Abelian characteristic is not join-closed

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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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Statement

It is possible to have a group G with Abelian characteristic subgroups (i.e., subgroups that are both Abelian and characteristic) H and K such that the join \langle H, K \rangle 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

Proof

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