Lie Properties of the Group Algebra and the Nilpotency Class of the Group of Units

We describe the upper and lower Lie nilpotency index of a modular group algebra Gof some metabelian groupGand apply these results to determine the nilpotency class of the group of units, extending certain results of Shalev without restriction to finite groups. A characterization of modular group algebras Gwith group of units of class 3 is given, which was obtained by Rao and Sandling for finite groupsG.