Large wreath products in modular group rings

Let K be a field of characteristic p > 0, and let G be a locally finite p-group. We show that, if the unit group of KG is not nilpotent, then it must involve arbitrarily large wreath products. This may be regarded as an asymptotic generalization of a theorem of D. B. Coleman and D. S. Passman concerning non-abelian unit groups. The proof relies on the following group-theoretic result, which extends a classical theorem of B. H. Neumann and J. Wiegold. Let G be any group in which every cyclic subgroup has not more than n conjugates. Then the derived subgroup of G is finite, and its order is bounded above in terms of n.