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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, .
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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 D is the dihedral group of order eight, given by the presentation:

,

and C is the cyclic group on two elements, with generator y.

Look at the subgroups and . We have the following:

• is a direct factor, and in particular, a central factor.
• K is an automorph of H under the automorphism of G given by . Thus, K is also a direct factor of G, 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 H and K 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.