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Elementary theory of groups

Published online by Cambridge University Press:  11 January 2010

Benjamin Fine
Affiliation:
Department of Mathematics, Fairfield University, Fairfield, Connecticut 06430, USA
Anthony M. Gaglione
Affiliation:
Department of Mathematics, U.S. Naval Academy, Anapolis, Maryland 1402, USA
Alexei Myasnikov
Affiliation:
Department of Mathematics, City College of CUNY, New York, New York 10031, USA
Gerhard Rosenberger
Affiliation:
Fachbereich Mathematik Universität, Dortmund, 44221 Dortmund, Germany
Dennis Spellman
Affiliation:
Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19132, USA
C. M. Campbell
Affiliation:
University of St Andrews, Scotland
E. F. Robertson
Affiliation:
University of St Andrews, Scotland
G. C. Smith
Affiliation:
University of Bath
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Summary

Abstract

Remarkable ties between group theory, logic and algebraic geometry have come to light via the positive solution of the Tarski conjecture. This has led to further work on the universal theory of groups. In this paper we describe and survey this material a large body of which is not familiar to most group theorists

Contents

  1. Introduction.

  2. First Order Languages and Model Theory.

  3. The Tarksi Problems.

  4. Residually Free and Universally Free Groups.

  5. Algebraic Geometry over Groups and Applications.

  6. The Positive Solution to the Tarski Problems.

  7. Discriminating, Co-discriminating and Squarelike Groups.

  8. Open Questions.

Introduction

The elementary theory of a group G consists of all the first-order or elementary sentences (see section 2) which are true in G. Although this is a concept which originated in formal logic, in particular model theory, it arises independently from the theory of equations within groups. Recall that an equation in a group G is a word W(x1,…, xn, g1,…, gk) in free variables x1,…, xn and constants g1,…, gk which are elements of G. A solution consists of an n-tuple (h1,…, hn) of elements from G which upon substitution for x1,…, xn make the word trivial in G. Hence an equation is a first-order sentence in the language L[G] consisting of the elementary language of group theory (again see section 2) augmented by allowing constants from the group G.

Type
Chapter
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Publisher: Cambridge University Press
Print publication year: 2003

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  • Elementary theory of groups
    • By Benjamin Fine, Department of Mathematics, Fairfield University, Fairfield, Connecticut 06430, USA, Anthony M. Gaglione, Department of Mathematics, U.S. Naval Academy, Anapolis, Maryland 1402, USA, Alexei Myasnikov, Department of Mathematics, City College of CUNY, New York, New York 10031, USA, Gerhard Rosenberger, Fachbereich Mathematik Universität, Dortmund, 44221 Dortmund, Germany, Dennis Spellman, Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19132, USA
  • Edited by C. M. Campbell, University of St Andrews, Scotland, E. F. Robertson, University of St Andrews, Scotland, G. C. Smith, University of Bath
  • Book: Groups St Andrews 2001 in Oxford
  • Online publication: 11 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542770.021
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  • Elementary theory of groups
    • By Benjamin Fine, Department of Mathematics, Fairfield University, Fairfield, Connecticut 06430, USA, Anthony M. Gaglione, Department of Mathematics, U.S. Naval Academy, Anapolis, Maryland 1402, USA, Alexei Myasnikov, Department of Mathematics, City College of CUNY, New York, New York 10031, USA, Gerhard Rosenberger, Fachbereich Mathematik Universität, Dortmund, 44221 Dortmund, Germany, Dennis Spellman, Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19132, USA
  • Edited by C. M. Campbell, University of St Andrews, Scotland, E. F. Robertson, University of St Andrews, Scotland, G. C. Smith, University of Bath
  • Book: Groups St Andrews 2001 in Oxford
  • Online publication: 11 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542770.021
Available formats
×

Send book to Google Drive

To send content items to your account, please confirm that you agree to abide by our usage policies. If this is the first time you use this feature, you will be asked to authorise Cambridge Core to connect with your account. Find out more about sending content to Google Drive.

  • Elementary theory of groups
    • By Benjamin Fine, Department of Mathematics, Fairfield University, Fairfield, Connecticut 06430, USA, Anthony M. Gaglione, Department of Mathematics, U.S. Naval Academy, Anapolis, Maryland 1402, USA, Alexei Myasnikov, Department of Mathematics, City College of CUNY, New York, New York 10031, USA, Gerhard Rosenberger, Fachbereich Mathematik Universität, Dortmund, 44221 Dortmund, Germany, Dennis Spellman, Department of Mathematics, Temple University, Philadelphia, Pennsylvania 19132, USA
  • Edited by C. M. Campbell, University of St Andrews, Scotland, E. F. Robertson, University of St Andrews, Scotland, G. C. Smith, University of Bath
  • Book: Groups St Andrews 2001 in Oxford
  • Online publication: 11 January 2010
  • Chapter DOI: https://doi.org/10.1017/CBO9780511542770.021
Available formats
×