On torsion-free nilpotent loops

TL;DR

This paper investigates torsion-free nilpotent loops, showing their left multiplication groups are also torsion-free nilpotent of at most the same class, and proves free loops are residually torsion-free nilpotent.

Contribution

It establishes that torsion-free nilpotent loops have torsion-free nilpotent left multiplication groups and proves free loops are residually torsion-free nilpotent, extending known residual properties.

Findings

Left multiplication groups are torsion-free nilpotent of at most the same class.

Free loops are residually torsion-free nilpotent.

The results extend residual nilpotency to torsion-free nilpotent loops.

Abstract

We show that a torsion-free nilpotent loop (that is, a loop nilpotent with respect to the dimension filtration) has a torsion-free nilpotent left multiplication group of, at most, the same class. We also prove that a free loop is residually torsion-free nilpotent and that the same holds for any free commutative loop. Although this last result is much stronger than the usual residual nilpotency of the free loop proved by Higman, it is proved, essentially, by the same method.