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

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 $G$ is a group and $p$ 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):

1. $G$ is a $p$-powering-injective group, i.e., $x \mapsto x^p$ is injective.
2. $G$ is a $p$-torsion-free group.
3. There exists an element $g \in G$ such that the equation $x^p = g$ has a unique solution for $x \in G$.
4. For every 2-generated subgroup $H$ of $G$, the center $Z(H)$ is a $p$-torsion-free group.
5. For every 2-generated subgroup $H$ of $G$, each of the successive quotients $Z^{i+1}(H)/Z^i(H)$ in the upper central series of $H$ is a $p$-torsion-free group.
6. For every 2-generated subgroup $H$ of $G$, all quotients of the form $Z^i(H)/Z^j(H)$ for $i > j$ are $p$-powering-injective groups, i.e., $x \mapsto x^p$ is injective in each such quotient group.
7. Every 2-generated nilpotent subgroup $H$ of $G$ is $p$-powering-injective, i.e., the map $x \mapsto x^p$ is injective when restricted to a map from $H$ 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.