Divisible nilpotent group
From Groupprops
This page describes a group property obtained as a conjunction (AND) of two (or more) more fundamental group properties: divisible group and nilpotent group
View other group property conjunctions OR view all group properties
Contents
Definition
A group is termed a divisible nilpotent group if it satisfies the following equivalent conditions:
-
is a divisible group.
- The abelianization of
is a divisible abelian group.
- For every positive integer
, the quotient group
of successive members of the lower central series is a divisible abelian group.
- For any two positive integers
, if
denote respectively the
and
members of the lower central series of
, then the quotient group
is a divisible group.
Relation with other properties
Stronger properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
rationally powered nilpotent group | |FULL LIST, MORE INFO | |||
divisible abelian group | can also be characterized as an injective object in the category of abelian groups | |FULL LIST, MORE INFO |
Weaker properties
Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|
nilpotent group | ||||
divisible group |
Prime set-parametrized version
- Nilpotent group that is divisible for a set of primes: Given a set of primes
, we can talk of the notion of a
-divisible nilpotent group.