Nilpotent group that is torsion-free for a set of primes

From Groupprops
Jump to: navigation, search
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: nilpotent group and torsion-free group for a set of primes
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: nilpotent group and powering-injective group for a set of primes
View other group property conjunctions OR view all group properties

Definition

Suppose \pi is a set of prime numbers. A group G is termed a \pi-torsion-free nilpotent group if it satisfies the following equivalent conditions:

  1. G is a \pi-powering-injective group, i.e., x \mapsto x^p is injective and each p \in \pi.
  2. G is a \pi-torsion-free group.
  3. For each p \in \pi, 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 is a \pi-torsion-free group.
  5. Each of the successive quotients Z^{i+1}(G)/Z^i(G) in the upper central series of G is a \pi-torsion-free group.
  6. All quotients of the form Z^i(G)/Z^j(G) for i > j are \pi-powering-injective groups, i.e., x \mapsto x^p is injective in each such quotient group and each p \in \pi.

Note that if we take \pi to be the set of all primes, this just becomes the same as torsion-free nilpotent group.

Equivalence of definitions

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