An explicit relation between knot groups in lens spaces and those in $S^3$.

We consider a cyclic covering map $(\Sigma,K) \to (\Sigma',K')$ between pairs of a 3-manifold and a knot, and describe the fundamental group $\pi_1(\Sigma \setminus K)$ in terms of $\pi_1(\Sigma' \setminus K')$. As a consequence, we give an alternative proof for the fact that a certain knot in $S^3$ cannot be represented as the preimage of any knot in a lens space. In our proofs, the subgroup of a group $G$ generated by the commutators and the $p$th power of each element of $G$ plays a key role.