Cyclic normal is not join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., cyclic normal subgroup) not satisfying a subgroup metaproperty (i.e., join-closed subgroup property).
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Since the join is always a normal subgroup, the particular thing that fails is that the join is cyclic.
Example of the quaternion group
Further information: quaternion group
In the quaternion group, the subgroup generated by and the subgroup generated by are both cyclic normal subgroups, but their join, which is the whole group, is not cyclic.
Example of a prime-cube order group for odd prime
Further information: prime-cube order group:p2byp
In the non-Abelian group of order obtained as a semidirect product of a cyclic group of order and a cyclic group of order , there are many different cyclic normal subgroups of order (in fact, there are of them). The join of any two of these is the whole group, which is not cyclic.