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Lie powers of free modules for certain groups of prime power order
Published online by Cambridge University Press: 09 April 2009
Abstract
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 K H-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.
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- Research Article
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- Copyright © Australian Mathematical Society 2001
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