Wrong.
You do need the center for them to work.. Cutting up the lens will ruin it.
A Fresnel lens is in essence a convex lens with most of the material removed. It is the curve of the lens that causes the magnification. The ridges that you feel on a fresnel lens are the curved portion with all the rest on the material removed. While you might get some magnification it wopuld be far less than optimum.

I have seen the credit cad size lenses in many book, magizine and novalty stores usually around .50.