This book is designed to be used as a one-semester text. It evolved from a set oflectures that was to bridge the gap between the group theory presented in an introductory graduate algebra course and a serious pursuit of the subject.
The book should be suitable for independent study. The overall approach and the methods of proof are varied
but standard, and no novelties are introduced. Students have had little adjustment in the transition from this treatment to either special topics or reference works. Exercises are included with the original intent of the book in mind. For the most part, they are not difficult. In particular, this is true of the proofs of theorems and corollaries that have been left to the reader to complete. Most of the notation is standard. Since tensor products are not introduced in this text, the symbol used for the direct product should not cause confusion.
Author(s): Homer Bechtell
Publisher: Pearson/Addison-Wesley
Year: 1971
Language: English
Pages: C+xii, 144
Cover
S Title
THE THEORY OF GROUPS
Copyright
© 1971 by Addison-Wesley Publishing Company
Dedication
PREFACE
CONTENTS
1 BASIC CONCEPTS AND NOTATION
2 PRODUCTS; DIRECT PRODUCTS; DIRECT PRODUCT WITH AMALGAMATED SUBGROUP; AND SUBDIRECT PRODUCTS
2.1 PRODUCTS AND DIRECT PRODUCTS
EXERCISES 2.1
2.2 DIRECT PRODUCT WITH AMALGAMATED SUBGROUP
EXERCISES 2.2
2.3 SUBDIRECT PRODUCTS
Exercises 2.3
3 SPLITTING EXTENSIONS; SEMIDIRECT AND WREATH PRODUCTS
3.1 PRODUCTS OF SUBGROUPS
EXERCISES 3.1
3.2 EXTENSIONS
EXERCISES 3.2
3.3 SPLITTING EXTENSIONS
EXERCISES 3.3
3.4 WREATH PRODUCTS
EXERCISES 3.4
4 THEOREMS ON SPLITTING; HALL SUBGROUPS
4.1 ON A THEOREM OF DIXON
EXERCISES 4.1
4.2 SPLITTING THEOREMS OF GASCHUTZ
EXERCISES 4.2
4.3 ON HALL n-SUBGROUPS
EXERCISES 4.3
4.4 ADDITIONAL COMMENTS
EXERCISES 4.4
5 NILPOTENT GROUPS; THE FRATTINI SUBGROUP
5.1 NILPOTENT GROUPS
EXERCISES 5.1
5.2 THE SYLOW STRUCTURE OF A NILPOTENT GROUP
EXERCISES 5.2
5.3 THE FRATTINI SUBGROUP
EXERCISES 5.3
5.4 ADDITIONAL REMARKS ON THE FRATTINI SUBGROUP
EXERCISES 5.4
6 THE FITTING SUBGROUP; SUPERSOLV ABLE GROUPS
6.1 THE FITTING SUBGROUP
EXERCISES 6.1
6.2 SUPERSOLVABLE GROUPS
EXERCISES 6.2
7 GENERAL EXTENSION THEORY
7.1 EXTENSIONS AND FACTOR SETS
EXERCISES 7.1
7.2 EQUIVALENT EXTENSIONS
EXERCISES 7.2
7.3 EXTENSIONS OF ABELIAN GROUPS
EXERCISES 7.3
7.4 CYCLIC EXTENSIONS
EXERCISES 7.4
7.5 FINITE EXTENSIONS OVER A CYCLIC MAXIMAL SUBGROUP OF PRIME POWER ORDER
EXERCISES 7.5
8 THE THEORY OF THE TRANSFER
8.1 THE TRANSFER
EXERCISES 8.1
8.2 BURNSIDE'S THEOREM
EXERCISES 8.2
8.3 THE THEOREMS OF GRUN
EXERCISES 8.3
8.4 SOME APPLICATIONS OF THE THEOREMS OF GRUN
EXERCISES 8.4
9 FREE GROUPS AND COPRODUCTS
9.1 FREE GROUPS
EXERCISES 9.1
9.2 FREE PRODUCTS AND COPRODUCTS IN Grp
EXERCISES 9.2
9.3 COPRODUCTS IN Ab
EXERCISES 9.3
APPENDIX SOME ELEMENTS OF CATEGORY THEORY
A.l CATEGORIES AND FUNCTORS
EXERCISES A.1
A.2 PRODUCTS AND COPRODUCTS
EXERCISES A.2
BIBLIOGRAPHY AND INDEX OF SPECIAL SYMBOLS
BIBLIOGRAPHY
INDEX OF SPECIAL SYMBOLS
INDEX
INDEX
