School of Computing and Mathematics

Faculty of Natural Sciences

For academic year: 2021/22 Last Updated: 24 September 2021

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 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: 1,3

derive Burnside¿s Lemma and use it in counting configurations: 1,3

demonstrate knowledge of group homomorphisms and the role of homomorphism as a unifying principle in Group Theory: 2,3

derive and apply the First Isomorphism Theorem: 2,3

demonstrate knowledge of conjugates, centralisers, the Class Equation and Sylow¿s theorems: 2,3

derive and apply Sylow¿s First Theorem: 2,3

derive Burnside¿s Lemma and use it in counting configurations: 1,3

demonstrate knowledge of group homomorphisms and the role of homomorphism as a unifying principle in Group Theory: 2,3

derive and apply the First Isomorphism Theorem: 2,3

demonstrate knowledge of conjugates, centralisers, the Class Equation and Sylow¿s theorems: 2,3

derive and apply Sylow¿s First Theorem: 2,3

Learning/teaching comprises 30 hours video lectures, and 5 hours flipped examples classes.

Independent study comprises 30 hours examples class preparation, 10 hours for completion of assignment, 20 hours preparation for examination, 53 hours consolidation of lecture material, and 2 hours final exam.

Independent study comprises 30 hours examples class preparation, 10 hours for completion of assignment, 20 hours preparation for examination, 53 hours consolidation of lecture material, and 2 hours final exam.

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Take-home assignment

Written assignment. The assignment consists of a set of questions with pre-allocated space for written solutions. Students should expect to spend 5 hours on the assessment.

Take-home Coursework

Written coursework. This consists of a set of questions with pre-allocated space for written solutions. Students should expect to spend 5 hours on the assessment.

2 HOUR CLOSED BOOK EXAM

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