Can someone please explain the proof of the "Factor group lemma" for Cayley graphs which is stated below.
Factor Group Lemma: Suppose that
1.$N$ is a cyclic, normal subgroup of a group $G$.
2.$(s_1,s_2,\ldots,s_m)$ is a hamiltonian cycle in $Cay(G/N;S)$.
3.The product $s_1s_2\cdots s_m$ generates $N$.
Then $Cay(G;S)$ has a Hamiltonian cycle.
Thanks a lot in advance.