## 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

## 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 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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