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

? bir grup olsun. ℤ? integral grup halkasındaki herhangi iki birimsel elemanın, ℚ? grup cebrindeki birimseller bakımından eşlenik olması durumunda rasyonel eşlenik olarak ifade edildiklerini anımsayalım. Zassenhaus, bir konjektür olarak sunmuştur ki ℤ?’deki herhangi bir sonlu mertebeli birimsel eleman, ? 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, ?3 = ⟨?, ?: ? 6 = 1, ? 3 = ? 2 , ??? −1 = ? −1⟩ iki-devirli grubunun ℤ?3 integral grup halkasındaki burulmalı birimsel elemanların yapısını, ikinci dereceden bir kompleks indirgenemez güvenilir temsil kullanarak karakterize etmektir. Birinci Zassenhaus konjektürü (ZC1) ile göstereceğiz ki ℤ?3 integral grup halkasındaki aşikar olmayan burulmalı birimsel elemanlar 3, 4 veya 6 mertebeli ve bunların her biri üç serbest parametre cinsinden ifade edilebilir.

On Torsion Units in Integral Group Ring of A Dicyclic Group

Let ? become an any group. We recall that any two elements of integral group ring ℤ? are rational conjugateprovided that they are conjugate in terms of units in ℚ?. Zassenhaus introduced as a conjecture that any unit offinite order in ℤ? is rational conjugate to an element of the group ?. 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 ℤ?3 ofdicyclic group ?3 = ⟨?, ?: ?6 = 1, ?3 = ?2, ???−1 = ?−1⟩ via utilizing a complex 2nd degree faithful andirreducible representation of ℤ?3 which is lifted from a representation of the group ?3. We show by ZC1 that nontrivial torsion units in ℤ?3 are of order 3, 4 or 6 and each of them can be stated by 3 free parameters.

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