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

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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.