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

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## Abstract

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 |
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Uncontrolled Keywords: | Free groups, relation modules, free Lie algebras, free metabelian Lie algebras |

Subjects: | MSC 2010, the AMS's Mathematics Subject Classification > 17 Nonassociative rings and algebras MSC 2010, the AMS's Mathematics Subject Classification > 20 Group theory and generalizations |

Depositing User: | Prof Ralph Stöhr |

Date Deposited: | 11 Feb 2011 |

Last Modified: | 20 Oct 2017 14:12 |

URI: | http://eprints.maths.manchester.ac.uk/id/eprint/1575 |

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