### 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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