Free groups and automorphism groups of infinite structures

by Luecke and Shelah. [LcSh:1014]
Forum Math Sigma, 2014
Let lambda be a cardinal with lambda = lambda^{aleph_0} and p be either 0 or a prime number. We show that there are fields K_0 and K_1 of cardinality lambda and characteristic p such that the automorphism group of K_0 is a free group of cardinality 2^lambda and the automorphism group of K_1 is a free abelian group of cardinality 2^lambda . This partially answers a question from [MR1736959] and complements results from [MR1934424], [MR2773054] and [MR1720580]. The methods developed in the proof of the above statement also allow us to show that the above cardinal arithmetic assumption is consistently not necessary for the existence of such fields and that it is necessary to use large cardinal assumptions to construct a model of set theory containing a cardinal lambda of uncountable cofinality with the property that no free group of cardinality greater than lambda is isomorphic to the automorphism group of a field of cardinality lambda .


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