School of Computing and Mathematics

Faculty of Natural Sciences

For academic year: 2019/20 Last Updated: 21 November 2019

MAT-30013 - Group Theory

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MAT-20025: Abstract Algebra

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This module builds on the Group Theory introduced in MAT - 20025 to develop some of the mathematics underlying the classification of finite groups. This culminates in a proof of Sylow's First Theorem which offers a partial converse to Lagrange's Theorem proved in MAT - 20025. The module also develops some applications of Group Theory, the natural setting for which is that of group actions. Several examples of applying group theoretic ideas to counting combinatorial configurations are presented.

The module develops the following Keele Graduate attributes:

1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought.

2. An appreciation of the development and value of Mathematics and the links between different areas of the subject.

4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand.

6. The ability to communicate clearly and effectively in written form.

The module develops the following Keele Graduate attributes:

1. An open and questioning approach to ideas, demonstrating curiosity and independence of thought.

2. An appreciation of the development and value of Mathematics and the links between different areas of the subject.

4. The ability creatively to solve problems using a range of different approaches and techniques, and to determine which techniques are appropriate for the problem at hand.

6. The ability to communicate clearly and effectively in written form.

The aim of this module is to develop some of the mathematics underlying the classification of finite groups and to develop some applications of Group Theory.

demonstrate knowledge of basic concepts such as abelian groups, normal subgroups, quotient groups and group actions: 1,2

derive Burnside¿s Lemma and use it in counting configurations: 1,2

demonstrate knowledge of group homomorphisms and the role of homomorphism as a unifying principle in Group Theory: 1,2

derive and apply the First Isomorphism Theorem: 1,2

demonstrate knowledge of conjugates, centralisers, the Class Equation and Sylow¿s theorems: 1,2

derive and apply Sylow¿s First Theorem: 1,2

derive Burnside¿s Lemma and use it in counting configurations: 1,2

demonstrate knowledge of group homomorphisms and the role of homomorphism as a unifying principle in Group Theory: 1,2

derive and apply the First Isomorphism Theorem: 1,2

demonstrate knowledge of conjugates, centralisers, the Class Equation and Sylow¿s theorems: 1,2

derive and apply Sylow¿s First Theorem: 1,2

Lectures: 30 hours

Preparation of coursework: 30 hours

Independent study: 88 hours

Unseen examination : 2 hours

Preparation of coursework: 30 hours

Independent study: 88 hours

Unseen examination : 2 hours

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Approximately 4 assignments set at regular intervals.

Continuous assessment will consist of written coursework, problem sheets, class tests, or any combination thereof.

2 HOUR UNSEEN EXAM

The examination paper will consist of no less than five and not more than eight questions all of which are compulsory.