# Characteristic subgroup of abelian group not implies divisibility-closed

From Groupprops

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) neednotsatisfy the second subgroup property (i.e., divisibility-closed subgroup)

View a complete list of subgroup property non-implications | View a complete list of subgroup property implications

Get more facts about characteristic subgroup of abelian group|Get more facts about divisibility-closed subgroup

EXPLORE EXAMPLES YOURSELF: View examples of subgroups satisfying property characteristic subgroup of abelian group but not divisibility-closed subgroup|View examples of subgroups satisfying property characteristic subgroup of abelian group and divisibility-closed subgroup

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) neednotsatisfy 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 and a characteristic subgroup of such that is not a divisibility-closed subgroup of . In other words, there exists a prime number such that is -divisible but is not.

## Related facts

### Opposite facts

## Proof

For any prime number :

- Let be the -quasicyclic group.
- Let be the subgroup comprising the elements of order 1 or .

Clearly:

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