Divisible 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: 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