Let ZG denote the integral group ring of a group G and let UZG denote its group of invertible elements. Let ε : ZG → Z be the augmentation homomorphism:

ε (∑ g∈G

agg ) = ∑ g∈G


For any unit u ∈ U(ZG) holds ε(u) = ±1. The subgroup of units with augmentation 1 will be denoted by U1ZG and called the group of normalized units. We have

UZG = ±U1ZG. Let C = Cn = 〈x〉 be the cyclic group of order n. We have

QC = ⊕∑ d|n Q(ξd),

where ξ = ξn is a primitive complex n-th root of unity. Under the above identification, we have the embedding

ZC ⊂ ⊕∑ d|n Z[ξd] =M,

whereM is the maximal order of the algebra QC. LetM× denote the group of invertible elements in M. Then

The rank of UZC = the rank of M× = ∑ 2<d|n

(ϕ(d) 2

− 1 ) .