Central factor is not finite-intersection-closed
This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor) not satisfying a subgroup metaproperty (i.e., finite-intersection-closed subgroup property).This also implies that it does not satisfy the subgroup metaproperty/metaproperties: Intersection-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 central factor|Get more facts about finite-intersection-closed subgroup propertyGet more facts about intersection-closed subgroup property|
An intersection of central factors need not be a central factor.
Further information: dihedral group:D8
We construct a counterexample as follows. Let where is the dihedral group of order eight, given by the presentation:
and is the cyclic group on two elements, with generator .
Look at the subgroups and . We have the following:
- is a direct factor, and in particular, a central factor.
- is an automorph of under the automorphism of given by . Thus, is also a direct factor of , and hence, a central factor.
- The intersection is given by . This is an abelian subgroup, but it is clearly not in the center, hence it cannot be a central factor.
Note that both and are direct factors, so the proof shows that an intersection of direct factors need not be a central factor. In fact, the same example shows that many related properties are not closed under finite intersections.