# Abelian normal is not join-closed

This article gives the statement, and possibly proof, of a subgroup property (i.e., Abelian normal subgroup)notsatisfying a subgroup metaproperty (i.e., join-closed subgroup property).

View all subgroup metaproperty dissatisfactions | View all subgroup metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for subgroup properties

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This article gives the statement, and possibly proof, of a group property (i.e., Abelian group)notsatisfying a group metaproperty (i.e., normal join-closed group property).

View all group metaproperty dissatisfactions | View all group metaproperty satisfactions|Get help on looking up metaproperty (dis)satisfactions for group properties

Get more facts about Abelian group|Get more facts about normal join-closed group property|

## Contents

## Statement

It is possible to have a group with Abelian normal subgroups such that the join is not an Abelian normal subgroup.

## Related facts

- Cyclic normal is not join-closed
- Abelian characteristic is not join-closed
- Nilpotent normal is finite-join-closed

A group obtained as a join of Abelian normal subgroups is termed a group generated by Abelian normal subgroups. Such groups have a number of nice properties.

## Proof

### Example of the dihedral group

{{further|[[Particular example::dihedral group:D8]}}

Let be the dihedral group of order eight:

.

Let be subgroups of given as follows:

.

Both and are Abelian normal subgroups, but the join of and , which is the whole group , is not an Abelian normal subgroup.

### Example of the quaternion group

`Further information: quaternion group`

In the quaternion group, the cyclic subgroups generated by and are both Abelian normal of order four, but their join, which is the whole group, is not Abelian.

### Any non-Abelian group of prime-cubed order

`Further information: prime-cube order group:p2byp, prime-cube order group:U3p`

If is an odd prime, the two non-Abelian -groups of order again offer examples of groups with Abelian normal subgroups whose join is not normal. In both cases, there are multiple subgroups of order , that are Abelian and normal, and whose join is the whole group, which is not normal.