Mathematics
MA343: Groups I (5 ECTS)
Introduction to Group Theory. Topics covered include the group axioms, symmetries, permutations, cyclic groups, dihedral groups, small groups of matrices, homomorphisms, normal subgroups, Isomorphism Theorems, automorphism groups, free groups, relators and presentations.
Taught in Semester(s) 1. Examined in Semester(s) 1.
Workload: 120 hours (24 Lecture hours, 12 Tutorial hours, 84 Self study hours).
Module Learning Outcomes.
On successful completion of this module the learner should be able to:
- Carry out calculations in abstract algebraic structures given by axioms;
- Work with homomorphisms, quotient structures and free groups;
- Determine the structure of small groups given by generators and relations or by generating matrices or by generating permutations;
- Describe symmetries of geometric objects in terms of permutations or matrices;
- Find and write proofs for abstract group theoretic facts at scholarly standard;
- Search, read, understand and make use of more advanced literature in the field.
Indicative Content
The material covered includes:
- Group Axioms: the group axioms as abstraction of properties of the symmetries of an object, permutations, rotations, reflections, translations, cyclic groups, the integers, groups of matrices.
- Basic notions: subgroup, order, cosets, Lagrange's Theorem, generators, many examples.
- Homomorphisms: structure preserving maps, quotient structures, kernels, normal subgroups, regular representation, conjugation representation.
- Theory: Isomorphism Theorems, simplicity of alternating groups, universal property of free groups, direct products, presentations, centre and commutator subgroup.
Module Resources
- J. J. Rotman, An Introduction to the Theory of Groups, Springer
- D. J. S. Robinson, A Course in the Theory of Groups, Springer
- P. M. Neumann & G. A. Stoy, Groups and Geometry, OUP
- G. Smith & O. Tabachnikova, Topics in Group Theory, Springer
- I. D. Macdonald, The Theory of Groups, Clarendon Press, Oxford
- R. C. Lyndon & P. E. Schupp, Combinatorial Group Theory, Springer
- W. Burnside, Theory of Groups of Finite Order, CUP
- B. Huppert & N. Blackburn, Finite Groups, Springer
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