ISBN: Ostalo
Godina izdanja: 1994
Jezik: Engleski
Oblast: Matematika
Autor: Strani

58388) GROUPS FOR UNDERGRADUATES , John Moody , World Scientific Publishing 1994 , 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.
Contents
Preface
Author`s message to the student
Chapter 0. Introduction
Representations (basic definitions)
Classification of permutation representations
Chapter 1. Number-Theoretic Preliminaries
Binomial theorem
The Chinese Remainder Theorem Sylow theorems
The structure of Z/nZX
Finite Fields
The conjugacy classes in Glm (Z/pZ)
Polynomial factorization and finite fields

Chapter 2. Semidirect Products
Chapter 3. Presentations
Presentations of semidirect products
Example: groups of order 75
The examples of Sn, Un, groups of order p3
Principle relating homomorphisms and presentations
Tietze moves
Chapter 4.
Calculation of a List of the Groups of Order 1 - 31
Chapter 5. Rewriting and Transfer
Rewriting
Example: An-Action on relation subgroup Transfer
Burnside`s p-complement theorem-Calculation of a list of groups of order 33-41 using transfer

Chapter 6. Classification of Certain Groups
Chapter 7. Cohomology
Chapter 8. More Cohomology
The coprime-order theorem
Application: the groups of order 42

Chapter 9. Linear Representations
The proof that they are unitary-Mashke`s theorem
Schur`s lemma
Unique decomposition into irreducibles
Induced representations

Chapter 10. Construction of the Linear Representations
Tensor products
Further cohomology
Classification theorem
Chapter 11. Further Cohomology
Vanishing lemma
Higher transfer
Chapter 12. Calculation of a List of the Representations of the Groups of Order 1-12

paperback, size 15 x 21,5 cm , English ; X +165 pages