Possibly trivial: do we know of a group (well, family of groups indexed by the security parameter) where the group operation can be computed in PPT, but you can't invert an element with a nonnegligible chance of success in PPT? What if I also want a PPT algorithm for picking an element of the group uniformly at random?

I can't seem to find an example of such a group. At the same time, I'm pretty convinced that any algorithms that operate on group elements as black boxes can't do inversion using only group operation, comparison for equality, and choose-a-random-element even on cyclic groups (and, I think, even if we also give them an operation that takes group element g and provides a pair (a,b) s.t. a+b=g, a!=e, b!=e that was chosen uniformly at random from all such pairs).