Group Theory Vol 1

Though these notes require no previous knowledge of group theory it is more suitable for a third year student than a first year one because it goes into the subject quite deeply. The opening chapters attempt to motivate group theory with many illustrative examples such as shuffling of cards, bell ringing and permutation puzzles. 

GroupsFairly early on the student is introduced to group presentations and the Todd-Coxeter algorithm. Although this algorithm has been around for about 80 years it is not normally taught. However as it underlies the modern approach to group theory it is well worth students knowing it. A new framework for the algorithm makes it accessible to third-year students.

Another novelty, for a first course in group theory, is the inclusion of a couple of chapters on representation theory. Although a full stand-alone treatment would be quite deep, by assuming the Fundamental Theorem of Characters (the irreducible characters forming an orthonormal basis) everything else that is needed can be easily proved. Students seem to enjoy the challenge of constructing character tables from a group presentation and the theory of characters reinforces the concepts and methods of linear algebra.

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