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

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.