arXiv subjects

Mahmut Kuzucuoğlu

Publications and source records attributed to Mahmut Kuzucuoğlu.

2 recordsLinked to original sources

arXiv2024

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

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 $κ$.

math.LO↗

arXiv2020

Automorphisms of $κ$-existentially closed groups

We investigate the automorphisms of some $κ$- existentially closed groups. In particular, we prove that $Aut(G)$ is the union of subgroups of level preserving automorphisms and $|Aut(G)|=2^κ$ whenever $κ$ is inaccessible and $G$ is the unique $κ$-existentially closed group of cardinality $κ$. Indeed, the latter result is a byproduct of an argument showing that, for any uncountable $κ$ and any group $G$ that is the limit of regular representation of length $κ$ with countable base, we have $|Aut(G)|=\beth_{κ+1}$, where $\beth$ is the beth function. Such groups are also $κ$-existentially closed if $κ$ is regular. Both results are obtained by an analysis and classification of level preserving automorphisms of such groups.

math.GR↗