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 (?), .
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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 $G = D \times C$ where $D$ is the dihedral group of order eight, given by the presentation:

$D = \langle a,x \mid a^4 = x^2 = e, axa^{-1} = x^{-1} \rangle$,

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

Look at the subgroups $H = \langle x,a \rangle$ and $K = \langle xy,a \rangle$. We have the following:

• $H = D \times 1$ is a direct factor, and in particular, a central factor.
• $K$ is an automorph of $H$ under the automorphism of $G$ given by $a \mapsto a, x \mapsto xy, y \mapsto y$. Thus, $K$ is also a direct factor of $G$, and hence, a central factor.
• The intersection $H \cap K$ is given by $\langle a \rangle$. 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.