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FREE GROUPS AND AUTOMORPHISM GROUPS OF INFINITE STRUCTURES

Part of: Special aspects of infinite or finite groups Structure and classification of infinite or finite groups Set theory

Published online by Cambridge University Press:  17 April 2014

PHILIPP LÜCKE  and
SAHARON SHELAH
PHILIPP LÜCKE
Affiliation:
Mathematisches Institut, Rheinische Friedrich-Wilhelms-Universität Bonn, Endenicher Allee 60, 53115 Bonn, Germanypluecke@uni-bonn.de
SAHARON SHELAH
Affiliation:
Einstein Institute of Mathematics, The Hebrew University of Jerusalem, Edmond J. Safra Campus, Givat Ram, Jerusalem 91904, Israelshelah@math.huji.ac.il Department of Mathematics, Hill Center-Busch Campus, Rutgers, The State University of New Jersey, 110 Frelinghuysen Road, Piscataway, NJ 08854-8019, USA

Abstract

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Given a cardinal $\lambda $ with $\lambda =\lambda ^{\aleph _0}$, we show that there is a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $. In the proof of this statement, we develop general techniques that enable us to realize certain groups as the automorphism group of structures of a given cardinality. They allow us to show that analogues of this result hold for free objects in various varieties of groups. For example, the free abelian group of rank $2^\lambda $ is the automorphism group of a field of cardinality $\lambda $ whenever $\lambda $ is a cardinal with $\lambda =\lambda ^{\aleph _0}$. Moreover, we apply these techniques to show that consistently the assumption that $\lambda =\lambda ^{\aleph _0}$ is not necessary for the existence of a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $. Finally, we use them to prove that the existence of a cardinal $\lambda $ of uncountable cofinality with the property that there is no field of cardinality $\lambda $ whose automorphism group is a free group of rank greater than $\lambda $ implies the existence of large cardinals in certain inner models of set theory.

MSC classification

Secondary: 03E75: Applications of set theory 20E05: Free nonabelian groups 20F29: Representations of groups as automorphism groups of algebraic systems
Type
Research Article
Information
Forum of Mathematics, Sigma , Volume 2 , 01 July 2014 , e8
Creative Commons
Creative Common License - CCCreative Common License - BY
The online version of this article is published within an Open Access environment subject to the conditions of the Creative Commons Attribution licence .
Copyright
© The Author(s) 2014

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