Algebraically closed groups of large cardinality

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Abstract
Let M be a countable algebraically closed group, κ an uncountable cardinal. We will prove in this paper the following theorems.Theorem 1. There is an algebraically closed group N of cardinality κ which is ∞ – ω-equivalent to M.Theorem 2. There is an algebraically closed group N of cardinality κ which is ∞ – ω-equivalent to M, and contains a free abelian group of cardinality κ.Theorem 3. There are 2κ nonisomorphic algebraically closed groups of cardinality κ which are ∞ – ω-equivalent to M.Theorem 4. There is an algebraically closed group N of cardinality κ which is ∞ – ω-equivalent to M and satisfies: Every subgroup of N of uncountable reqular cardinality contains a free subgroup of the same cardinality.Theorems 2 and 4 illustrate Theorem 3 by exhibiting two groups N ≡ ∞ωM of cardinality κ which are nonisomorphic by obvious reasons. We state and prove Theorem 1 separately in order to give an easy example of our principal tool: the use of automorphisms instead of indiscernibles (see §2).

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