Completely divisibility-closed normal subgroup

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
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Definition

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

1. $H$ is both a normal subgroup of $G$ and a completely divisibility-closed subgroup of $G$.
2. $H$ is a normal subgroup of $G$, and for any prime number $p$ such that $G$ is $p$-divisible, the quotient group $G/H$ is $p$-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