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
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: 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.

Equivalence of definitions

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