# Equivalence of definitions of locally nilpotent group that is torsion-free for a set of primes

From Groupprops

This article gives a proof/explanation of the equivalence of multiple definitions for the term locally nilpotent group that is torsion-free for a set of primes

View a complete list of pages giving proofs of equivalence of definitions

## Statement

### For an arbitrary (not necessarily locally nilpotent) group and a prime

Suppose is a group and is a prime number. We have the implications (1) implies (2a) implies (3) implies (4) implies (5) implies (6) and also (1) implies (2b) implies (3) implies (4) implies (5) implies (6):

- is a -powering-injective group, i.e., is injective.
- is a -torsion-free group.
- There exists an element such that the equation has a unique solution for .
- For every 2-generated subgroup of , the center is a -torsion-free group.
- For every 2-generated subgroup of , each of the successive quotients in the upper central series of is a -torsion-free group.
- For every 2-generated subgroup of , all quotients of the form for are -powering-injective groups, i.e., is injective in each such quotient group.
- Every 2-generated nilpotent subgroup of is -powering-injective, i.e., the map is injective when restricted to a map from to itself.

## Proof

The proof is similar to that for equivalence of definitions of nilpotent group that is torsion-free for a set of primes.