Modular Lie representations of groups of prime order

Let K be a field of prime characteristic p and let G be a group of order p. For any finite-dimensional K G-module V and any positive integer n let L n(V) denote the nth homogeneous component of the free Lie K -algebra generated by (a basis of) V. Then Ln(V) can be considered as a K G-module, called the n th Lie power of V. The main result of the paper is a formula which describes the module structure of Ln(V) up to isomorphism.