[Turkmath:4691] Seminar on December 2nd, 2020, Department of Math, Istanbul University

[TEMHA ERKOÇ YILMAZTÜRK]

Dear  all,

On 02.12.2020  at 13:00 p.m,  Prof. Mahmut Kuzucuoğlu from METU will give a
talk  at weekly seminars in Mathematics Department of Istanbul University.
The title and the abstract are below. The seminar will be held online via
the Zoom program. Those who want to participate should send an e-mail to "
huseyinuysal at istanbul.edu.tr"  in order to receive the Zoom meeting ID and
Passcode.

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*Title :*  *What can we do with Cayley's Theorem*


*Abstract:  *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 G1 with |G1| ≥ 3
embed G1 into
Sym(G1) = G2 by Cayley’s theorem and continue like this by embedding G2 into
Sym(G2) = G3 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 κ, then the direct limit group is the unique κ-existentially
closed group of cardinality κ, see [2]. The ℵ0-existentially closed groups
are introduced by W. R. Scott in 1951, see [3]. For the existence of
κ-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.



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Best wishes


Temha

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