Shuffle the plane

Abstract

We prove that any permutation p of the plane can be obtained as a composition of a fixed number (209) of simple transformations of the form (x, y) → (x, y + f(x)) and (x,y) → (x + g(y),y), where f and g are arbitrary R → R functions. As a corollary we get that the full symmetric group acting on a set of continuum cardinal is a product of finitely many (209) copies of two isomorphic Abelian subgroups.