Unit Group of Integral Group Ring Z(G X C 3 )

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 xC3) by showing the existence of a subgroup as (F55 & rtimes; F3) & rtimes; (S & lowast;3 xC2) where F rho denotes a free group of rank rho. Later, we introduce an explicit structure of the unit group in Z(S3 x C3) in terms of semi-direct product of torsion-free normal complement of S3 and the group of units in RS3 where R = Z[w] is the complex integral domain since w 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(GxC3) for an arbitrary group G depends on torsion-free normal complement V (G) of Gin U(Z(G xC3)) in an implicit form. As a consequence, a conjecture which is found in [21] is solved.