On Torsion Units in Integral Group Ring of A Dicyclic Group


Küsmüş Ö.

Bitlis Eren Üniversitesi Fen Bilimleri Dergisi, vol.9, no.2, pp.609-614, 2020 (Peer-Reviewed Journal) identifier

  • Publication Type: Article / Article
  • Volume: 9 Issue: 2
  • Publication Date: 2020
  • Journal Name: Bitlis Eren Üniversitesi Fen Bilimleri Dergisi
  • Journal Indexes: TR DİZİN (ULAKBİM)
  • Page Numbers: pp.609-614
  • Van Yüzüncü Yıl University Affiliated: Yes

Abstract

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.