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Algebraically closed groups of large cardinality

Published online by Cambridge University Press:  12 March 2014

Saharon Shelah  and
Martin Ziegler
Saharon Shelah
Affiliation:
Hebrew University, Jerusalem, Israel
Martin Ziegler
Affiliation:
Technische Universität, Berlin, Federal Republic of Germany
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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).

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 44 , Issue 4 , December 1979 , pp. 522 - 532
Copyright
Copyright © Association for Symbolic Logic 1979

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References

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