Let k be a knot in S 3 . There is an epimorphism from π 1 (S 3 −k) onto a free product of two nontrivial cyclic groups sending a meridian to an element of length two iff k has property Q (Topology of Manifolds, Markham, Chicago, IL, 1970, pp. 195–199) that is if there is a closed surface F in S 3 containing k, such that k is imprimitive in H 1 (X) and in H 1 (Y) where X and Y are the closures of the components of S 3 −F. We give answers to questions of Simon (1970) about properties Q, Q∗ and Q∗∗. Epimorphisms from knot groups onto torus knot groups are also studied and some results on property P and surgery are included.