Yıl: 2020 Cilt: 9 Sayı: 2 Sayfa Aralığı: 609 - 614 Metin Dili: İngilizce İndeks Tarihi: 19-11-2020

On Torsion Units in Integral Group Ring of A Dicyclic Group

Öz:
Let 𝐺 become an any group. We recall that any two elements of integral group ring ℤ𝐺 are rational conjugateprovided that they are conjugate in terms of units in ℚ𝐺. Zassenhaus introduced as a conjecture that any unit offinite order in ℤ𝐺 is rational conjugate to an element of the group 𝐺. This is known as the first conjecture ofZassenhaus [4]. We denote this conjecture by ZC1 throughout the article. ZC1 has been satisfied for some typesof solvable groups and metacyclic groups. Besides one can see that there exist some counterexamples in metabeliangroups. In this paper, the main aim is to characterize the structure of torsion units in integral group ring ℤ𝑇3 ofdicyclic group 𝑇3 = ⟨𝑎, 𝑏: 𝑎6 = 1, 𝑎3 = 𝑏2, 𝑏𝑎𝑏−1 = 𝑎−1⟩ via utilizing a complex 2nd degree faithful andirreducible representation of ℤ𝑇3 which is lifted from a representation of the group 𝑇3. We show by ZC1 that nontrivial torsion units in ℤ𝑇3 are of order 3, 4 or 6 and each of them can be stated by 3 free parameters.
Anahtar Kelime:

İki-devirli Bir Grubun İntegral Grup Halkasındaki Burulmalı Birimsel Elemanlar Üzerine

Öz:
𝐺 bir grup olsun. ℤ𝐺 integral grup halkasındaki herhangi iki birimsel elemanın, ℚ𝐺 grup cebrindeki birimseller bakımından eşlenik olması durumunda rasyonel eşlenik olarak ifade edildiklerini anımsayalım. Zassenhaus, bir konjektür olarak sunmuştur ki ℤ𝐺’deki herhangi bir sonlu mertebeli birimsel eleman, 𝐺 grubunun bir elemanı ile rasyonel eşleniktir. Bu, Zassenhaus’un ilk konjektürü olarak bilinir [4]. Biz bu konjektürü makale boyunca ZC1 ile göstereceğiz. ZC1, çözülebilir ve meta-devirli grupların bazı sınıfları için çözülmüştür. Bunun yanı sıra, biri görebilir ki metabelyen gruplarda bazı aksine örnekler vardır. Bu makalede temel amaç, 𝑇3 = ⟨𝑎, 𝑏: 𝑎 6 = 1, 𝑎 3 = 𝑏 2 , 𝑏𝑎𝑏 −1 = 𝑎 −1⟩ iki-devirli grubunun ℤ𝑇3 integral grup halkasındaki burulmalı birimsel elemanların yapısını, ikinci dereceden bir kompleks indirgenemez güvenilir temsil kullanarak karakterize etmektir. Birinci Zassenhaus konjektürü (ZC1) ile göstereceğiz ki ℤ𝑇3 integral grup halkasındaki aşikar olmayan burulmalı birimsel elemanlar 3, 4 veya 6 mertebeli ve bunların her biri üç serbest parametre cinsinden ifade edilebilir.
Anahtar Kelime:

Belge Türü: Makale Makale Türü: Araştırma Makalesi Erişim Türü: Erişime Açık
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APA kusmus o (2020). On Torsion Units in Integral Group Ring of A Dicyclic Group. , 609 - 614.
Chicago kusmus omer On Torsion Units in Integral Group Ring of A Dicyclic Group. (2020): 609 - 614.
MLA kusmus omer On Torsion Units in Integral Group Ring of A Dicyclic Group. , 2020, ss.609 - 614.
AMA kusmus o On Torsion Units in Integral Group Ring of A Dicyclic Group. . 2020; 609 - 614.
Vancouver kusmus o On Torsion Units in Integral Group Ring of A Dicyclic Group. . 2020; 609 - 614.
IEEE kusmus o "On Torsion Units in Integral Group Ring of A Dicyclic Group." , ss.609 - 614, 2020.
ISNAD kusmus, omer. "On Torsion Units in Integral Group Ring of A Dicyclic Group". (2020), 609-614.
APA kusmus o (2020). On Torsion Units in Integral Group Ring of A Dicyclic Group. Bitlis Eren Üniversitesi Fen Bilimleri Dergisi, 9(2), 609 - 614.
Chicago kusmus omer On Torsion Units in Integral Group Ring of A Dicyclic Group. Bitlis Eren Üniversitesi Fen Bilimleri Dergisi 9, no.2 (2020): 609 - 614.
MLA kusmus omer On Torsion Units in Integral Group Ring of A Dicyclic Group. Bitlis Eren Üniversitesi Fen Bilimleri Dergisi, vol.9, no.2, 2020, ss.609 - 614.
AMA kusmus o On Torsion Units in Integral Group Ring of A Dicyclic Group. Bitlis Eren Üniversitesi Fen Bilimleri Dergisi. 2020; 9(2): 609 - 614.
Vancouver kusmus o On Torsion Units in Integral Group Ring of A Dicyclic Group. Bitlis Eren Üniversitesi Fen Bilimleri Dergisi. 2020; 9(2): 609 - 614.
IEEE kusmus o "On Torsion Units in Integral Group Ring of A Dicyclic Group." Bitlis Eren Üniversitesi Fen Bilimleri Dergisi, 9, ss.609 - 614, 2020.
ISNAD kusmus, omer. "On Torsion Units in Integral Group Ring of A Dicyclic Group". Bitlis Eren Üniversitesi Fen Bilimleri Dergisi 9/2 (2020), 609-614.