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Algebraically closed groups of large cardinality

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah
Hebrew University, Jerusalem, Israel
Martin Ziegler
Technische Universität, Berlin, Federal Republic of Germany


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).

Research Article
Copyright © Association for Symbolic Logic 1979

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