# Cyclic normal is not join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., cyclic normal subgroup)notsatisfying 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.