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

From Groupprops

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 is a set of prime numbers. A group is termed a -**torsion-free nilpotent group** if it satisfies the following equivalent conditions:

- is a -powering-injective group, i.e., is injective and each .
- is a -torsion-free group.
- For each , there exists an element (possibly dependent on ) such that the equation has a unique solution for .
- The center is a -torsion-free group.
- Each of the successive quotients in the upper central series of is a -torsion-free group.
- All quotients of the form for are -powering-injective groups, i.e., is injective in each such quotient group and each .

Note that if we take 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`