Abstract: In joint work with Slawomir Kwasik and Reinhardt Schultz from many years ago, we proved that a finite group acting smoothly on the three-sphere with an isolated fixed point must have exactly two such points. The techniques that we used were unusual for three-dimensional topology, and further consideration of the underlying ideas suggests that the methods might be more widely applicable. Even though my attempts to extend the methods were unsuccessful ultimately, I think that they were sufficiently interesting to be worth presenting.