Finite not implies divisibility-closed in abelian group
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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It is possible to have a group (in fact, we can choose to be an abelian group) and a finite subgroup of such that is not a divisibility-closed subgroup of . In other words, there exists a prime number such that is a -divisible group but is not a -divisible group.
In fact, we can construct an example of this sort for each prime number .
For any prime number :
- Let be the -quasicyclic group.
- Let be the subgroup comprising the elements of order 1 or .
- is a finite subgroup. In fact, it has order .
- However, is not divisibility-closed: is -divisible, but is not.