Finite not implies divisibility-closed in abelian group

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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., finite subgroup) need not satisfy the second subgroup property (i.e., divisibility-closed subgroup)
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Statement

It is possible to have a group G (in fact, we can choose G to be an abelian group) and a finite 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 a p-divisible group but H is not a p-divisible group.

In fact, we can construct an example of this sort for each prime number p.

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 finite subgroup. In fact, it has order p.
  • However, H is not divisibility-closed: G is p-divisible, but H is not.