On the lower bound of the derived length of the unit group of a nontorsion group algebra (Q2180616)

Suppose $G$ is a non-abelian nilpotent group which contains a $p$-element and $F$ is a field of characteristic $p\not= 2$ such that the unit group $U(FG)$ of the group ring $FG$ is solvable. The paper under review provides a lower bound for the derived length of $U(FG)$ and the main result (Theorem 1) states the following: Let $F$ be a field of characteristic $p>2$ and $G$ a nilpotent group such that the commutator group $G'$ is not a finite $p$-group, but $FG$ is modular. If $U(FG)$ is solvable, then the derived length of $U(FG)$ is strictly greater than $log_2(p + 1)$. Two similar results, namely Theorems 2 and 3, are proved when $G'$ is a finite $p$-group. The presented results correct recent ones in the same subject obtained by \textit{G. T. Lee} et al. in [Algebr. Represent. Theory 17, No. 5, 1597--1601 (2014; Zbl 1309.16025)] when $G$ is non-torsion and $G'$ is a finite $p$-group.