On periodic normal subgroups of the multiplicative group of a group algebra (Q798743)

Let \({\mathcal U}_ 1(k,G)\) be the group of normalized units in the group algebra k[G] of a group G over a field k, \(\Delta =\Delta (G)=\{x\in G| [G:{\mathcal C}(x)]<\infty\}\) and let \(\Delta^+=\Delta^+(G)\) be the torsion subgroup of \(\Delta\) (G). \textit{C. Polcino-Milies} [Commun. Algebra 9, 699-712 (1981; Zbl 0463.16011)] has investigated nilpotent and FC-groups G, such that the elements of finite order in \({\mathcal U}_ 1(k,G)\) form a subgroup. The author gives an extension and refinement of these results. He proves Th1: If the multiplicative group of k or the group G contains an element having infinite order, then the elements of finite order in \({\mathcal U}_ 1(k,{\mathcal G})\) form a subgroup F and the following is true: (1) for a Sylow p-subgroup \(P\leq\Delta^+\) there is \(P\triangleleft G\) and \(\Delta^+/P\) is Abelian (in case char k\(=0\) assume \(P=(1))\); (2) all idempotents in \(k[\Delta^+]\) are central modulo \(\omega\) P; (3) for char k\(=0\) and for G with periodic subgroups of any finitely generated subgroup being finite there is \(F=\Delta^+\).