Completely divisibility-closed normal subgroup

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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 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