Central factor is not finite-intersection-closed

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

Proof
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.