Let G be a countable Abelian group with Zd as a subgroup so that G/Zd 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.
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