Abelian normal is not join-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., Abelian normal subgroup) not satisfying 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
Get more facts about Abelian normal subgroup|Get more facts about join-closed subgroup property|
This article gives the statement, and possibly proof, of a group property (i.e., Abelian group) not satisfying 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.