Have I embarassed my supervisors by solving a problem that a PhD student in my group was working on without success?
First, if you haven't already, I suggest you have a discussion with your advisor about what to do with the result.  Is it worth writing up?  Is it worth trying to publish or try to go further?  If so, you should write it up and make sure there are no errors, and hopefully your advisor will be willing to help check over this.


  Is it possible they will try to bully me out of first-authorship to protect the reputation of the PhD student?


Anything it possible, but nothing you have said gives any indication anyone else thinks they should be a co-author for this result, assuming it is publishable.  It's common in math that a problem seems difficult from one perspective, but is easy from another.  This can always be a little embarrassing, but usually it's no big deal if you're dealing with reasonable mathematicians.  However, it may mean (for a variety of possible reasons) that the problem is less interesting to them than originally thought, and this could be part of what you're sensing.  I've often gotten excited about discovering things, only to realize later that they weren't so novel, or been interested in problems because I thought my methods could solve them then lost interest when I found out other methods are much better.

Anyway, hopefully an open chat with your advisor about this will clear things up.  I would not mention the PhD student in this discussion, just focus on the questions in my first paragraph.

Edit: I forgot to say, your situation may indicate that this department might not be your best option for a PhD (there may be a trend of weak grad students or not-on-the-ball advisors).