You can lock down your privacy settings but if you join an open group, anyone can see the posts you make there. But you can block her, so that as far as she can see, you don't exist on Facebook at all. The only way someone can tell you've blocked them is if, on some discussion thread (on a mutual friend's wall, or an open group), someone posts after you, "I agree, Turtledove, glad you said that" (or some such thing).
And yes, it's weird to join a group you are clearly not part of. We had a high school reunion group going for a specific grad class, and some guy who says he went to that school 30 years before us joined and would "like" and comment. He seemed pretty harmless, and probably just lonely, but I found it weird that he would post in that group at all. Nobody seemed to know him IRL either.