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2008.117: Lie powers of relation modules for groups

2008.117: L. G. Kovács and Ralph Stöhr (2008) Lie powers of relation modules for groups. Journal of Algebra, 326 (1). pp. 192-200. ISSN 0021-8693

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DOI: 10.1016/j.jalgebra.2009.10.007


Motivated by applications to abstract group theory, we study Lie powers of relation modules. The relation module associated to a free presentation $G=F/N$ of a group $G$ is the abelianization $N_{ab}=N/[N,N]$ of $N$, with $G$-action given by conjugation in $F$. The degree $n$ Lie power is the homogeneous component of degree $n$ in the free Lie ring on $N_{ab}$ (equivalently, it is the relevant quotient of the lower central series of $N$). We show that after reduction modulo a prime $p$ this becomes a projective $G$-module, provided $n>1$ and $n$ is not divisible by $p$.

Item Type:Article
Uncontrolled Keywords:Free groups, relation modules, free Lie algebras, free metabelian Lie algebras
Subjects:MSC 2000 > 17 Nonassociative rings and algebras
MSC 2000 > 20 Group theory and generalizations
MIMS number:2008.117
Deposited By:Prof Ralph Stöhr
Deposited On:11 February 2011

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