Torsion-free nilpotent group

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This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: torsion-free group and nilpotent group
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: group in which every power map is injective and nilpotent group
View other group property conjunctions OR view all group properties

Definition

A group G is termed a torsion-free nilpotent group if G is a nilpotent group and it satisfies the following equivalent conditions:

  1. G is a group in which every power map is injective.
  2. G is a torsion-free group.
  3. For each prime number p, there exists an element g \in G (possibly dependent on p) such that the equation x^p = g has a unique solution for x \in G.
  4. The center Z(G) is a torsion-free abelian group.
  5. Each of the successive quotients Z^{i+1}(G)/Z^i(G) in the upper central series of G is a torsion-free abelian group.
  6. All quotients of the form Z^i(G)/Z^j(G) for i > j are [[group in which every power map is injective|groups in which every power map is injective], i.e., x \mapsto x^p is injective in each such quotient group for all prime numbers p.

Equivalence of definitions

Further information: equivalence of definitions of nilpotent group that is torsion-free for a set of primes

Relation with other properties

Stronger properties

Property Meaning Proof of implication Proof of strictness (reverse implication failure) Intermediate notions
free nilpotent group |FULL LIST, MORE INFO
rationally powered nilpotent group |FULL LIST, MORE INFO

Prime set-parametrized version