Divisible nilpotent group

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

Definition

A group $G$ is termed a divisible nilpotent group if it satisfies the following equivalent conditions:

1. $G$ is a divisible group.
2. The abelianization of $G$ is a divisible abelian group.
3. For every positive integer $i$, the quotient group $\gamma_i(G)/\gamma_{i+1}(G)$ of successive members of the lower central series is a divisible abelian group.
4. For any two positive integers $i < j$, if $\gamma_i(G),\gamma_j(G)$ denote respectively the $i^{th}$ and $j^{th}$ members of the lower central series of $G$, then the quotient group $\gamma_i(G)/\gamma_j(G)$ 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 $\pi$, we can talk of the notion of a $\pi$-divisible nilpotent group.