Torsion units in integral group rings

Abstract

Let G = ⟨ a ⟩ ⋊ X G = \left \langle a \right \rangle \rtimes X where ⟨ a ⟩ \left \langle a \right \rangle is a cyclic group of order n , X n,X is an abelian group of order m m , and ( n , m ) = 1 (n,m) = 1 . We prove that if Z G \mathbb {Z}G is the integral group ring of G G and H H is a finite group of units of augmentation one of Z G \mathbb {Z}G , then there exists a rational unit γ \gamma such that H γ ⊆ G {H^\gamma } \subseteq G .