On Torsion Units in Integral Group Ring of A Dicyclic Group

Abstract

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