Torsion-free nilpotent group

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 $$p$$.

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.