Limit Groups and Automorphisms of $κ$-Existentially Closed Groups
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arXiv
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| Auteurs principaux: | , , , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866929482084909056 |
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| author | Kaya, Burak Kuzucuoğlu, Mahmut Longobardi, Patrizia Maj, Mercede |
| author_facet | Kaya, Burak Kuzucuoğlu, Mahmut Longobardi, Patrizia Maj, Mercede |
| contents | The structure of automorphism groups of $κ$-existentially closed groups are studied
by Kaya-Kuzucuoğlu in 2022.
It was proved that Aut(G) is the union of subgroups of level preserving automorphisms and $|Aut(G)|=2^κ$
whenever $κ$ is an inaccessible cardinal and $G$ is the unique
$κ$-existentially closed group of
cardinality $κ$. The cardinality of the automorphism group of a $κ$-existentially
closed group of cardinality $λ>κ$ is asked in Kourovka Notebook Question 20.40.
Here we answer positively the promised case $κ=λ$ namely:
If $G$ is a $κ$-existentially closed group of cardinality $κ$,
then $|Aut(G)|=2^κ$. We also answer Kegel's question
on universal groups, namely: For any uncountable cardinal $κ$,
there exist universal groups of cardinality $κ$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_00545 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Limit Groups and Automorphisms of $κ$-Existentially Closed Groups Kaya, Burak Kuzucuoğlu, Mahmut Longobardi, Patrizia Maj, Mercede Logic Group Theory (2020): Primary: 20B27, Secondary: 20B35, 20E36, 20F28 The structure of automorphism groups of $κ$-existentially closed groups are studied by Kaya-Kuzucuoğlu in 2022. It was proved that Aut(G) is the union of subgroups of level preserving automorphisms and $|Aut(G)|=2^κ$ whenever $κ$ is an inaccessible cardinal and $G$ is the unique $κ$-existentially closed group of cardinality $κ$. The cardinality of the automorphism group of a $κ$-existentially closed group of cardinality $λ>κ$ is asked in Kourovka Notebook Question 20.40. Here we answer positively the promised case $κ=λ$ namely: If $G$ is a $κ$-existentially closed group of cardinality $κ$, then $|Aut(G)|=2^κ$. We also answer Kegel's question on universal groups, namely: For any uncountable cardinal $κ$, there exist universal groups of cardinality $κ$. |
| title | Limit Groups and Automorphisms of $κ$-Existentially Closed Groups |
| topic | Logic Group Theory (2020): Primary: 20B27, Secondary: 20B35, 20E36, 20F28 |
| url | https://arxiv.org/abs/2409.00545 |