Abelian characteristic is not join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., Abelian characteristic subgroup) not satisfying a subgroup metaproperty (i.e., join-closed subgroup property).
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Since the join is always a characteristic subgroup, the particular thing that can fail is that the join need not be Abelian.
- Characteristicity is strongly join-closed
- Abelian normal is not join-closed
- Cyclic normal is not join-closed