Groups for Undergraduates
About this book
In these lecture notes the student learns basic theorems of the subject (due to Sylow, Burnside, Schur and Frobenius). More importantly, the student learns to use the theorems in various combinations, to discover for himself the groups of reasonably small order. In examples, presentations of the groups of order 1–31 and 33–42 are constructed.
Once the groups are presented abstractly, the problem is not done: one needs to know how each abstract group may arise as a group of permutations or matrices. Theorems and techniques of representation theory are given which can do this for any group the student may have constructed in the earlier chapters — and the student ends up building the actual representations (not only the characters). In a series of examples, which the student may carry further, all the matrix representations are constructed for the groups of order less than 13.
For students who are already familiar with homomorphisms, cosets, Lagrange's theorem, and finite abelian groups, the text may be used alone. For any group theory course, at least one text such as this one, containing lots of examples, is strongly recommended.
The book is written in a lucid, straightforward style. The subject matter is presented from a student's perspective and constantly demands the student's involvement. Both these strategies are highly appropriate for a book of lecture notes and guarantee the student's understanding of the mathematical concepts.
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Information
Table of contents
- Contents
- Preface
- Author's message to the student
- 0. Introduction
- 1. Number-Theoretic Preliminaries
- 2. Semidirect Products
- 3. Presentations
- 4. The Groups of Order 1-31
- 5. Rewriting and Transfer
- 6. Classification of Certain Groups up to Isomorphism
- 7. Group Cohomology
- 8. More Cohomology: the Coprime-Order Theorem
- 9. Linear Representations
- 10. Construction of the Linear Representations
- 11. Further Cohomology— a Vanishing Lemma, Higher Transfer
- 12. The Representations of the Groups of Order 1-12