Unit group of integral group ring ℤ(G × C3)

Abstract

Presenting an explicit descryption of unit group in the integral group ring of a given non-abelian group is a classical and open problem. Let S3 be a symmetric group of order 6 and C3 be a cyclic group of order 3. In this study, we firstly explore the commensurability in unit group of integral group ring ℤ(S3 × C3) by showing the existence of a subgroup as (F55 ⋊ F3) ⋊ (S3∗× C2) where Fρ denotes a free group of rank ρ. Later, we introduce an explicit structure of the unit group in ℤ(S3 × C3) in terms of semi-direct product of torsion-free normal complement of S3 and the group of units in RS3 where R = ℤ[ω] is the complex integral domain since ω is the primitive 3rd root of unity. At the end, we give a general method that determines the structure of the unit group of ℤ(G × C3) for an arbitrary group G depends on torsion-free normal complement V (G) of G in U(ℤ(G × C3)) in an implicit form. As a consequence, a conjecture which is found in [21] is solved.