# Central factor is not finite-intersection-closed

From Groupprops

This article gives the statement, and possibly proof, of a subgroup property (i.e., central factor)notsatisfying a subgroup metaproperty (i.e., finite-intersection-closed subgroup property).This also implies that it doesnotsatisfy the subgroup metaproperty/metaproperties: Intersection-closed subgroup property (?), .

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## Statement

An intersection of central factors need not be a central factor.

## Proof

`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.