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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Contents
Statement
It is possible to have a group with two cyclic normal subgroups (i.e., subgroups that are both cyclic and normal) whose join is not a cyclic normal subgroup.
Since the join is always a normal subgroup, the particular thing that fails is that the join is cyclic.
Related facts
- Abelian normal is not join-closed
- Abelian characteristic is not join-closed
- Normality is strongly join-closed
Proof
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.