feel free to reply to other students if they are able to.

  • Week 1: Fundamental definitions

    This week's lectures will cover the following fundamental definitions of group theory: groups; subgroups; the order of a group; the order of an element. We will also introduce Cayley tables, introduce the idea of a subgroup generated by a set of elements, and consider some simple examples of groups.

    This week's classes will correspond roughly to Sections 1.0 to 1.3 of the lecture notes.

    At the end of this week you will be able to use the Week 1 quiz. This quiz does not carry credit, and may be taken any number of times.

  • Week 2 : Examples of groups

    This week's focus will be on familiarising ourselves with examples of groups, including: cyclic groups; the Quaternion group; groups defined by multiplication modulo an integer; dihedral groups; and some examples of groups of matrices. We may also begin to investigate the symmetric group. 

    This week's material will correspond approximately to sections 1.4 to 2.5 of the printed notes. 

    At the end of this week you will be able to use the Week 2 quiz. This quiz does not carry credit, and may be taken any number of times.

  • Module Description

    This module is about the theory of groups, continuing from the basics taught in Introduction to Algebra. Groups describe `symmetry’ and for that matter they arise all over mathematics. We will delve further into the study of groups (from a rather abstract viewpoint), and in so doing meet many more examples of groups including ones arising in geometry and from other algebraic objects such as number systems and matrices. In many scenarios groups operate on certain sets (think of the group of symmetry operations of some geometric shape for instance). The theory of group actions deals with this kind of situation. Another highlight of the module are Sylow's Theorems which provide precise information on the number of subgroups of certain orders that a group must have.

    Throughout this module there is a strong emphasis on abstract thinking and proof.

  • Syllabus

    1. Introduction and basics
      Definition of group, subgroup, order, generators, and basic properties.
    2. Examples
      Cyclic gropus, the quaternion group of order 8, symmetric groups, alternating groups, symmetry group of geometric objects, matrix groups, group of units modulo n.
    3. Cosets and Conjugacy
      Cosets, Lagrange's Theorem, conjugacy, normal subgroups, quotient groups, products of subgroups, the commutator subgroup.
    4. Homomorphisms
      Definitions, image and kernel, the Isomorphism Theorems, the Correspondence Theorem, automorphisms, inner and outer automorphism groups.
    5. Actions
      Definitions, orbits, stabilisers, the Orbit–Stabiliser Theorem, centralisers and normalisers, the Orbit-Counting Lemma and applications.
    6. Simple groups and composition series
      Definitions, simple abelian groups, simplicity of alternating groups, composition series, statement of Jordan-Hölder.
    7. p-groups
      Sylow p-subgroups, the Sylow theorems, finite p-groups.

  • Learning outcomes

    At the end of this module, students should be able to:

    • Define groups, homomorphisms, (normal) subgroups, quotient groups and related objects.
    • Define and compute with examples, including the quaternion group, dihedral groups, the symmetric group and matrix groups.
    • State and prove basic properties of groups, including Lagrange's Theorem and the Isomorphism Theorems.
    • State and (with guidance) prove more advanced results concerning groups.
    • Define group actions, work with examples, and prove basic results.
    • Create examples to illustrate the underlying theory, and work with these examples.
    • Recount and explain concepts of group theory.

  • Quizzes (unassessed)

    These quizzes don't count towards your final mark - they're just to help you learn.  They all have multiple-choice or numerical answers which are marked automatically, and you can attempt them as many times as you want.  So please keep trying until you get all the questions right!

    Some of the questions are deliberately quite tough, because you're allowed several attempts, so don't get disheartened if you find it difficult - the best way to learn is to struggle with difficult tasks.

    • End of week 1 quiz

      This quiz will test you on the concepts taught during week 1.  It will become available at the end of the week.

  • Lecture notes

    Lecture notes for this module are available here.

  • Q-Review

    Unfortunately this module is taught in rooms which are not equipped with Q-Review.

  • Assessment Information

    Assessment Pattern - 20% in-term assessments and 80% final exam in Jan 2025.

    Format and dates for the in-term assessments - There will be one in-term assessment  contributing 20% to your overall module mark. This will be set to  open  in week 7.  Assessment 1 will require submission via QM Plus. 

    Format of final assessment - The final assessment will be online, require submission via QM Plus. We will announce all related details in due time. 

    Link to past papers - mid term past papers and final exams past papers with solutions will be added to "Past Papers" tab under Assessment information section of the module content page.

    Description of Feedback - In term and final assessments will be marked with written feedback provided. In addition there will be weekly exercises in the form of online quizzes and exam style questions (like the final exam) at the end of weekly lecture slides with solutions available the following week on QM Plus.

     


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