Unitary subgroup of integral group rings (Q5905582)

One of the main topics in recent research on group ring units has been the construction of large, finitely generated subgroups of the unit group \(U(\mathbb{Z} G)\) in the integral group ring of a finite group \(G\). Of particular interest is the group \(B_ 2\) of bicyclic units \(1+(1- a)b\widehat a\), \(a,b\in G\) [see, e.g., \textit{J. Ritter} and \textit{S. K. Sehgal} in Trans. Am. Math. Soc. 324, 603-621 (1991; Zbl 0723.16016)]. The paper under review presents the following result: If \(G\) is dihedral and if \(f:G\to\{\pm 1\}\) is a homomorphism, then \(B_ 2\) is of finite index in \(U_ f(\mathbb{Z} G)=\{u=\sum a_ xx: u^{-1}=\sum f(a_ x)x^{- 1}\}\). The case when \(G\) is cyclic has been considered by \textit{A. Bovdi} [Mat. Zametki 41, 469-474 (1987; Zbl 0632.16010)]; the case of an abelian group \(G\) by Hoechsmann and Sehgal.