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| The present volume is the first of three that will be published under the general title Lectures in lbstract fllgebra. These vol-umes are based on lectures which the author has given during the past ten years at the University of North Carolina, at The Johns Hopkins University, and at Yale University. The general plan of the work is as follows.The present first volume gives an introduction to abstract algebra and gives an account of most of the important algebraic concepts. In a treatment of this type it is impossible to give a comprehensive account of the topics which are introduced. Nevertheless we have tried to go beyond ?the foundations and elementary properties of the algebraic sys-tems. This has necessitated a certain amount of selection and omission. We feel that even at the present stage a deeper under-standing of a few topics is to be preferred to a superficial under-standing of many. |
| INTRODUCTION￡oCONCEPTS FROM SET THEORY THE SYSTEM OF NATURAL NUMBERS SECTION 1. Operations on sets 2. Product sets, mappings 3. Equivalence relations 4. The natural numbers 5. The system of integers 6. The division process in I CHAPTER I￡oSEMI-GROUPS AND GROUPS 1 ￡?Definition and examples of semi-groups 2. Non-associative binary compositions 3. Generalized associative law. Powers 4. Commutativity 5. Identities and inverses 6. Definition and examples of groups 7. Subgroups 8. Isomorphism 9. Transformation groups 10. Realization of a group as a transformation group 11. Cyclic groups. Order of an element 12. Elementary properties of permutations 13. Coset decompositions of a group 14. Invariant subgroups and factor groups 15. Homomorphism of groups 16. The fundamental theorem of homomorphism for groups 17. Endomorphisms, automorphisms, center of a group 18. Conjugate classes …… |
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