Characteristic subgroup of abelian group not implies divisibility-closed

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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties. That is, it states that every subgroup satisfying the first subgroup property (i.e., characteristic subgroup of abelian group) need not satisfy the second subgroup property (i.e., divisibility-closed subgroup)
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This article gives the statement and possibly, proof, of a non-implication relation between two subgroup properties, when the big group is a abelian group. That is, it states that in a abelian group, every subgroup satisfying the first subgroup property (i.e., characteristic subgroup) need not satisfy the second subgroup property (i.e., divisibility-closed subgroup)
View all subgroup property non-implications | View all subgroup property implications

Statement

It is possible to have an abelian group G and a characteristic subgroup H of G such that H is not a divisibility-closed subgroup of G. In other words, there exists a prime number p such that G is p-divisible but H is not.

Related facts

Opposite facts

Proof

For any prime number p:

  • Let G be the p-quasicyclic group.
  • Let H be the subgroup comprising the elements of order 1 or p.

Clearly:

  • H is a characteristic subgroup. In fact, it is a fully invariant subgroup of G.
  • However, H is not divisibility-closed: G is p-divisible, but H is not.