<div dir="ltr">Dear  all,<div><br></div><div>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 "<a href="mailto:huseyinuysal@istanbul.edu.tr" target="_blank">huseyinuysal@istanbul.edu.tr</a>"  in order to receive the Zoom meeting ID and Passcode.<br></div><div><br></div><div>---------------------------------------------------------------------------------------------------------------</div><div><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;text-align:justify;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;vertical-align:baseline;font-size:11pt;font-family:Calibri,sans-serif"><b style="color:rgb(0,0,0)"><span lang="EN-GB" style="font-size:9pt;border:1pt none windowtext;padding:0cm">Title :</span></b><span lang="EN-GB" style="color:rgb(0,0,0);font-size:9pt;font-family:Helvetica,sans-serif"> </span><span style="color:red;font-size:9pt;font-family:Helvetica,sans-serif;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"> </span><font color="#000000"><span style="font-size:9pt;font-family:Helvetica,sans-serif;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial"><b>What can we do with Cayley's Theorem</b></span><span lang="EN-GB" style="font-family:Helvetica,sans-serif;font-size:9pt"></span></font></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;text-align:justify;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;vertical-align:baseline;font-size:11pt;font-family:Calibri,sans-serif"><font color="#000000"><br></font></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;text-align:justify;line-height:normal;font-size:11pt;font-family:Calibri,sans-serif"><font color="#000000"><span style="font-size:11pt"><span lang="EN-GB" style="font-size:9pt;line-height:13.8px"><b>Abstract:  </b></span></span><span style="font-size:9pt;line-height:13.8px;font-family:Helvetica,sans-serif">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.</span></font></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;line-height:16.8667px;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;vertical-align:baseline;font-size:11pt;font-family:Calibri,sans-serif"><font color="#000000"><span style="font-size:9pt;line-height:13.8px;font-family:Helvetica,sans-serif"> 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 </span><span style="font-size:9pt;line-height:13.8px;font-family:"Cambria Math",serif">ℵ</span>_{<span style="font-size:9pt;line-height:13.8px;font-family:Helvetica,sans-serif">0</span>}<span style="font-size:9pt;line-height:13.8px;font-family:Helvetica,sans-serif">-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:</span></font></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;line-height:16.8667px;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;vertical-align:baseline;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:9pt;line-height:13.8px;font-family:Helvetica,sans-serif"><font color="#000000"> </font></span></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;line-height:16.8667px;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;vertical-align:baseline;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:9pt;line-height:13.8px;font-family:Helvetica,sans-serif"><font color="#000000"><b>Theorem 1 (Kaya-Kegel-K [1])</b> Let κ ≤ λ be uncountable cardinals. If λ is a successor cardinal, then there exists a κ-existentially closed group of cardinality λ.</font></span></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;line-height:16.8667px;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;vertical-align:baseline;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:9pt;line-height:13.8px;font-family:Helvetica,sans-serif"><font color="#000000"> </font></span></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;line-height:16.8667px;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;vertical-align:baseline;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:9pt;line-height:13.8px;font-family:Helvetica,sans-serif"><font color="#000000"><b>References</b></font></span></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;line-height:16.8667px;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;vertical-align:baseline;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:9pt;line-height:13.8px;font-family:Helvetica,sans-serif"><font color="#000000">[1] Burak Kaya, Otto H. Kegel and Mahmut Kuzucuoğlu; On the existence of k-existentially closed groups, Arch. Math. (Basel), 111, 225- 229, (2018).</font></span></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;line-height:16.8667px;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;vertical-align:baseline;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:9pt;line-height:13.8px;font-family:Helvetica,sans-serif"><font color="#000000">[2] Otto H. Kegel and Mahmut Kuzucuğlu, κ-existentially closed groups, J. Algebra 499, 298–310, (2018).</font></span></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;text-align:justify;line-height:normal;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;vertical-align:baseline;font-size:11pt;font-family:Calibri,sans-serif"></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;line-height:16.8667px;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;vertical-align:baseline;font-size:11pt;font-family:Calibri,sans-serif"><span style="font-size:9pt;line-height:13.8px;font-family:Helvetica,sans-serif"><font color="#000000">[3] William R. Scott, Algebraically closed groups, Proc. Amer. Math. Soc. 2 (1951), 118–121.</font></span></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;line-height:16.8667px;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;vertical-align:baseline;font-size:11pt;font-family:Calibri,sans-serif"><font color="#000000"><span style="font-size:9pt;line-height:13.8px;font-family:Helvetica,sans-serif"><br></span></font></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;line-height:16.8667px;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;vertical-align:baseline;font-size:11pt;font-family:Calibri,sans-serif"><font color="#000000"><span style="font-size:9pt;line-height:13.8px;font-family:Helvetica,sans-serif"><br></span></font></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;line-height:16.8667px;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;vertical-align:baseline;font-size:11pt;font-family:Calibri,sans-serif"><font color="#000000"><span style="font-size:9pt;line-height:13.8px;font-family:Helvetica,sans-serif">----------------------------------------------------------------------------------------------------------------------------</span></font></p><p class="MsoNormal" style="margin:0cm 0cm 0.0001pt;line-height:16.8667px;background-image:initial;background-position:initial;background-size:initial;background-repeat:initial;background-origin:initial;background-clip:initial;vertical-align:baseline;font-size:11pt;font-family:Calibri,sans-serif"><font color="#000000"><span style="font-size:9pt;line-height:13.8px;font-family:Helvetica,s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