FREE GROUPS AND AUTOMORPHISM GROUPS OF INFINITE STRUCTURES

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.

Keywords

• primary 03E75
• 20E05
• 20F29; secondary 03E35.