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## 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 Level Undergraduate 3 First Sub Session 15 credits (7.5 ECTS credits) None. No Dr Assaf Libman

### Qualification Prerequisites

• Either Programme Level 3 or Programme Level 4

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### 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

### Contact Teaching Time

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

### Teaching Breakdown

Details, including assessments, may be subject to change until 31 August 2023 for 1st half-session courses and 22 December 2023 for 2nd half-session courses.

### Summative Assessments

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).

### 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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