MX3020: GROUP THEORY (2018-2019)

• 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

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

Information on contact teaching time is available from the course guide.

More Information about Week Numbers

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).

Informal assessment of weekly homework through discussions in tutorials.

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.