# Completely divisibility-closed normal subgroup

From Groupprops

This page describes a subgroup property obtained as a conjunction (AND) of two (or more) more fundamental subgroup properties: completely divisibility-closed subgroup and normal subgroup

View other subgroup property conjunctions | view all subgroup properties

## Definition

Suppose is a group and is a subgroup of . We say that is a **completely divisibility-closed normal subgroup** of if the following equivalent conditions are satisfied:

- is both a normal subgroup of and a completely divisibility-closed subgroup of .
- is a normal subgroup of , and for any prime number such that is -divisible, the quotient group is -torsion-free.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

kernel of a bihomomorphism | kernel of a bihomomorphism implies completely divisibility-closed | |||

intersection of kernels of bihomomorphisms | intersection of kernels of bihomomorphisms implies completely divisibility-closed |