Factor Group Lemma: Suppose that
1.$N$ is a cyclic, normal subgroup of 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.
According to the above lemma why is it necessary that $N$ should be cyclic? Can someone explain how the cyclic nature applies in determining hamiltonicity? Is there a proof of this lemma so that I will be able to understand where the cyclic property is used.
What will happen if $N$ was a abelian normal subgroup? Is there a similar lemma or theorem that can be applied?