This is one of the most commonly asked questions. In short, the answer is yes, you can manipulate the location of a pivot point.
Here's the bare bones explanation:
A pivot point for a set (note I didn't say object) is determined by the first object or objects added to an empty set. Objects added later do not affect a set's pivot point unless you use the "recenter pivot" command.
Here's what you need to do to get the most control:
Add a cube to an empty set. The pivot point for the set is now established as the center of the cube. Then, in a separate step, add the objects you want to control to the same set. At this point, the pivot point for the set follows the cube. When you have it where you want it, either turn off visibility for the cube (this allows you the option of repositioning the pivot point) or remove it from the set.