Symmetry of knots and cyclic surgery

If a nontorus knot K admits a symmetry which is not a strong inversion, then there exists no nontrivial cyclic surgery on K. No surgery on a symmetric knot can produce a fake lens space or a 3-manifold M with |π₁(M)|=2.This generalizes the result of Culler-Gordon-Luecke-Shalen- Bleiler-Scharlemann and supports the conjecture thatno nontrivial surgery on a nontrivial knot yields a 3-manifoldM with |π₁(M)|+5.