| Autor: |
PHILIPP LÜCKE, SAHARON SHELAH |
| Jazyk: |
angličtina |
| Rok vydání: |
2014 |
| Předmět: |
|
| Zdroj: |
Forum of Mathematics, Sigma, Vol 2 (2014) |
| Druh dokumentu: |
article |
| ISSN: |
2050-5094 |
| DOI: |
10.1017/fms.2014.9 |
| Popis: |
Given a cardinal $\lambda $ with $\lambda =\lambda ^{\aleph _0}$ , we show that there is a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $ . In the proof of this statement, we develop general techniques that enable us to realize certain groups as the automorphism group of structures of a given cardinality. They allow us to show that analogues of this result hold for free objects in various varieties of groups. For example, the free abelian group of rank $2^\lambda $ is the automorphism group of a field of cardinality $\lambda $ whenever $\lambda $ is a cardinal with $\lambda =\lambda ^{\aleph _0}$ . Moreover, we apply these techniques to show that consistently the assumption that $\lambda =\lambda ^{\aleph _0}$ is not necessary for the existence of a field of cardinality $\lambda $ whose automorphism group is a free group of rank $2^\lambda $ . Finally, we use them to prove that the existence of a cardinal $\lambda $ of uncountable cofinality with the property that there is no field of cardinality $\lambda $ whose automorphism group is a free group of rank greater than $\lambda $ implies the existence of large cardinals in certain inner models of set theory. |
| Databáze: |
Directory of Open Access Journals |
| Externí odkaz: |
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