Group rings with metabelian unit groups
Let \(G\) be a group, \(F\) a field of positive characteristic \(p\) and denote by \(FG\) the group algebra of \(G\) over \(F\) and by \(U(FG)\) the unit group of \(FG\). Describing the structure of \(U(FG)\) is a notoriously difficult problem. Though in a series of papers the cases when \(U(FG)\) is solvable have been characterized, the last contribution being \textit{A. Bovdi}'s [Commun. Algebra 33, No. 10, 3725--3738 (2005; Zbl 1082.16036)], the derived length of \(U(FG)\) in these cases is not that well understood. For \(p\) odd and \(G\) finite \textit{A. Shalev} described when \(U(FG)\) is metabelian [J. Pure Appl. Algebra 72, No. 3, 295--302 (1991; Zbl 0735.16020)] and this has been generalized by \textit{F. Catino} and the second author to the case that \(G\) is a torsion group [J. Group Theory 13, No. 4, 577--588 (2010; Zbl 1205.16030)]. The result is that \(U(FG)\) is metabelian if and only if \(p=3\) and the derived subgroup \(G'\) is central of order \(p\). In this paper the authors generalize this description to all infinite groups, obtaining a somehow surprising result. Denote by \(\mathrm{Syl}_p(G)\) the maximal \(p\)-subgroup of \(G\), which is unique when \(G\) is nilpotent. Again assume that \(p\) is odd. Then \(U(FG)\) is metabelian if and only if \(G\) is nilpotent of class \(2\) and either \(p = 3\) and \(G'\) has order \(p\) or \(p = 3\) and \(G' = \mathrm{Syl}_p(G)\) is elementary abelian of order \(p^2\) or \(p = 5\) and \(G' = \mathrm{Syl}_p(G)\) has order \(p\). This is proven by employing general bounds on the derived length of \(U(FG)\) and then constructing elements in \(U(FG)\) which show that the derived length is bigger than \(2\) in the cases not included in the theorem switching calculations between group commutators in \(U(FG)\) and Lie commutators in \(FG\).
scientific article; zbMATH DE number 7461214
scientific article