# Nilpotent group that is torsion-free for a set of primes

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

## Definition

Suppose $\pi$ is a set of prime numbers. A group $G$ is termed a $\pi$-torsion-free nilpotent group if it satisfies the following equivalent conditions:

1. $G$ is a $\pi$-powering-injective group, i.e., $x \mapsto x^p$ is injective and each $p \in \pi$.
2. $G$ is a $\pi$-torsion-free group.
3. For each $p \in \pi$, 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 is a $\pi$-torsion-free group.
5. Each of the successive quotients $Z^{i+1}(G)/Z^i(G)$ in the upper central series of $G$ is a $\pi$-torsion-free group.
6. All quotients of the form $Z^i(G)/Z^j(G)$ for $i > j$ are $\pi$-powering-injective groups, i.e., $x \mapsto x^p$ is injective in each such quotient group and each $p \in \pi$.

Note that if we take $\pi$ to be the set of all primes, this just becomes the same as torsion-free nilpotent group.