T.H.
Well it is hard, but think of it as a skill she has to learn, not that you are abandoning her! I would start by just getting out something new or something she hasn't seen in a while and tell her that she can play with that for a bit but you are going to fold laundry (or whatever). I would let her play with it and anything else for about 20 minutes while you are actively doing something else. If she fusses explain that when you're done with this specific task you will play with her again...then actually do it. ;) This way you can break into the idea of her playing by herself and she can still see you and interact with you but is basically entertaining herself. It might not work right away, but probably will eventually.
My son who will be 3 in a few weeks plays by himself quite a bit. My daughter, who is older, is also very good at it and honestly I think he just learned from her, so he's maybe not the norm, but it is really nice for both of us! Plus, don't feel bad about letting her watch a little TV. A show or two a day is fine plus she really will learn stuff if you pick a quality show and it's okay for you to want a break to sit down or make dinner!
Also, keep up the playgroups. At this age they still don't really play with others as much as near others, but maybe before you go tell her that she will need to play with the other kids or by herself but you'd like to talk to the other mommies for a while. You can always jump in if you feel like she's feeling anxious, but give her a chance!