Unit groups of completed modular group algebras and the isomorphism problem

Abstract

In this paper it is shown that isomorphism of group algebras of finite $p$-groups over the field of $p$ elements implies isomorphism of the groups, if one of the groups has a normal complement in the group of normalized units of the group algebra. Furthermore, a class of groups satisfying this condition is provided, and it is shown that the associated graded Lie-$p$-algebra of the group of normalized units of the Magnus algebra of $G,G$ being a residually ’nilpotent $p$-group of bounded exponent’, is a split extension of the one associated to $G$.