Lie dimension subgroups, lie nilpotency indices, and the exponent of the group of normalized units

Let K be a field of characteristic p > 0, and let G be a finite p-group. Let U be the group of normalized units of the modular group algebra KG. In this paper we study the relation between exp (U) and exp (G). The main result shows that, if p ≥ 7 and exp(G)³ > G, then G and U have the same exponent. We also show that, in general, exp(U) cannot be bounded above by any fixed function of exp(G). The method involves a reduction to problems in Lie nilpotency indices, which are solved via an extensive study of Lie dimension subgroups. Some results for smaller p are also given.