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FREE GROUPS AND AUTOMORPHISM GROUPS OF INFINITE STRUCTURES
Published online by Cambridge University Press: 17 April 2014
Abstract
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.
- Type
- Research Article
- Information
- Creative Commons
- 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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