Virtually abelian not implies FZ

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., virtually abelian group) need not satisfy the second group property (i.e., FZ-group)
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Statement

It is possible to have a group $G$ that is a virtually abelian group (i.e., it has an abelian normal subgroup that has finite index) but such that $G$ is not a FZ-group -- the center of $G$ has infinite index.

Proof

The infinite dihedral group is an example. It has an abelian normal subgroup of index two -- its cyclic maximal subgroup which is an infinite cyclic group. However, it is a centerless group, so its center has infinite index.