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Varieties of Groups

  • Hanna┬áNeumann
Book

Part of the Ergebnisse der Mathematik und ihrer Grenzgebiete book series (MATHE2, volume 37)

Table of contents

  1. Front Matter
    Pages II-XI
  2. Hanna Neumann
    Pages 1-37
  3. Hanna Neumann
    Pages 38-76
  4. Hanna Neumann
    Pages 77-109
  5. Hanna Neumann
    Pages 145-180
  6. Back Matter
    Pages 181-193

About this book

Introduction

Varieties of algebras are equationally defined classes of algebras, or "primitive classes" in MAL'CEV'S terminology. They made their first explicit appearance in the 1930's, in Garrett BIRKHOFF'S paper on "The structure of abstract algebras" and B. H. NEUMANN'S paper "Identical relations in groups I". For quite some time after this, there is little published evidence that the subject remained alive. In fact, however, as part of "universal algebra", it aroused great interest amongst those who had access, directly or indirectly, to PHILIP HALL'S lectures given at Cambridge late in the 1940's. More recently, category theory has provided a general setting since varieties, suitably interpreted, are very special examples of categories. Whether their relevance to category theory goes beyond this, I do not know. And I doubt that the category theoretical approach to varieties will be more than a fringe benefit to group theory. Whether or not my doubts have substance, the present volume owes its existence not to the fact that varieties fit into a vastly more general pattern, but to the benefit group theory has derived from the classification of groups by varietal properties. It is this aspect of the study of varieties that seems to have caused its reappearance in the literature in the 1950's.

Keywords

Abstract algebra Group theory Gruppe (Math.) Varieties algebra category theory equation

Authors and affiliations

  • Hanna┬áNeumann
    • 1
  1. 1.The School of General Studies Department of Pure MathematicsThe Australian National UniversityCanberraAustralia

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-642-88599-0
  • Copyright Information Springer-Verlag Berlin Heidelberg 1967
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-88601-0
  • Online ISBN 978-3-642-88599-0
  • Series Print ISSN 0071-1136
  • Buy this book on publisher's site