Limit Groups and Automorphisms of $κ$-Existentially Closed Groups

Burak Kaya, Mahmut Kuzucuoğlu, Patrizia Longobardi, Mercede Maj
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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 $\kappa$.
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κ$-存在封闭群的极限群和自动形
Kaya-Kuzucuo\u{g}lu 在 2022 年研究了$\kappa$-存在封闭群的自变群结构。研究证明,Aut(G)是保留自变量的级子群的联合,并且当$\kappa$是不可访问的心数且$G$是心数为$\kappa$的唯一的$\kappa$-存在封闭群时,$|Aut(G)|=2^\kappa$。在库洛夫卡笔记本问题 20.40 中提出了心性为 $\lambda>\kappa$ 的 $\kappa$-existentially closed group 的自变群的心性问题。在此,我们对承诺的$\kappa=\lambda$情况作正面回答:如果$G$是一个心性为$\kappa$的存在封闭群,那么$|Aut(G)|=2^{\kappa}$。我们还回答了凯格尔关于普遍群的问题,即对于任何不可数的心数 $\kappa$, 都存在心数为 $\kappa$ 的普遍群。
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