Group rings with solvable unit groups of minimal derived length. (Q471771)

Let \(FG\) be a noncommutative modular group algebra of prime characteristic \(p>2\). By \textit{C. Bagiński} [Commun. Algebra 30, No. 10, 4905-4913 (2002; Zbl 1017.16022)] and \textit{Z. Balogh} and \textit{Y. Li} [J. Algebra Appl. 6, No. 6, 991-999 (2007; Zbl 1168.16016)], if \(G\) is finite then the derived length of the group of units \(U(FG)\) has lower bound \(\log_2(p+1)\) attained if and only if \(G'\) is of order \(p\). \textit{F. Catino} and \textit{E. Spinelli} [J. Group Theory 13, No. 4, 577-588 (2010; Zbl 1205.16030)] extended this result to nilpotent torsion groups \(G\). The authors extend this result to non-torsion groups \(G\). New methods are required in presence of elements of infinite order as indicated by \textit{A. Bovdi} [Commun. Algebra 33, No. 10, 3725-3738 (2005; Zbl 1082.16036)].