Torsion-free nilpotent group
From Groupprops
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
Contents
Definition
A group is termed a torsion-free nilpotent group if
is a nilpotent group and it satisfies the following equivalent conditions:
-
is a group in which every power map is injective.
-
is a torsion-free group.
- For each prime number
, there exists an element
(possibly dependent on
) such that the equation
has a unique solution for
.
- The center
is a torsion-free abelian group.
- Each of the successive quotients
in the upper central series of
is a torsion-free abelian group.
- All quotients of the form
for
are [[group in which every power map is injective|groups in which every power map is injective], i.e.,
is injective in each such quotient group for all prime numbers
.
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
- Nilpotent group that is torsion-free for a set of primes: For a set of primes
, we can talk of the notion of
-torsion-free nilpotent group, which is a nilpotent group that has no
-torsion.