Mod \(2\) group algebras with metabelian unit groups
The authors classify the finite groups \(G\) such that the group of units of the group ring \(\mathbb{F} G\) is metabelian in case the characteristic of \(\mathbb{F}\) is 2. \textit{A. Shalev} gave [in J. Pure Appl. Algebra 72, No. 3, 295-302 (1991; Zbl 0735.16020)] the odd characteristic case. The authors obtain that the unit group of \(\mathbb{F} G\) is metabelian in case the characteristic of \(\mathbb{F}\) is 2 in exactly the following cases: Either \(G\) is nilpotent of class less or equal to 2 and the derived subgroup of \(G\) is a subgroup of a Klein four group, or \(\mathbb{F}\) is of cardinality 2 and \(G\) is a split extension of an elementary Abelian 3-group and a cyclic group of order 2 which acts by inversion. The authors prove first that for a 2-group \(G\) the unit group of \(\mathbb{F} G\) is not metabelian for large classes of groups \(G\). They essentially look at double commutators of certain elements in the group of units. This leaves as only possibility the case mentioned in the statement. The case of a non-2 group is less labourous.
scientific article; zbMATH DE number 1308221