A relationship between central-by-finite groups and FC-groups A group is said FC-group if for all $x\in G$ is true that the set $x^G$ is finite. Equivalently, $G$ is a FC-group if $|G:C_G(x)|$ is finite for all $x \in G$.
A group is said a central-by-finite if the center of $G$ has finite index $G$.
Is clear that if $G$ is a central-by-finite group then $G$ is a FC-group (in fact, $G$ is a BFC-group - look for the definition of BFC-grup in D. Robinson - Finiteness conditions and generalized soluble groups.
It is also true that: If G is a finitely generated FC-group, then G/Z(G) and Tor(G) are both finite. 
So I do my question:
Let $G$ be a locally finite group. Suppose that $G$ is a FC-group. Then $G$ is central-by-finite? 
Is true? If not, would like a counterexample.

A central product of countably infinitely many copies of $D_8$ (or more generally of any extraspecial group) is a counterexample. This group has the presentation
$$\langle x_i,y_i,z\ (i \in {\mathbb N}) \mid z^2=1,\ x_i^2=y_i^2=1,\ [x_i,y_i]=z\ (i \in {\mathbb N}),\ 
{\rm all\ other\ pairs\ of\ generators\ commute}\  \rangle.$$
Its centre is the finite subgroup $\langle z \rangle$. The conjugacy classes all have size $1$ or $2$, so it is a BFC-group.
