Some remarks on the unit groups of integral group rings
Let G be a group. We denote by \(U_ 1({\mathbb{Z}}G)\) the set of units in \({\mathbb{Z}}G\) which have augmentation one. This paper studies the Lie residual nilpotence of \(\Delta_{{\mathbb{Z}}}(U_ 1{\mathbb{Z}}G)\). It is first shown that a group G is residually (nilpotent with derived group torsion free) if and only if \(U_ 1{\mathbb{Z}}G\) is. Then it is shown that, for an arbitrary group G, we have that \(\cap_{n}\Delta_{{\mathbb{Z}}}^{(n)}(G)=0\) if and only if \(\cap_{n}\Delta_{{\mathbb{Z}}}^{(n)}(U_ 1{\mathbb{Z}}G)=0\). Finally, the author deals with the residual solvability of U\({\mathbb{Z}}G\), a question raised by \textit{I. Musson} and \textit{A. Weiss} [Arch. Math. 38, 514-530 (1982; Zbl 0476.16010)]. He shows that if G is a group satisfying certain conditions on its commutator subgroup G' then \(U_ 1{\mathbb{Z}}G\) is residually solvable.
scientific article; zbMATH DE number 3920706