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One can use direct limit method to obtain new groups from the given ones with some prescribed properties. Recall that Cayley's theorem states that every group $G$  can be embedded by right regular representation into the symmetric group $Sym(G)$. By using Cayley's theorem,   the famous Hall's universal locally finite group can be obtained as a direct limit of finite symmetric groups. Indeed start with a group  $G_1$ with $|G_1|\geqslant 3$ embed $G_1$ into $Sym(G_1)=G_2$  by Cayley's theorem and continue like this by embedding  $G_2$ into $Sym(G_2)=G_3$ until infinity. The direct limit of these groups forms the Hall's universal locally finite group. We will discuss the basic properties of this group. Moreover we will continue to talk on existentially closed groups, their basic properties and mention the joint work with Burak Kaya and Otto H. Kegel.

If we forget to stop at level  $w$ in the construction of Hall's universal group and continue to apply Cayley's theorem until the first inaccessible cardinal say $\kappa$, then the direct limit group is the unique  $\kappa$-existentially closed group of cardinality $\kappa$, see [2]. The $\aleph_0$-existentially closed groups are introduced by W. R. Scott in 1951, see [3].  For the existence of $\kappa$-existentially closed groups, we prove the following:

Theorem 1 (Kaya-Kegel-K [1]) Let κ ≤ λ be uncountable cardinals. If λ is a successor cardinal, then there exists a κ-existentially closed group of cardinality λ.

References

[1] Burak Kaya, Otto H. Kegel and Mahmut Kuzucuoğlu; On the existence of k-existentially closed groups, Arch. Math. (Basel), 111, 225- 229, (2018).

[2] Otto H. Kegel and Mahmut Kuzucuğlu, κ-existentially closed groups, J. Algebra 499, 298–310, (2018).

[3] William R. Scott, Algebraically closed groups, Proc. Amer. Math. Soc. 2 (1951), 118–121