Burak Kaya, Mahmut Kuzucuoğlu, Patrizia Longobardi, Mercede Maj
{"title":"κ$-存在封闭群的极限群和自动形","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":"88 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"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\":\"88 1\",\"pages\":\"\"},\"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}","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}
Limit Groups and Automorphisms of $κ$-Existentially Closed Groups
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 $\kappa$.