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MX3020: GROUP THEORY (2017-2018)

Last modified: 23 Aug 2017 15:32


Course Overview

Group theory concerns the study of symmetry. The course begins with the group axioms, which provide an abstract setting for the study of symmetry. We proceed to study subgroups, normal subgroups, and group actions in various guises. Group homomorphisms are introduced and the related isomorphism theorems are proved. Composition series are introduced and the Jordan-Holder theorem is proved. Sylow p-subgroups are introduced and the three Sylow theorems are proved. Throughout symmetric groups are consulted as a source of examples.




Course Details

Study Type Undergraduate Level 3
Session First Sub Session Credit Points 15 credits (7.5 ECTS credits)
Campus None. Sustained Study No
Co-ordinators
  • Professor Ben Martin

Qualification Prerequisites

  • Either Programme Level 3 or Programme Level 4

What courses & programmes must have been taken before this course?

What other courses must be taken with this course?

None.

What courses cannot be taken with this course?

None.

Are there a limited number of places available?

No

Course Description

  • Group axioms, subgroups, examples of groups.
  • Cosets of a subgroup
  • Lagrange's Theorem.
  • Homomorphisms, isomorphisms, normal subgroups, quotient groups.
  • Calculations in symmetric and alternating groups.
  • Group actions.
  • Sylow's Theorems.

 

Syllabus

  • Lagrange’s theorem
  • Symmetric groups and alternating groups.
  • Cyclic groups and their subgroups.
  • Homomorphisms and isomorphisms.
  • Normal subgroups and their quotient groups.
  • The three isomorphism theorems.
  • Simple groups and composition series. The Jordan-Holder theorem.
  • Group actions. The orbit-stabiliser theorem.
  • Sylow's theorems.
  • The classification of finite abelian groups

Further Information & Notes

Course Aims
Group theory is the science of symmetry. This course will develop the basic ideas of group theory and the general notion of groups acting on sets.

 

Learning Objectives
By the end of the course the student should:
- know and be able to reproduce the definitions of group and subgroup, to derive standard consequences of the axioms and to perform group theoretic computations,
- be able to establish that suitably defined sets with a binary operation are groups and have a familiarity with examples of groups, particularly matrix groups and symmetry     groups of finite sets,
- be able to define the order of an element in a group and establish the properties of powers of an element, particularly in a finite group,
- know and be able to establish the main properties of finite cyclic groups, their subgroups and generators,
- be able to state and prove Lagrange’s theorem and to calculate cosets of subgroups,
- understand and be able to illustrate with examples the concept of an isomorphism,
- be able to define normal subgroups and their quotient groups and calculate examples,
- understand and be able to define the concept of a homomorphism, establish that the kernel is a normal subgroup, that the image is a subgroup and to state and use the isomorphism theorems,
- be able to calculate in finite symmetric and alternating groups,
- be able to define the concept of a group acting on a set and the concepts of stabiliser and orbit and to state, apply the Orbit Stabiliser theorem and Burnside’s theorem, and to reproduce ideas of the proofs of these theorems
- be able to calculate stabilisers and orbits in examples,
- be able to state and apply Cauchy’s theorem and Sylow’s theorems and to reproduce ideas of the proofs of these theorems

Degree Programmes for which this Course is Prescribed

  • BSc Applied Mathematics
  • BSc Education (Secondary) - Mathematics
  • BSc Mathematics
  • BSc Mathematics with Gaelic
  • MA Applied Mathematics
  • MA Mathematics
  • Mathematics Joint
  • Mathematics Major

Contact Teaching Time

32 hours

This is the total time spent in lectures, tutorials and other class teaching.

Teaching Breakdown


Assessment

1st Attempt: 1 two-hour written examination (80%); in-course assessment (20%). Resit: 1 two-hour examination (maximum of 100% resit and 80% resit with 20% in-course assessment). Only the marks obtained on first sitting can be used for Honours classification.

Formative Assessment

Informal assessment of weekly homework through discussions in tutorials.

Feedback

In-course assignments will normally be marked within one week and feedback provided to students in tutorials. Students will be invited to contact Course Coordinators for feedback on the final examination.

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