Let G be a countable Abelian group with Z^d as a subgroup so that G/Z^d is a locally finite group. (An Abelian group is locally finite if every element has finite order.) We can construct a rank one action of G so that the Z-subaction is 2-simple, 2-mixing and only commutes with the other transformations in the action of G.

Applications of this construction include a transformation with square roots of all orders but no infinite square root chain, a transformation with countably many nonisomorphic square roots, a new proof of an old theorem of Baxter and Akcoglu on roots of transformations, and a simple map with no prime factors. The last example, originally constructed by del Junco, was the inspiration for this work.

The author was supported by NSERC, the University of Toronto, and the State University of New York -- College at Potsdam.