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Online course materials for MATH32001

Group Theory


Unit code: MATH32001
Credit Rating: 10
Unit level: Level 3
Teaching period(s): Semester 1
Offered by School of Mathematics
Available as a free choice unit?: N

Requisites

Prerequisite

Additional Requirements

Please note

Students are not permitted to take more than one of MATH32001 or MATH42001 for credit in the same or different undergraduate year.  Students are not permitted to take MATH42001 and MATH62001 for credit in an undergraduate programme and then a postgraduate programme.

Aims

This lecture course unit aims to introduce students to some more sophisticated concepts and results of group theory as an essential part of general mathematical culture and as a basis for further study of more advanced mathematics.

Overview

The ideal aim of Group Theory is the classification of all groups (up to isomorphism). It will be shown that this goal can be achieved for finitely generated abelian groups. In general, however, there is no hope of a similar result as the situation is far too complex, even for finite groups. Still, since groups are of great importance for the whole of mathematics, there is a highly developed theory of outstanding beauty. It takes just three simple axioms to define a group, and it is fascinating how much can be deduced from so little. The course is devoted to some of the basic concepts and results of Group Theory.

Learning outcomes

On successful completion of this course unit students will have acquired

  • a sound understanding of the classification of finitely generated abelian groups,
  • knowledge of some fundamental results and techniques from the theory of finite groups,
  • knowledge of group actions on sets, simple groups, Sylow's theorems and various applications of Sylow's theorems.

Assessment methods

  • Other - 10%
  • Written exam - 90%

Assessment Further Information

  • Coursework: in-class test weighting 10%
  • End of semester examination: two hours, weighting 90%

Syllabus

    1. Revision of basic notions (subgroups and factor groups, homomorphisms and isomorphisms), generating sets, commutator subgroups. [2 lectures]
    2. Abelian groups, the Fundamental Theorem on finitely generated abelian groups. [4]
    3. The Isomorphism Theorems. [3]
    4. Simple groups, the simplicity of the alternating groups. [3]
    5. Composition series, the Jordan-Hölder Theorem. [2]
    6. Group actions on sets, orbits, stabilizers, the number of elements in an orbit, Burnside's formula for the number of orbits, conjugation actions, centralizers and normalizers. [5]
    7. Sylow's Theorems, groups of order pq, pqr. [3]

 

For MATH42001 the lectures will be enhanced by additional reading on related topics.

Recommended reading

Recommended text:

John B Fraleigh, A First Course in Abstract Algebra, (5th edition), 1967, Addison-Wesley.

Feedback methods

Feedback tutorials will provide an opportunity for students' work to be discussed and provide feedback on their understanding.  Coursework or in-class tests (where applicable) also provide an opportunity for students to receive feedback.  Students can also get feedback on their understanding directly from the lecturer, for example during the lecturer's office hour.
 

Study hours

  • Lectures - 22 hours
  • Tutorials - 11 hours
  • Independent study hours - 67 hours

Teaching staff

Peter Rowley - Unit coordinator

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