Burak Kaya, Mahmut Kuzucuoğlu, Patrizia Longobardi, Mercede Maj
{"title":"Limit Groups and Automorphisms of $κ$-Existentially Closed Groups","authors":"Burak Kaya, Mahmut Kuzucuoğlu, Patrizia Longobardi, Mercede Maj","doi":"arxiv-2409.00545","DOIUrl":null,"url":null,"abstract":"The structure of automorphism groups of $\\kappa$-existentially closed groups\nare studied by Kaya-Kuzucuo\\u{g}lu in 2022. It was proved that Aut(G) is the union of subgroups of level preserving\nautomorphisms and $|Aut(G)|=2^\\kappa$ whenever $\\kappa$ is an inaccessible cardinal and $G$ is the unique $\\kappa$-existentially closed group of cardinality $\\kappa$. The cardinality of the automorphism group of a\n$\\kappa$-existentially closed group of cardinality $\\lambda>\\kappa$ is asked in Kourovka Notebook\nQuestion 20.40. Here we answer positively the promised case $\\kappa=\\lambda$ namely: If $G$ is a $\\kappa$-existentially closed group of cardinality $\\kappa$, then $|Aut(G)|=2^{\\kappa}$. We also answer Kegel's question on universal groups, namely: For any uncountable cardinal $\\kappa$, there exist universal groups of cardinality $\\kappa$.","PeriodicalId":501306,"journal":{"name":"arXiv - MATH - Logic","volume":null,"pages":null},"PeriodicalIF":0.0000,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"arXiv - MATH - Logic","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/arxiv-2409.00545","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
引用次数: 0
Abstract
The structure of automorphism groups of $\kappa$-existentially closed groups
are studied by Kaya-Kuzucuo\u{g}lu in 2022. It was proved that Aut(G) is the union of subgroups of level preserving
automorphisms and $|Aut(G)|=2^\kappa$ whenever $\kappa$ is an inaccessible cardinal and $G$ is the unique $\kappa$-existentially closed group of cardinality $\kappa$. The cardinality of the automorphism group of a
$\kappa$-existentially closed group of cardinality $\lambda>\kappa$ is asked in Kourovka Notebook
Question 20.40. Here we answer positively the promised case $\kappa=\lambda$ namely: If $G$ is a $\kappa$-existentially closed group of cardinality $\kappa$, then $|Aut(G)|=2^{\kappa}$. We also answer Kegel's question on universal groups, namely: For any uncountable cardinal $\kappa$, there exist universal groups of cardinality $\kappa$.