Torsion-free nilpotent group

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: torsion-free group and nilpotent group
View other group property conjunctions OR view all group properties

This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: group in which every power map is injective and nilpotent group
View other group property conjunctions OR view all group properties

Definition

A group $G$ is termed a torsion-free nilpotent group if $G$ is a nilpotent group and it satisfies the following equivalent conditions:

1. $G$ is a group in which every power map is injective.
2. $G$ is a torsion-free group.
3. For each prime number $p$, there exists an element $g\in G$ (possibly dependent on $p$) such that the equation $x^{p}=g$ has a unique solution for $x\in G$.
4. The center $Z(G)$ is a torsion-free abelian group.
5. Each of the successive quotients $Z^{i+1}(G)/Z^{i}(G)$ in the upper central series of $G$ is a torsion-free abelian group.
6. All quotients of the form $Z^{i}(G)/Z^{j}(G)$ for $i>j$ are [[group in which every power map is injective|groups in which every power map is injective], i.e., $x\mapsto x^{p}$ is injective in each such quotient group for all prime numbers $p$.

Equivalence of definitions

Further information: equivalence of definitions of nilpotent group that is torsion-free for a set of primes

Relation with other properties

Stronger properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
free nilpotent group				|FULL LIST, MORE INFO
rationally powered nilpotent group				|FULL LIST, MORE INFO

Prime set-parametrized version

• Nilpotent group that is torsion-free for a set of primes: For a set of primes $\pi$, we can talk of the notion of $\pi$-torsion-free nilpotent group, which is a nilpotent group that has no $\pi$-torsion.