Medicine is applied science where as pure Mathemetics is a highly theoretical field.
Well, I don't think there is a single overriding reason for doing a MD/PhD or for such programs to exist, there is definitely some value in attacting people who are interested in interfacing between fields.
Translational medicine is certain one kind of interface, which is characterized as an interface between medicine and basic science - mainly biology.
You are probably not going find Sylow groups useful in medicine (but maybe), I definitely know of some applications of metrics and manifolds in medicine. The best case I can think of off the top of my head is Steve Alchuler and Lani Wu's laboratory at UT Southwestern (a biomedical campus with no math department at all):
http://www.hhmi.swmed.edu/Labs/ms/people.html
Steve and Lani are techinically in the Pharmacology department by the way.
There are pure mathematicians who have crossed over to biology to answer some very interesting questions. Professor Alchuler has defined a way of generating high dimensional metrics between pharmasueticals which he can use to predict new applications of drugs. In his research you can see things such as hyperplanes defined by his metrics that clearly separate different classes of drugs and predict novel functionality. Professor Wu is working on cell polarity and using differential equations to evaluate some of the consequences of molecular signalling pathways.
To be frank, I don't think there is a lot of mathematics in medicine at the moment just because there are not a lot of mathematicians working at this interface. If you are interested, I encourage you to pursue this path as the medicine as a field is in dire need for a rigorous mathematical approach.
While you might not be able to do a pure mathematics PhD in some programs, you will that you can find something very close by joining a Biophysics program (physics is applied math in many ways, and biophysics is just its interface to biology) or a Computational Biology program (the more theoretical the closer to math it is).