Lie powers of free modules for certain groups of prime power order
Bryant, Roger M.
Michos, Ioannis C.
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Let G be a finite group of order pk, where p is a prime and k ≥ 1, such that G is either cyclic, quaternion or generalised quaternion. Let V be a finite-dimensional free KG-module where K is a field of characteristic p. The Lie powers Ln(V) are naturally KG-modules and the main result identifies these modules up to isomorphism. There are only two isomorphism types of indecomposables occurring as direct summands of these modules, namely the regular KG-module and the indecomposable of dimension pk - pk-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 the multiplicities of these indecomposables in Ln(V). This extends and utilises work of the first author and R. Stöhr concerned with the case where G has order p.