Lie powers of free modules for certain groups of prime power order

Let G be a finite group of order p(k), where p is a prime and k greater tha n or equal to 1, such that G is either cyclic, quaternion or generalised qu aternion. Let V be a finite-dimensional free KG-module where K is a field o f characteristic p. The Lie powers L-n( V) are naturally KG-modules and the main result identifies. these modules up to isomorphism. There are only tw o isomorphism types of indecomposables occurring as direct summands of thes e modules, namely the regular KG-module and the indecomposable of dimension p(k) - p(k-1) induced from the indecomposable KH-module of dimension p - 1 , where H is the unique subgroup of G of order p. Formulae are given for th e multiplicities of these indecomposables. in L-n( V). This extends and uti lises work of the first author and R. Stohr concerned with the case where G has order p.