Discovering Group Theory: A Transition to Advanced Mathematics presents the usual material that is found in a first course on groups and then does a bit more. The book is intended for students who find the kind of reasoning in abstract mathematics courses unfamiliar and need extra support in this transition to advanced mathematics.
The book gives a number of examples of groups and subgroups, including permutation groups, dihedral groups, and groups of integer residue classes. The book goes on to study cosets and finishes with the first isomorphism theorem.
Very little is assumed as background knowledge on the part of the reader. Some facility in algebraic manipulation is required, and a working knowledge of some of the properties of integers, such as knowing how to factorize integers into prime factors.
The book aims to help students with the transition from concrete to abstract mathematical thinking.
Table of Contents
1. Proof; 2 Sets; 3. Binary operations; 4. Integers; 5. Groups ; 6. Subgroups; 7. Cyclic groups; 8. Products of groups; 9. Functions; 10. Composition of functions; 11. Isomorphisms; 12. Permutations; 13. Dihedral groups; 14. Cosets; 15. Groups of orders up to 8; 16. Equivalence relations; 17. Quotient groups; 18. Homomorphisms; 19. The First Isomorphism Theorem; Answers; Index
Tony Barnard has lectured at King's College London on abstract algebra for over 35 years. His research activity was initially in abstract algebra and more recently has been in the psychology of mathematics education. He has served on several consultative committees of the UK government and learned societies, advising on matters relating to the school mathematics curriculum and university mathematics teaching.
Hugh Neill started as a school teacher, moved into mathematics teaching at the University of Durham and then became the senior mathematics inspector in schools in Inner London until the Inner London Education Authority was abolished in 1990. During this time he was heavily involved in the design and assessment of mathematics courses for future mathematics teachers. Since 1990 he has been writing mathematics books.
There are numerous textbooks designed to help students transition into more in-depth mathematics comprehension other than calculating and applying algorithms. These transitional works are typically the first time students are required to prove a mathematical statement. Many textbooks designed for this use sets, relations, functions, and some elementary number theory as platforms. In this text, Barnard (King's College London, UK) and Neill, an experienced mathematician, use group theory as the content for transitioning to proof writing and more advanced mathematics. The most common proof techniques are presented and applied. Since a group is a set, basic set theory is included, along with functions for permutation groups and isomorphisms. The text also presents number theory and applies it to group theory. The authors provide detailed solutions to all the exercises in their book, as well as excellent explanations of their thought process related to the presented proofs. The text culminates with the first isomorphism theorem; as a result, undergraduate majors will need more abstract algebra than is offered in this work. However, this text offers students a superb introduction to both proof writing and group theory.
--J. T. Zerger, Catawba College, Choice magazine 2016
"These days group theory is an essential part of many university courses in physics as well as mathematics. This book now goes to the top of my list of recommendations for students who want to encounter groups before starting such courses, and it will also prove a useful companion in the early stages at university."
--Owen Toller, St Paul’s School, London