In this paper, normal forms are established for group elements in the solvable Baumslag-Solitar groups to classify all test elements in these groups. These normal forms are used to identify two general types of endomorphisms and automorphisms are identified through these types. Test elements are then identified as elements whose total exponents on one of the generators is zero. Finally, we show that Turner's Retract Theorem does not hold for these groups by giving a specific counterexample.

The author would like to especially thank a former student, Tom McCaleb, for pointing out several typographical errors in the initial writeup of this work and to Ted Turner, for asking the initial question years ago. He would also like to thank the referee for pointing out several errors and suggested changes which have improved the flow of the paper.