# Abelian characteristic is not join-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., Abelian characteristic subgroup)notsatisfying a subgroup metaproperty (i.e., join-closed subgroup property).

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## Statement

It is possible to have a group with Abelian characteristic subgroups (i.e., subgroups that are both Abelian and characteristic) and such that the join is not an Abelian characteristic subgroup.

Since the join is always a characteristic subgroup, the particular thing that can fail is that the join need not be Abelian.

## Related facts

- Characteristicity is strongly join-closed
- Abelian normal is not join-closed
- Cyclic normal is not join-closed

## Proof

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